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10.5: Polygons, Perimeter, and Circumference

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Learning Objectives

After completing this section, you should be able to:

• Identify polygons by their sides.
• Identify polygons by their characteristics.
• Calculate the perimeter of a polygon.
• Calculate the sum of the measures of a polygon’s interior angles.
• Calculate the sum of the measures of a polygon’s exterior angles.
• Calculate the circumference of a circle.
• Solve application problems involving perimeter and circumference.

In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose.

Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design.

When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle.

Identifying Polygons

A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See Figure 10.82 for some examples. We show only up to eight-sided polygons, but there are many, many more.

If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons . However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in Figure 10.83 and Figure 10.83, are both pentagons because they each have five sides and five vertices. The third shape Figure 10.83 is a hexagon because it has six sides and six vertices. We should note here that the hexagon in Figure 10.83 is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than 180 ∘ 180 ∘ . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out.

While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See Figure 10.84.

Example 10.24

Identify each polygon.

• This shape has six sides. Therefore, it is a hexagon.
• This shape has four sides, so it is a quadrilateral. It has two pairs of parallel sides making it a parallelogram.
• This shape has eight sides making it an octagon.
• This is an equilateral triangle, as all three sides are equal.
• This is a rhombus; all four sides are equal.
• This is a regular octagon, eight sides of equal length and equal angles.

Your Turn 10.24

Example 10.25

Determining multiple polygons.

What polygons make up Figure 10.85?

Shapes 1 and 5 are hexagons; shapes 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, and 16 are triangles; shapes 8 and 17 are parallelograms; and shape 11 is a trapezoid.

Your Turn 10.25

Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.

A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula:

The formula for the perimeter P P of a rectangle is P = 2 L + 2 W P = 2 L + 2 W , twice the length L L plus twice the width W W .

For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus,

24 = 2 l + 2 ( 4 ) = 2 l + 8 24 − 8 = 2 l 16 = 2 l 8 = l 24 = 2 l + 2 ( 4 ) = 2 l + 8 24 − 8 = 2 l 16 = 2 l 8 = l

The length is 8 units.

The perimeter of a regular polygon with n n sides is given as P = n ⋅ s P = n ⋅ s . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is P = 3 ( 7 ) = 21 cm P = 3 ( 7 ) = 21 cm .

Example 10.26

Finding the perimeter of a pentagon.

Find the perimeter of a regular pentagon with a side length of 7 cm (Figure 10.87).

A regular pentagon has five equal sides. Therefore, the perimeter is equal to P = 5 ( 7 ) = 35 cm P = 5 ( 7 ) = 35 cm .

Your Turn 10.26

Example 10.27, finding the perimeter of an octagon.

Find the perimeter of a regular octagon with a side length of 14 cm (Figure 10.88).

A regular octagon has eight sides of equal length. Therefore, the perimeter of a regular octagon with a side length of 14 cm is P = 8 ( 14 ) = 112 cm P = 8 ( 14 ) = 112 cm .

Your Turn 10.27

Sum of Interior and Exterior Angles

To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.

The sum of the interior angles of a polygon with n n sides is given by

S = ( n − 2 ) 180 ∘ . S = ( n − 2 ) 180 ∘ .

For example, if we want to find the sum of the interior angles in a parallelogram, we have

S = ( 4 − 2 ) 180 ∘ = 2 ( 180 ) = 360 ∘ . S = ( 4 − 2 ) 180 ∘ = 2 ( 180 ) = 360 ∘ .

Similarly, to find the sum of the interior angles inside a regular heptagon, we have

S = ( 7 − 2 ) 180 ∘ = 5 ( 180 ) = 900 ∘ . S = ( 7 − 2 ) 180 ∘ = 5 ( 180 ) = 900 ∘ .

To find the measure of each interior angle of a regular polygon with n n sides, we have the following formula.

The measure of each interior angle of a regular polygon with n n sides is given by

a = ( n − 2 ) 180 ∘ n . a = ( n − 2 ) 180 ∘ n .

For example, find the measure of an interior angle of a regular heptagon, as shown in Figure 10.90. We have

a = ( 7 − 2 ) 180 ∘ 7 = 128.57 ∘ . a = ( 7 − 2 ) 180 ∘ 7 = 128.57 ∘ .

Example 10.28

Calculating the sum of interior angles.

Find the measure of an interior angle in a regular octagon using the formula, and then find the sum of all the interior angles using the sum formula.

An octagon has eight sides, so n = 8 n = 8 .

Step 1: Using the formula a = ( n − 2 ) 180 ∘ 8 a = ( n − 2 ) 180 ∘ 8 :

a = ( 8 − 2 ) 180 ∘ 8 = ( 6 ) 180 ∘ 8 = 135 ∘ . a = ( 8 − 2 ) 180 ∘ 8 = ( 6 ) 180 ∘ 8 = 135 ∘ .

So, the measure of each interior angle in a regular octagon is 135 ∘ 135 ∘ .

Step 2: The sum of the angles inside an octagon, so using the formula:

S = ( n − 2 ) 180 ∘ = ( 8 − 2 ) 180 ∘ = 6 ( 180 ) = 1,080 ∘ . S = ( n − 2 ) 180 ∘ = ( 8 − 2 ) 180 ∘ = 6 ( 180 ) = 1,080 ∘ .

Step 3: We can test this, as we already know the measure of each angle is 135 ∘ 135 ∘ . Thus, 8 ( 135 ∘ ) = 1,080 ∘ 8 ( 135 ∘ ) = 1,080 ∘ .

Your Turn 10.28

Example 10.29, calculating interior angles.

Use algebra to calculate the measure of each interior angle of the five-sided polygon (Figure 10.91).

Step 1: Let us find out what the total of the sum of the interior angles should be. Use the formula for the sum of the angles in a polygon with n n sides: S = ( n − 2 ) 180 ∘ S = ( n − 2 ) 180 ∘ . So, S = ( 5 − 2 ) 180 ∘ = 540 ∘ S = ( 5 − 2 ) 180 ∘ = 540 ∘ .

Step 2: We add up all the angles and solve for x x :

5 ( x + 7 ) + 120 + ( 6 x + 25 ) + 5 ( 2 x + 5 ) + 5 ( 3 x − 5 ) = 540 5 x + 6 x + 10 x + 15 x + 180 = 540 36 x = 360 x = 10 5 ( x + 7 ) + 120 + ( 6 x + 25 ) + 5 ( 2 x + 5 ) + 5 ( 3 x − 5 ) = 540 5 x + 6 x + 10 x + 15 x + 180 = 540 36 x = 360 x = 10

Step 3: We can then find the measure of each interior angle:

m ∡ A = 5 ( 10 + 7 ) = 85 ∘ m ∡ B = 120 ∘ m ∡ C = 6 ( 10 ) + 25 = 85 ∘ m ∡ D = 5 ( 2 * 10 + 5 ) = 125 ∘ m ∡ E = 5 ( 3 * 10 − 5 ) = 125 ∘ m ∡ A = 5 ( 10 + 7 ) = 85 ∘ m ∡ B = 120 ∘ m ∡ C = 6 ( 10 ) + 25 = 85 ∘ m ∡ D = 5 ( 2 * 10 + 5 ) = 125 ∘ m ∡ E = 5 ( 3 * 10 − 5 ) = 125 ∘

Your Turn 10.29

An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with n n sides is found using the following formula:

To find the measure of an exterior angle of a regular polygon with n n sides we use the formula

b = 360 ∘ n . b = 360 ∘ n .

In Figure 10.93, we have a regular hexagon ABCDEF ABCDEF . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, b = 360 ∘ 6 = 60 ∘ b = 360 ∘ 6 = 60 ∘ .

Now, an important point is that the sum of the exterior angles of a regular polygon with n Figure 10.93, we multiply the measure of each exterior angle, 60 ∘ 60 ∘ , by the number of sides, six. Thus, the sum of the exterior angles is 6 ( 60 ∘ ) = 360 ∘ . 6 ( 60 ∘ ) = 360 ∘ .

Example 10.30

Calculating the sum of exterior angles.

Find the sum of the measure of the exterior angles of the pentagon (Figure 10.94).

Each individual angle measures 360 5 = 72 ∘ . 360 5 = 72 ∘ . Then, the sum of the exterior angles is 5 ( 72 ∘ ) = 360 ∘ . 5 ( 72 ∘ ) = 360 ∘ .

Your Turn 10.30

Circles and Circumference

The perimeter of a circle is called the circumference . To find the circumference, we use the formula C = π d , C = π d , where d d is the diameter, the distance across the center, or C = 2 π r , C = 2 π r , where r r is the radius.

The circumference of a circle is found using the formula C = π d , C = π d , where d d is the diameter of the circle, or C = 2 π r , C = 2 π r , where r r is the radius.

The radius is ½ of the diameter of a circle. The symbol π = 3.141592654 … Figure 10.96.

Let the radius be equal to 3.5 inches. Then, the circumference is

C = 2 π ( 3.5 ) = 21.99 in . C = 2 π ( 3.5 ) = 21.99 in .

Example 10.31

Finding circumference with diameter.

Find the circumference of a circle with diameter 10 cm.

If the diameter is 10 cm, the circumference is C = 10 π = 31.42 cm . C = 10 π = 31.42 cm .

Your Turn 10.31

Example 10.32, finding circumference with radius.

Find the radius of a circle with a circumference of 12 in.

If the circumference is 12 in, then the radius is

12 = 2 π r 12 2 π = r = 1.91 in . 12 = 2 π r 12 2 π = r = 1.91 in .

Your Turn 10.32

Example 10.33, calculating circumference for the real world.

You decide to make a trim for the window in Figure 10.97. How many feet of trim do you need to buy?

The trim will cover the 6 feet along the bottom and the two 12-ft sides plus the half circle on top. The circumference of a semicircle is ½ the circumference of a circle. The diameter of the semicircle is 6 ft. Then, the circumference of the semicircle would be 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft . 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft .

Therefore, the total perimeter of the window is 6 + 12 + 12 + 9.4 = 39.4 ft . 6 + 12 + 12 + 9.4 = 39.4 ft . You need to buy 39.4 ft of trim.

Your Turn 10.33

People in Mathematics

The overwhelming consensus is that Archimedes (287–212 BCE) was the greatest mathematician of classical antiquity, if not of all time. A Greek scientist, inventor, philosopher, astronomer, physicist, and mathematician, Archimedes flourished in Syracuse, Sicily. He is credited with the invention of various types of pulley systems and screw pumps based on the center of gravity. He advanced numerous mathematical concepts, including theorems for finding surface area and volume. Archimedes anticipated modern calculus and developed the idea of the “infinitely small” and the method of exhaustion. The method of exhaustion is a technique for finding the area of a shape inscribed within a sequence of polygons. The areas of the polygons converge to the area of the inscribed shape. This technique evolved to the concept of limits, which we use today.

One of the more interesting achievements of Archimedes is the way he estimated the number pi, the ratio of the circumference of a circle to its diameter. He was the first to find a valid approximation. He started with a circle having a diameter of 1 inch. His method involved drawing a polygon inscribed inside this circle and a polygon circumscribed around this circle. He knew that the perimeter of the inscribed polygon was smaller than the circumference of the circle, and the perimeter of the circumscribed polygon was larger than the circumference of the circle. This is shown in the drawing of an eight-sided polygon. He increased the number of sides of the polygon each time as he got closer to the real value of pi. The following table is an example of how he did this.

Archimedes settled on an approximation of π ≈ 3.1416 π ≈ 3.1416 after an iteration of 96 sides. Because pi is an irrational number, it cannot be written exactly. However, the capability of the supercomputer can compute pi to billions of decimal digits. As of 2002, the most precise approximation of pi includes 1.2 trillion decimal digits.

The Platonic Solids

The Platonic solids (Figure 10.100) have been known since antiquity. A polyhedron is a three-dimensional object constructed with congruent regular polygonal faces. Named for the philosopher, Plato believed that each one of the solids is associated with one of the four elements: Fire is associated with the tetrahedron or pyramid, earth with the cube, air with the octahedron, and water with the icosahedron. Of the fifth Platonic solid, the dodecahedron, Plato said, “… God used it for arranging the constellations on the whole heaven.”

Plato believed that the combination of these five polyhedra formed all matter in the universe. Later, Euclid proved that exactly five regular polyhedra exist and devoted the last book of the Elements to this theory. These ideas were resuscitated by Johannes Kepler about 2,000 years later. Kepler used the solids to explain the geometry of the universe. The beauty and symmetry of the Platonic solids have inspired architects and artists from antiquity to the present.

Check Your Understanding

Section 10.4 Exercises

10.4 Polygons, Perimeter, and Circumference

Learning objectives.

After completing this section, you should be able to:

• Identify polygons by their sides.
• Identify polygons by their characteristics.
• Calculate the perimeter of a polygon.
• Calculate the sum of the measures of a polygon’s interior angles.
• Calculate the sum of the measures of a polygon’s exterior angles.
• Calculate the circumference of a circle.
• Solve application problems involving perimeter and circumference.

In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose.

Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design.

When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle.

Identifying Polygons

A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See Figure 10.63 for some examples. We show only up to eight-sided polygons, but there are many, many more.

If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons . However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in Figure 10.64 and Figure 10.64 , are both pentagons because they each have five sides and five vertices. The third shape Figure 10.64 is a hexagon because it has six sides and six vertices. We should note here that the hexagon in Figure 10.64 is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than 180 ∘ 180 ∘ . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out.

While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See Figure 10.65 .

Example 10.24

Identify each polygon.

• This shape has six sides. Therefore, it is a hexagon.
• This shape has four sides, so it is a quadrilateral. It has two pairs of parallel sides making it a parallelogram.
• This shape has eight sides making it an octagon.
• This is an equilateral triangle, as all three sides are equal.
• This is a rhombus; all four sides are equal.
• This is a regular octagon, eight sides of equal length and equal angles.

Your Turn 10.24

Example 10.25, determining multiple polygons.

What polygons make up Figure 10.66 ?

Shapes 1 and 5 are hexagons; shapes 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, and 16 are triangles; shapes 8 and 17 are parallelograms; and shape 11 is a trapezoid.

Your Turn 10.25

Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.

A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula:

The formula for the perimeter P P of a rectangle is P = 2 L + 2 W P = 2 L + 2 W , twice the length L L plus twice the width W W .

For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus,

The length is 8 units.

The perimeter of a regular polygon with n n sides is given as P = n ⋅ s P = n ⋅ s . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is P = 3 ( 7 ) = 21 cm P = 3 ( 7 ) = 21 cm .

Example 10.26

Finding the perimeter of a pentagon.

Find the perimeter of a regular pentagon with a side length of 7 cm ( Figure 10.67 ).

A regular pentagon has five equal sides. Therefore, the perimeter is equal to P = 5 ( 7 ) = 35 cm P = 5 ( 7 ) = 35 cm .

Your Turn 10.26

Example 10.27, finding the perimeter of an octagon.

Find the perimeter of a regular octagon with a side length of 14 cm ( Figure 10.68 ).

A regular octagon has eight sides of equal length. Therefore, the perimeter of a regular octagon with a side length of 14 cm is P = 8 ( 14 ) = 112 cm P = 8 ( 14 ) = 112 cm .

Your Turn 10.27

Sum of interior and exterior angles.

To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.

The sum of the interior angles of a polygon with n n sides is given by

For example, if we want to find the sum of the interior angles in a parallelogram, we have

Similarly, to find the sum of the interior angles inside a regular heptagon, we have

To find the measure of each interior angle of a regular polygon with n n sides, we have the following formula.

The measure of each interior angle of a regular polygon with n n sides is given by

For example, find the measure of an interior angle of a regular heptagon, as shown in Figure 10.69 . We have

Example 10.28

Calculating the sum of interior angles.

Find the measure of an interior angle in a regular octagon using the formula, and then find the sum of all the interior angles using the sum formula.

An octagon has eight sides, so n = 8 n = 8 .

Step 1: Using the formula a = ( n − 2 ) 180 ∘ 8 a = ( n − 2 ) 180 ∘ 8 :

So, the measure of each interior angle in a regular octagon is 135 ∘ 135 ∘ .

Step 2: The sum of the angles inside an octagon, so using the formula:

Step 3: We can test this, as we already know the measure of each angle is 135 ∘ 135 ∘ . Thus, 8 ( 135 ∘ ) = 1,080 ∘ 8 ( 135 ∘ ) = 1,080 ∘ .

Your Turn 10.28

Example 10.29, calculating interior angles.

Use algebra to calculate the measure of each interior angle of the five-sided polygon ( Figure 10.70 ).

Step 1: Let us find out what the total of the sum of the interior angles should be. Use the formula for the sum of the angles in a polygon with n n sides: S = ( n − 2 ) 180 ∘ S = ( n − 2 ) 180 ∘ . So, S = ( 5 − 2 ) 180 ∘ = 540 ∘ S = ( 5 − 2 ) 180 ∘ = 540 ∘ .

Step 2: We add up all the angles and solve for x x :

Step 3: We can then find the measure of each interior angle:

Your Turn 10.29

An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with n n sides is found using the following formula:

To find the measure of an exterior angle of a regular polygon with n n sides we use the formula

In Figure 10.71 , we have a regular hexagon ABCDEF ABCDEF . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, b = 360 ∘ 6 = 60 ∘ b = 360 ∘ 6 = 60 ∘ .

Now, an important point is that the sum of the exterior angles of a regular polygon with n n sides equals 360 ∘ . 360 ∘ . This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get 360 ∘ . 360 ∘ . For the example in Figure 10.71 , we multiply the measure of each exterior angle, 60 ∘ 60 ∘ , by the number of sides, six. Thus, the sum of the exterior angles is 6 ( 60 ∘ ) = 360 ∘ . 6 ( 60 ∘ ) = 360 ∘ .

Example 10.30

Calculating the sum of exterior angles.

Find the sum of the measure of the exterior angles of the pentagon ( Figure 10.72 ).

Each individual angle measures 360 5 = 72 ∘ . 360 5 = 72 ∘ . Then, the sum of the exterior angles is 5 ( 72 ∘ ) = 360 ∘ . 5 ( 72 ∘ ) = 360 ∘ .

Your Turn 10.30

Circles and circumference.

The perimeter of a circle is called the circumference . To find the circumference, we use the formula C = π d , C = π d , where d d is the diameter, the distance across the center, or C = 2 π r , C = 2 π r , where r r is the radius.

The circumference of a circle is found using the formula C = π d , C = π d , where d d is the diameter of the circle, or C = 2 π r , C = 2 π r , where r r is the radius.

The radius is ½ of the diameter of a circle. The symbol π = 3.141592654 … π = 3.141592654 … is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See Figure 10.73 .

Let the radius be equal to 3.5 inches. Then, the circumference is

Example 10.31

Finding circumference with diameter.

Find the circumference of a circle with diameter 10 cm.

If the diameter is 10 cm, the circumference is C = 10 π = 31.42 cm . C = 10 π = 31.42 cm .

Your Turn 10.31

Example 10.32, finding circumference with radius.

Find the radius of a circle with a circumference of 12 in.

If the circumference is 12 in, then the radius is

Your Turn 10.32

Example 10.33, calculating circumference for the real world.

You decide to make a trim for the window in Figure 10.74 . How many feet of trim do you need to buy?

The trim will cover the 6 feet along the bottom and the two 12-ft sides plus the half circle on top. The circumference of a semicircle is ½ the circumference of a circle. The diameter of the semicircle is 6 ft. Then, the circumference of the semicircle would be 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft . 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft .

Therefore, the total perimeter of the window is 6 + 12 + 12 + 9.4 = 39.4 ft . 6 + 12 + 12 + 9.4 = 39.4 ft . You need to buy 39.4 ft of trim.

Your Turn 10.33

People in mathematics.

The overwhelming consensus is that Archimedes (287–212 BCE) was the greatest mathematician of classical antiquity, if not of all time. A Greek scientist, inventor, philosopher, astronomer, physicist, and mathematician, Archimedes flourished in Syracuse, Sicily. He is credited with the invention of various types of pulley systems and screw pumps based on the center of gravity. He advanced numerous mathematical concepts, including theorems for finding surface area and volume. Archimedes anticipated modern calculus and developed the idea of the “infinitely small” and the method of exhaustion. The method of exhaustion is a technique for finding the area of a shape inscribed within a sequence of polygons. The areas of the polygons converge to the area of the inscribed shape. This technique evolved to the concept of limits, which we use today.

One of the more interesting achievements of Archimedes is the way he estimated the number pi, the ratio of the circumference of a circle to its diameter. He was the first to find a valid approximation. He started with a circle having a diameter of 1 inch. His method involved drawing a polygon inscribed inside this circle and a polygon circumscribed around this circle. He knew that the perimeter of the inscribed polygon was smaller than the circumference of the circle, and the perimeter of the circumscribed polygon was larger than the circumference of the circle. This is shown in the drawing of an eight-sided polygon. He increased the number of sides of the polygon each time as he got closer to the real value of pi. The following table is an example of how he did this.

Archimedes settled on an approximation of π ≈ 3.1416 π ≈ 3.1416 after an iteration of 96 sides. Because pi is an irrational number, it cannot be written exactly. However, the capability of the supercomputer can compute pi to billions of decimal digits. As of 2002, the most precise approximation of pi includes 1.2 trillion decimal digits.

The Platonic Solids

The Platonic solids ( Figure 10.76 ) have been known since antiquity. A polyhedron is a three-dimensional object constructed with congruent regular polygonal faces. Named for the philosopher, Plato believed that each one of the solids is associated with one of the four elements: Fire is associated with the tetrahedron or pyramid, earth with the cube, air with the octahedron, and water with the icosahedron. Of the fifth Platonic solid, the dodecahedron, Plato said, “… God used it for arranging the constellations on the whole heaven.”

Plato believed that the combination of these five polyhedra formed all matter in the universe. Later, Euclid proved that exactly five regular polyhedra exist and devoted the last book of the Elements to this theory. These ideas were resuscitated by Johannes Kepler about 2,000 years later. Kepler used the solids to explain the geometry of the universe. The beauty and symmetry of the Platonic solids have inspired architects and artists from antiquity to the present.

Check Your Understanding

Section 10.4 exercises.

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[FREE] Fun Math Games & Activities Packs

Always on the lookout for fun math games and activities in the classroom? Try our ready-to-go printable packs for students to complete independently or with a partner!

In order to access this I need to be confident with:

Here you will learn what perimeter is, how to find perimeter of various shapes and how to apply perimeter to real world scenarios.

Students will first learn how to find perimeter as part of their work in measurement and data in 3 rd grade and 4 th grade and expand their knowledge as they progress through middle and high school.

What is the perimeter?

The perimeter of a two-dimensional shape is the sum of the lengths of each of the sides.

Perimeter of squares

Let’s find the perimeter of the square.

The perimeter is found by summing the lengths of all the sides of the square.

Since a square has four side lengths that are the same measurement, you can add:

The perimeter of this square is 28 units.

Another way to find the perimeter of a square is to multiply one side length by 4 .

\begin{aligned}& P=4(7) \\\\ & P=28\end{aligned}

The perimeter is 28 units.

Step-by-step guide: Perimeter of squares

Perimeter of triangles

Let’s find the perimeter of a triangle.

The perimeter is found by summing the lengths of all the sides of the triangle.

The perimeter of the triangle is 20 units.

Find a missing side using the perimeter

You can also figure out the missing side length of a polygon when you are given the perimeter.

For example, find the missing side of the triangle.

In order to find the missing side length, think about what number can be put in place of the “?” to make the equation true.

\begin{aligned}& 9+12 \; + \; ?=26 \\\\ & 21 \; + \; ?=26\end{aligned}

Replacing the “?” with 5 will make the equation true.

5 inches is the length of the missing side.

Step-by-step guide: Perimeter of a triangle

[FREE] Perimeter Worksheet (Grade 3 to 4)

Use this quiz to check your grade 3 to 4 students’ understanding of perimeter. 10+ questions with answers covering a range of 3rd and 4th grade perimeter topics to identify learning gaps!

Perimeter of rectangles

Let’s find the perimeter of the rectangle.

The perimeter is found by summing the lengths of all the sides of the rectangle.

The perimeter of the rectangle is 22 units.

You can also use a formula to find the perimeter of a rectangle.

\begin{aligned}& P=(2 \times \text { length })+(2 \times \text { width }) \\\\ & P=2 l+2 w \end{aligned}

Applying the formula to find the perimeter:

\begin{aligned}& \text { length }=8 \text { units } \\\\ & \text { width }=3 \text { units }\end{aligned}

\begin{aligned}& P=2(8)+2(3) \\\\ & P=16+6 \\\\ & P=22\end{aligned}

The perimeter is 22 units.

Find a missing dimension of a rectangle using the perimeter

Find the width of the rectangle.

P=\text { length }+ \text { length }+ \text { width }+ \text { width }

Think about the number that makes the equation true.

\begin{aligned}& 54=10+10 \; + \; ? \; + \; ? \\\\ & 54=20 \; + \; ? \; + \; ? \\\\ & 54=20+34\end{aligned}

34 represents the sum of both sides, which means 17 is the measurement of the width because:

17 \, cm is the width of the rectangle.

You can also use the formula for the perimeter of a rectangle to find the width (which is a 7 th grade skill).

\begin{aligned}& P=2 l+2 w \\\\ & 54=2(10)+2 w \\\\ & 54=20+2 w \\\\ & 54-20=20-20+2 w \\\\ & 34=2 w \\\\ &\cfrac{34}{2}=\cfrac{2 w}{2} \\\\ & 17=w\end{aligned}

The width is 17 \, cm .

Step-by-step guide:   Perimeter of a rectangle

Common Core State Standards

How does this relate to 3 rd grade – 7 th grade math?

• Grade 3 – Measurement and data (3.MD.D.8) Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
• Grade 4 – Measurement and data (4.MD.A.3) Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
• Grade 5 – Geometry (5.G.B.4) Classify two-dimensional figures in a hierarchy based on properties.
• Grade 6 – Geometry (6.G.A.3) Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
• Grade 7 – Expressions and equations (7.EE.B.4a) Solve word problems leading to equations of the form px+q=r and p \, (x + q)=r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 \, cm . Its length is 6 \, cm . What is its width?

How to find perimeter

There are several strategies to find the perimeter of 2 D shapes. For more specific step-by-step guides, check out the individual pages linked in the “What is Perimeter?” section above or read through the examples below.

In order to calculate the perimeter of a shape:

Add all the side lengths or apply the formula.

Write the final answer with the correct units.

Step-by-step guide :  How to find perimeter

How to find perimeter examples

Example 1: equilateral triangle.

What is the perimeter of the triangle?

The length of each side is 1 \cfrac{1}{5} \mathrm{~cm}.

To find the perimeter, add up all the side lengths.

\begin{aligned}& P=1 \cfrac{1}{5}+1 \cfrac{1}{5}+1 \cfrac{1}{5} \\\\ & P=3 \cfrac{3}{5}\end{aligned}

OR since all the sides are the same length, you can multiply 1 \cfrac{1}{5} by 3.

\begin{aligned}& P=3 \times 1 \cfrac{1}{5} \\\\ & P=\cfrac{3}{1} \times \cfrac{6}{5} \\\\ & P=\cfrac{18}{5}=3 \cfrac{3}{5} \end{aligned}

2 Write the final answer with the correct units.

The perimeter of the triangle is 3 \cfrac{3}{5} {~cm} .

Example 2: square

Find the perimeter of the square.

All sides of a square are congruent (equal) so each side’s length is 17.5 inches.

17.5+17.5+17.5+17.5=70

OR since all the side lengths are the same, you can multiply 17.5 by 4 .

4 \times 17.5=70

The perimeter of the square is 70 inches.

Example 3: perimeter of rectangle

What is the perimeter of the rectangle on the coordinate graph?

Counting the units, the length of the rectangle is 6 units and the width of the rectangle is 8 units.

You can use the formula to find the perimeter of a rectangle; P=2 l+2 w .

l=6 \text { and } w=8

\begin{aligned}& P=2(6)+2(8) \\\\ & P=12+16 \\\\ & P=28\end{aligned}

The perimeter of the rectangle on the coordinate graph is 28 units.

Example 4: perimeter of the parallelogram

What is the perimeter of the parallelogram?

Similar to a rectangle, the opposite sides of a parallelogram are congruent (equal). So, there are two sides that measure 13 {~cm} and two sides that measure 28 {~cm} .

13+13+28+28=82

The perimeter of the parallelogram is 82 {~cm} .

Example 5: word problem

A rectangular vegetable garden has a perimeter of 34 feet. The width of the garden is 11 {~ft} . Find the length of the garden.

A rectangle has congruent (equal) opposite sides.

\begin{aligned}& 34=11+11 \; + \; ? \; + \; ? \\\\ & 34=22 \; + \; ? \; + \; ?\end{aligned}

What number will make the equation true?

12 makes the equation true.

Since both sides of the rectangle are equal, 12 is the sum of both sides.

OR you can use the formula to find the length, which is a 7 th grade skill.

You can use the formula to find out the length of the garden.

\begin{aligned}& P=2 l+2 w \\\\ & 34=2 l+2(11) \\\\ & 34=2 l+22 \\\\ & 34-22=2 l+22-22 \\\\ & 12=2 l \\\\ & \cfrac{12}{2}=\cfrac{2 l}{2} \\\\ & 6=l\end{aligned}

The length of the vegetable garden is 6 {~ft} .

Example 6: perimeter of irregular shape

What is the perimeter of the driveway?

Add all the side lengths.

Choose one of the sides of the driveway to start, and go around the driveway, adding all the side lengths together.

58+1.5+1+5+10+8+7+2.5+22+5+9.5+8+8+13+12+1=171.5

Each side of the driveway is measured in feet.

The perimeter of the driveway is 171.5 feet.

Teaching tips for how to find the perimeter

• Use active learning activities such as having students measure the perimeter of the classroom or a room in their home. When students are physically active in learning, they tend to retain concepts.
• Choose worksheets that provide a mixture of regular and irregular shapes, both gridded and non-gridded, so students can practice all perimeter solving strategies. This also challenges them to decide which strategies are useful for each type of shape.
• Have students try to identify patterns when finding perimeters of regular polygons.

Easy mistakes to make

• Confusing the concept of area with the concept of perimeter When students first learn perimeter and area, they can easily mix up the two concepts. Having students do performance tasks that require them to link the concepts to real world scenarios helps build understanding.
• Thinking that the formula for perimeter of a rectangle works for all polygons Formulas are helpful when making calculations. However, if they are applied haphazardly, it could lead to errors. Having students develop an understanding of formulas through investigative activities prevents them from haphazardly applying formulas to get answers.
• Converting units In order to find the perimeter, the side lengths have to be the same unit. If the sides are given in different units, convert them all to the same unit before finding the perimeter.

Practice perimeter questions

1. What is the perimeter of the square?

The perimeter is the distance around the square which is the same as summing the four sides. Since a square has four congruent (equal) sides, you can add the four sides up or multiply one side by 4.

\begin{aligned}& P=5 \cfrac{1}{2}+5 \cfrac{1}{2}+5 \cfrac{1}{2}+5 \cfrac{1}{2} \\\\ & P=22\end{aligned}

\begin{aligned}& P=4 \times 5 \cfrac{1}{2} \\\\ & P=\cfrac{4}{1} \times \cfrac{11}{2} \\\\ & P=\cfrac{44}{2}=22\end{aligned}

The perimeter of the square is 22 {~cm} .

2. The perimeter of an equilateral triangle is 30 units. Find the length of the sides.

An equilateral triangle has 3 congruent (equal) sides. To find the perimeter, you would sum the three sides together. In this case, the perimeter is 30 units.

So, ? \; + \; ? \; + \; ?=30 where the numbers added have to be the same numbers.

Think about the possible side lengths. You know that 10+10+10=30 , so the side lengths of the equilateral triangle are 10 units.

Since the three sides have to be equal, another way to think about it, is what number multiplied by three is equal to 30 .

3 \; \times \; ?=30

The number that makes this equation true is   10 .

Either way you think about it,   10 units is the measure of the side lengths.

3. What is the perimeter of the irregular polygon?

The perimeter is the distance around the entire   2 D shape, which is the same as summing all the sides.

This irregular 2 D shape is a hexagon which means it has six sides. To find the perimeter, sum the six sides.

\begin{aligned}& P=2+2+4+3+14+9 \\\\ & P=34\end{aligned}

The perimeter of the 2 D shape is 34 inches.

4. What is the perimeter of the octagon?

All sides of a regular pentagon are the same length. So the octagon has eight sides that are all 6 {~cm} .

To find the perimeter, find the distance around the octagon which is the same as summing up the side lengths.

6+6+6+6+6+6+6+6=48

Another way to think about it is to multiply one side length by 8 since they are all the same length.

8 \times 6=48

The perimeter of the octagon is 48 {~cm } .

5. The graph below shows a rectangular field on a ranch. The ranch owner wants to fence in the rectangular field. How many yards of fencing does he need? (The coordinates are represented in yards).

Count the number of spaces from one point to the next. Notice that each grid line is in increments of 10.

The perimeter is the distance around the rectangle, which is the same as summing the sides.

\begin{aligned}& P=100+100+30+30 \\\\ & P=260\end{aligned}

You can apply the formula for perimeter of the rectangle.

l=30 \text { and } w=100

\begin{aligned}& P=2l+2w \\\\ & P=2(30)+2(100) \\\\ & P=260\end{aligned}

The perimeter is 260 yards which means the rancher needs a total of 260 yards of fencing.

6. This is the pool in the yard at Elvis Presley’s Palm Springs home. Find the perimeter of the pool.

The perimeter is the distance around the figure, which is the same as summing up the sides.

\begin{aligned}& P=8.5+14+12.5+16+15.5 \\\\ & 66.5=8.5+14+12.5+16+15.5\end{aligned}

The perimeter of Elvis’s pool is 66.5 {~ft} .

Perimeter FAQs

Yes, you can find the perimeter or distance around any 2 D shape with straight edges or curved edges. Typically, the perimeter of a circle is called the circumference of a circle. You will learn how to find the circumference of circles in middle school. You will also learn how to find the perimeter of ellipses in high school.

Perimeter is the distance around a rectangle and is measured in units. Area is the space within a rectangle and is measured with square units. They represent different parts of a rectangle, but both require knowing the dimensions of the length and width to solve.

Yes, the side lengths of a polygon can be whole numbers, fractions, or decimals. You would add the fractions together using the rules for adding fractions.

Yes, you can find the perimeter of any quadrilateral or any polygon by summing the side lengths which is finding the total distance around the figure.

The next lessons are

• Angles in polygons
• Congruence and similarity
• Prism shape

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Perimeter of Polygon

The perimeter of a polygon is defined as the sum of the length of the boundary of the polygon. In other words, we say that the total distance covered by the sides of any polygon gives its perimeter. In this lesson, we will learn to find the perimeter of polygons, and find the difference between the area and perimeter of the polygons in detail.

What is the Perimeter of Polygon?

The perimeter of a polygon is the measure of the total length of the boundary of the polygon. As polygons are closed plane shapes , thus, the perimeter of the polygons also lies in a two-dimensional plane. The perimeter of a polygon is always expressed in linear units like meters, centimeters, inches, feet, etc. For example, if the sides of a triangle are given as 4 cm, 6 cm, and 7 cm, then its perimeter will be, 4 + 6 + 7 = 17 cm. This basic formula applies to all polygons.

Difference Between Area and Perimeter of Polygon

The area and perimeter of polygons can be calculated if the lengths of the sides of the polygon are known. The following table shows the difference between the area and perimeter of polygons.

There is one similarity between the area and perimeter of a polygon. Both depend directly on the length of the sides of the shape and not directly on the interior angles or the exterior angles of the polygon.

Formula for Perimeter of Polygon

We can categorize a polygon as a regular or irregular polygon based on the length of its sides. The perimeter formula of some known polygons is given as follows:

• Perimeter of a triangle = a + b + c, where, a, b, and c are the length of its sides.
• Perimeter of a rectangle = 2 × (length + width)

Before calculating the perimeter of the polygon, we first find out whether the given polygon is a regular polygon or an irregular polygon. After that, the appropriate formula is used to find the perimeter of the polygon.

Perimeter of Regular Polygons

A polygon that is equilateral and equiangular is known as a regular polygon. Thus, we calculate the perimeter of regular polygons using the formulas associated with each polygon. The formulas of some commonly used regular polygons are:

Therefore, the formula to find the perimeter of a regular polygon is: Perimeter of regular polygon = (number of sides) × (length of one side)

Example: Find the perimeter of a regular hexagon whose each side is 6 inches long.

Solution: Given, the length of one side = 5 inches and the number of sides = 6 (as it is a hexagon).

Thus, the perimeter of the regular hexagon = (number of sides) × (length of one side) = (6 × 5) = 30 inches. Therefore, the perimeter of the regular hexagon is 30 inches.

Perimeter of Irregular Polygons

Polygons that do not have equal sides and equal angles are referred to as irregular polygons. Thus, in order to calculate the perimeter of irregular polygons, we add the lengths of all sides of the polygon.

Example: Find the perimeter of the given polygon.

Thus, the perimeter of the irregular polygon will be the sum of the lengths of all its sides. The perimeter of ABCD = AB + BC + CD + AD ⇒ Perimeter of ABCD = (7 + 8 + 3 + 5) = 23 units

Therefore, the perimeter of ABCD is 23 units.

Perimeter of Polygon with Coordinates

The perimeter of a polygon with coordinates can be found using the following steps:

• Step 1: Find the distance between all the points using the distance formula, D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates.
• Step 2: Once the dimensions of the polygon are known, we need to find whether the given polygon is a regular polygon or not.
• Step 3: If the polygon is a regular polygon we use the formula, perimeter of regular polygon = (number of sides) × (length of one side); while if the polygon is an irregular polygon we just add the lengths of all sides of the polygon.

Example: What is the perimeter of the polygon formed by the coordinates A(0,0), B(0, 3), C(3, 3), and D(3, 0)?

Solution: On plotting the coordinates A(0,0), B(0, 3), C(3, 3), and D(3, 0) on an XY plane and joining the dots we get a four-sided polygon as shown below.

In order to understand whether it is a regular polygon or not, we need to find the distance between all the points using the distance formula , D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates. After substituting the values in the formula, the length of sides AB, BC, CD and DA can be calculated as shown below.

• Length of AB = \(\sqrt{({0 - 0})^2 + ({3 - 0})^2}\) = 3 units. This was calculated with \(x_1\) = 0, \(x_2\) = 0, \( y_1\) = 0, \( y_2\) = 3
• Length of BC = \(\sqrt{({3 - 0})^2 + ({3 - 3})^2}\) = 3 units. This was calculated with \(x_1\) = 0, \(x_2\) = 3, \( y_1\) = 3, \( y_2\) = 3
• Length of CD = \(\sqrt{({3 - 3})^2 + ({0 - 3})^2}\) = 3 units. This was calculated with \(x_1\) = 3, \(x_2\) = 3, \( y_1\) = 3, \( y_2\) = 0
• Length of DA = \(\sqrt{({0 - 3})^2 + ({0 - 0})^2}\) = 3 units. This was calculated with \(x_1\) = 3, \(x_2\) = 0, \( y_1\) = 0, \( y_2\) = 0

Now, we know that the length of all sides of the given four-sided polygon is the same. This shows that it is a square. Thus, the perimeter of the polygon ABCD (square) can be calculated with the formula, Perimeter = number of sides) × (length of one side). After substituting the values in the formula, we get, perimeter = 4 × 3 = 12 units. Hence, the perimeter of the polygon with coordinates (0,0), (0, 3), (3, 3), and (3, 0) is 12 units.

Perimeter of Polygons Examples

Example 1: Find the missing length FA of the polygon shown below if the perimeter of polygon is 18.5 units.

Solution: It can be seen that the given polygon is an irregular polygon. The perimeter of the given polygon is 18.5 units. The given lengths of the sides of polygon are AB = 3 units, BC = 4 units, CD = 6 units, DE = 2 units, EF = 1.5 units; and let FA = x units.

Given that, the perimeter of the polygon ABCDEF = 18.5 units ⇒ Perimeter of polygon ABCDEF = AB + BC + CD + DE + EF + FA = 18.5 units ⇒ (3 + 4 + 6 + 2 + 1.5 + x) = 18.5. Thus, x = 18.5 - (3 + 4 + 6 + 2 + 1.5) = 2 units

Therefore, the missing length FA of the polygon ABCDEF is 2 units.

Example 2: Find the length of the side of an equilateral triangle i f its perimeter is 27 units.

Solution: Given, the perimeter of polygon (equilateral triangle) = 27 units. Let the length of the side of the equilateral triangle be "a" units. Now, the length of the side of the equilateral triangle can be calculated using the formula:

The perimeter of equilateral triangle = 3 × a ⇒ Perimeter of equilateral triangle = 3 × a = 27 units. Thus, a = 27/3 = 9 units

Therefore, the length of the side of the equilateral triangle is 9 units.

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Practice Questions on Perimeter of Polygons

Faqs on perimeter of polygons.

The perimeter of a polygon is defined as the total length of the boundary of the polygon in a two-dimensional plane. The perimeter of a polygon is expressed in linear units like meters, centimeters, inches, feet, etc.

How to Find the Perimeter of a Polygon?

The perimeter of a polygon can be found by using the following steps:

• Step 1: Find whether the given polygon is a regular polygon or not.
• Step 2: If it is a regular polygon, the perimeter can be calculated using the formula, Perimeter of regular polygon = (number of sides) × (length of one side). In case, if it is an irregular polygon, then its perimeter can be calculated by adding the lengths of all its sides.
• Step 3: Once the perimeter of the polygon is obtained, we need to mention the unit along with the value of the perimeter.

What is the Difference Between the Area and Perimeter of Polygons?

The perimeter of a polygon is the total length of its boundary, whereas, the area of a polygon is the space enclosed by the polygon. We can find the perimeter of a polygon by adding the length of all its sides. The area of a polygon is calculated by using the appropriate formulas or by reducing the polygon into smaller regular polygons. The area of a polygon is always expressed in square units, like meter 2 , centimeter 2 , while the perimeter of a polygon is always expressed in linear units like meters, inches, and so on.

How to Find the Perimeter of Polygons with Vertices?

We can find the perimeter of polygons with given vertices using the following steps:

• Step 1: First, we need to calculate the distance between all the points using the distance formula , D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates of the vertices.
• Step 2: Once the dimensions of the polygons are known, we need to check whether the given polygon is a regular polygon or an irregular polygon.
• Step 3: The perimeter of a regular polygon can be found by using the formula, perimeter of regular polygon = (number of sides) × (length of one side), whereas, if the polygon is an irregular one, we can simply add the lengths of all its sides.

How to Find the Perimeter of Regular Polygons?

The perimeter of regular polygons can be found using the following steps:

• Step 1: Count the number of sides of the polygon.
• Step 2: Note the length of one side.
• Step 3: Use the values obtained in Step 1 and Step 2 to find the value of perimeter using the formula, Perimeter of a regular polygon = (number of sides) × (length of one side).

How to Find the Perimeter of Irregular Polygons?

In order to calculate the perimeter of an irregular polygon we use the following steps:

• Step 1: Note the length of each side of the given polygon.
• Step 2: Once the length of all the sides is obtained, the perimeter is found by adding all the sides.

Is the Perimeter of a Regular Polygon Directly Proportional to the Length of Side?

Yes, the perimeter of a regular polygon is directly proportional to its side length. We know that the perimeter of a regular polygon is calculated by the formula, Perimeter = (number of sides) × (length of one side). Thus, if the length of the side is increased, the value of the perimeter also increases. For example, a square with a side length of 4 units will have a larger perimeter as compared to a square with a side length of 2 units.

How to Find the Missing Side Length When the Perimeter of Polygon is Given?

We can find the missing side length when the perimeter of the polygon is given in the following way:

• Step 2: If the given polygon is a regular polygon, then we use the formula, Perimeter of regular polygon = (number of sides) × (length of one side) to find the missing side length. In case, if the given polygon is an irregular polygon, then we add the lengths of all the given sides and subtract it from the perimeter to get the missing side.

What is the Formula of the Perimeter of Polygon?

The formula that is used to calculate the perimeter of a polygon is simple to understand because 'perimeter' means the sum of the length of all its sides and hence, the formula is expressed as, Perimeter = Sum of the sides. If it is a regular polygon, it means that all the sides are equal. In that case, to make things easier, the formula is expressed as, Perimeter = number of sides × length of one side.

How to Find the Perimeter of Polygons with Coordinates?

The perimeter of polygons with coordinates can be calculated by using the following steps.

• First, the length of the sides of the polygon can be calculated using the distance formula. The given coordinates, \((x_1, y_1)\) and \((x_2, y_2) \) are substituted in the distance formula , D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\).
• After the length is known, we should find out if the polygon is a regular polygon or an irregular one. Accordingly, the formula for the perimeter is used to calculate the perimeter.
• If it is a regular polygon, the formula that is used is, Perimeter = number of sides × length of one side. If it is an irregular polygon, then the sides can be added to find the perimeter using the formula, Perimeter = Sum of the sides.

o Perimeter

o Equilateral

o Equiangular

o Regular polygon

o Recognize polygons and their associated nomenclature

o Calculate the perimeter of a polygon

o Identify the special characteristics of regular polygons

Obviously, the more complex polygons (such as the two bottom figures above) have no apparent simple formulas for calculating such properties as the enclosed area or the length of diagonal line segment. As a result, we will limit our discussion to mainly simple convex polygons. (The terms simple, complex, concave, and convex have the same definitions for polygons as they do for quadrilaterals.) Note, however, that complex and concave polygons can be subdivided into more manageable polygons (such as triangles) for the purposes of calculating area or other parameters.

Polygon Nomenclature

Polygons have a moderately systematic nomenclature, and understanding this nomenclature can be helpful. For instance, you may have heard of the Pentagon (the home of the United States Department of Defense), which is a five-sided building, or you may have heard other terms such as octagon or hexagon . Again, these names are largely chosen in a systematic fashion. The common names of polygons with five to ten sides are listed below; note that the terms triangle and quadrilateral do not follow this nomenclature.

pentagon five-sided polygon

hexagon six-sided polygon

heptagon seven-sided polygon

octagon eight-sided polygon

nonagon nine-sided polygon

decagon ten-sided polygon

More complicated prefixes apply to polygons with more than 10 sides, but for the sake of simplicity, we will simply refer to these using the n -gon terminology (for instance, we will use 11-gon to refer to an 11-sided polygon).

Properties of Simple Convex Polygons

A hexagon can thus be divided into four triangles, for a total of 720°. We can see a pattern beginning to develop: with each additional side, a polygon gains 180° in its sum of interior angles. Let's derive a general formula for the number of degrees in an n -gon. First, we know that an n -gon has n sides and n vertices. As we have done above, we divide the n -gon into triangles by picking one vertex and then drawing dividing segments to each non-adjacent vertex. So, if we pick one vertex of an n -gon, we are left with n – 1 remaining vertices. Two of these vertices are adjacent to the chosen vertex, leaving us with n – 3 vertices to which we will draw dividing segments. Also notice from the above examples that if we draw m dividing segments in a polygon, we end up with m + 1 triangles. Thus, drawing n – 3 dividing segments yields ( n – 3) + 1, or n – 2, triangles. Each triangle has 180°, so the formula for the number of degrees in an n -gon is

In addition, we may also be interested in determining the perimeter of a polygon, which is the total length of all the sides in the figure. Obviously, then, we can calculate the perimeter by adding the lengths of all the sides. For instance, a square with sides of length 2 meters has four sides of 2 meters each, for a perimeter of 8 meters.

Practice Problem : How many degrees are in a decagon?

Solution : A decagon is a polygon with 10 sides (in other words, an n -gon with n = 10). Thus, we can calculate the number of degrees in the figure using our formula from earlier.

Thus, the sum of the measures of the interior angles of a decagon is 1,440°. If you ever have trouble recalling the formula, draw an arbitrary decagon (make it as simple as possible!) and divide it into triangles. Then, multiply the number of triangles (eight in this case) by 180°. This approach, although slightly more involved, is equally as valid as using the formula.

Regular n -gons

If all of the sides of a polygon are congruent (that is, the polygon is equilateral ) and all the angles are congruent (that is, the polygon is equiangular ), then it is called a regular polygon . Because all the angles of a regular n -gon are congruent, the measure of each of those angles is simply the total number of degrees in the n -gon divided by the total number of angles ( n ).

For an n -gon with sides of length x , the perimeter is simply nx (since the figure has n sides of length x ). Calculating the area of a regular n -gon can be done, but deriving a general formula for the area of a regular n -gon requires trigonometric analysis. Although this analysis is not fundamentally difficult, we have not yet studied the prerequisite concepts.

Practice Problem : Prove that the triangles ADE and ABC inside the regular pentagon ABCDE are congruent.

Solution : Recall that a regular n -gon is both equilateral and equiangular. Thus, we know that segments EA, AB, BC, and DE are all congruent. Furthermore, we also know that angles E and B are congruent. Let's show these facts in the diagram.

By the SAS (side-angle-side) condition, we can then conclude that the triangles ADE and ABC are congruent.

Solution : A regular quadrilateral is a square (since it is equilateral and equiangular). Let's draw a diagram of the square and its diagonal in this case. We'll also mark the sides as having length x .

Thus, each side of the square has a length of 3 feet. The perimeter is the sum of the lengths of all the sides-in this case, the perimeter is 12 feet.

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Curriculum  /  Math  /  3rd Grade  /  Unit 5: Shapes and Their Perimeter  /  Lesson 7

Shapes and Their Perimeter

Lesson 7 of 16

Criteria for Success

Anchor tasks.

Problem Set

Target Task

Additional practice.

Find perimeter of shapes with all side lengths labeled.

Common Core Standards

Core standards.

The core standards covered in this lesson

Measurement and Data

3.MD.D.8 — Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

Foundational Standards

The foundational standards covered in this lesson

Operations and Algebraic Thinking

3.OA.D.8 — Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Find the perimeter of a shape in which all side lengths are labeled.
• Write an equation to represent the perimeter of a shape (optionally including simplified equations, such as a multiplication equation for a regular polygon) (MP.2).

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Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

What do you notice? What do you wonder?

Guiding Questions

IM Grade 3 Unit 7 Lesson 8 Activity 1 Problem 1, accessed on Nov. 14, 2022, 4:44 p.m., is licensed by Illustrative Mathematics under either the  CC BY 4.0  or  CC BY-NC-SA 4.0 . For further information, contact Illustrative Mathematics .

Find the perimeter of each shape below. Show or explain your work.

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

• What multiplication equation can you write to represent the perimeter of the shape in #1(b)? In #1(f)?
• Did anyone use any mental strategies to find the perimeter in #2? Which side lengths did you add together first to make the computation easier? Why?
• Which shape has a greater perimeter in #3? How do you know? Did you calculate the perimeter differently for each shape? Why?
• Who is correct in #4, Michael, Jackson, both of them, or neither of them? Why? 

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Heather’s rectangular patio, shown below, is 10 meters long and 18 meters wide. What is the perimeter of her patio? 

Darryl’s patio has a different shape, shown below. 

a.   What is the perimeter of Darryl’s patio?

b.   Darryl says his patio has a larger perimeter than Heather’s patio. Is he correct? Explain how you know.

Student Response

An example response to the Target Task at the level of detail expected of the students.

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer keys for Problem Sets and Extra Practice Problems are available with a Fishtank Plus subscription.

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Topic A: Attributes of Two-Dimensional Shapes

Compare and classify polygons.

Compare and classify quadrilaterals.

Draw shapes with specified attributes.

Reason about composing and decomposing polygons using tangrams.

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Topic B: Understanding Perimeter

Understand perimeter to be the boundary around a two-dimensional shape. Compare perimeters of various polygons using concrete non-standard units.

Find perimeter of polygons whose sides are marked with unit length marks by counting the unit lengths.

Find perimeter by measuring side lengths in whole number units.

Find perimeter of regular polygons and of rectangles when some side lengths are not given.

Solve word problems involving finding perimeter given the side lengths.

Find side lengths of regular polygons and of rectangles when given perimeter and other side lengths.

Solve more complex word problems involving perimeter, such as finding a missing side length given perimeter and other side lengths.

Topic C: Distinguishing Between Area and Perimeter

Find rectangles with the same area and different perimeters.

Find rectangles with the same perimeter and different areas.

Solve a variety of word problems involving area and perimeter.

Reason about composing and decomposing polygons with various areas and perimeters using tetrominoes.

3.G.A.1 3.MD.D.8

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Perimeter of a Polygon

Last updated: Oct 1, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

The perimeter of a polygon is the combined length of all its straight-line sides. In many geometry problems, we are asked to find the perimeter of an irregular shape, where not all of the side lengths are given, In many cases, we can use the properties of simple polygons like triangles or rectangles to find the perimeter of such irregular polygons, as in this example.

Find the perimeter of the following shape, created by cutting a triangle from a square with side 5 units, as in the following drawing.

The irregular shape was created by taking a square, where all four side are equal, and replacing one of the sides. Since the original shape was a square, the three sides that remain of the original shape are all equal and measure 5 units each.

The hint for finding out the two other sides is in the given angle - 30 °. In the original shape, which is a square, the interior angles all have a measure of 90°. so if we draw back the missing side of the square, the resulting triangle will have two angles measuring 60°. But if two of the triangle's angles are 60°, then by the sum of angles in a triangle , the third angle must also measure 60° (180° - 60°-60°=60°).

So this triangle is an equilateral triangle , and all its sides have equal length. The missing side which we drew back measures 5 units, since it was originally a side of a square with side length 5 units, and so all sides of the equilateral triangle must also measure 5 units .

So the perimeter is 5+5+5+5+5=25 units.

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Perimeter of Polygons (Grade 3)

Suggested learning targets.

• I can identify polygons.
• I can define perimeter.
• I can find the perimeter of polygons when give the lengths of all sides.
• I can find the unknown side lengths of polygons when given the perimeter.
• I can show how rectangles with the same perimeter can have different areas and show rectangles with the same area can have different perimeters.
• I can solve word problems.

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Course: 3rd grade   >   Unit 11

• Area and perimeter situations
• Comparing areas and perimeters of rectangles
• Compare area and perimeter

Area and perimeter word problems

• Perimeter: FAQ
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

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Perimeter Of Polygons Math Games

In this series of games, your students will learn to solve real world and mathematical problems involving perimeters of polygons. The Perimeter Of Polygons l earning objective — based on CCSS  and state standards — delivers improved student engagement and academic performance in your classroom, as demonstrated by research . This learning objective directly references 3.MD.D.8 as written in the common core national math standards.

Scroll down for a preview of this learning objective’s games and the concepts.

Concepts Covered

A perimeter is the boundary of any 2-dimensional shape. The length of the perimeter of any polygon is the sum of all its sides. Given a rectangle with 2 adjacent sides given, find the perimeter by multiplying each side by 2 and adding the products. Given a rectangle with one side and perimeter labeled, find the missing side by doubling the side given, subtracting the product from the perimeter, then dividing the difference by 2. Given a rectangle with the area and one side labeled, find the missing side by dividing the area by the given side.

Create rectangles with different areas but the same perimeter. Create rectangles with the same perimeter but different areas. Given a rectangle with both sides and perimeter, identify other rectangles that have the same perimeter but a different length and width. Given the perimeter of a polygon and one missing side length, determine the missing side length. Identify the figure that matches the perimeter given.

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Perimeter of Polygons Lesson Plan: Measurement and Data

*Click to open and customize your own copy of the  Perimeter of Polygons Lesson Plan . 

This lesson accompanies the BrainPOP topic Perimeter of Polygons , and supports the standard of solving real world and mathematical problems involving perimeters of polygons. Students demonstrate understanding through a variety of projects.

Step 1: ACTIVATE PRIOR KNOWLEDGE

Prompt students to imagine they’re using a map to plan a run. Display this image:

Ask students:

• How can you calculate the total distance you’d run if you follow the red path? 

Step 2: BUILD KNOWLEDGE

• Read the description on the Perimeter of Polygons topic page .
• Play the Movie , pausing to check for understanding.

Step 3: APPLY and ASSESS 

Assign Perimeter of Polygons Challenge and Quiz , prompting students to apply essential literacy skills while demonstrating what they learned about this topic.

Step 4: DEEPEN and EXTEND

Students express what they learned about finding the perimeter of polygons while practicing essential literacy skills with one or more of the following activities. Differentiate by assigning ones that meet individual student needs.

• Make-a-Movie : Produce a tutorial explaining how to determine the perimeter of your classroom.
• Make-a-Map : Create a concept map sequencing the steps to determine the perimeter of a playground.
• Creative Coding : Code a math problem about finding the perimeter of a polygon. Challenge a classmate to solve. 

More to Explore 

Area Builder : Create shapes using colorful blocks and explore the perimeter and area.

Related BrainPOP Topics : Deepen understanding of measurement with these topics: Intro to Area , Polygons , and Area of Polygons .

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• Pause Point Overview : Video tutorial showing how Pause Points actively engage students to stop, think, and express ideas.  
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MAFS.3.MD.4.8 Archived Standard

13 Lesson Plans

5 Formative Assessments

2 Virtual Manipulatives

2 Original Student Tutorials

• STEM Lessons - Model Eliciting Activity 2
• MFAS Formative Assessments 5
• Original Student Tutorials Mathematics - Grades K-5 2

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

• Assessment Limits : For items involving area, only polygons that can be tiled with square units are allowable. Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10. Items are not required to have a graphic, but sufficient dimension information must be given.
• Test Item #: Sample Item 1

Ben is planning a garden. Which measurement describes the perimeter of his garden?

• Difficulty: N/A
• Type: MC: Multiple Choice
• Test Item #: Sample Item 2

Ben has a rectangular garden with side lengths of 2 feet and 5 feet. What is the perimeter, in feet, of Ben's garden?

• Type: EE: Equation Editor
• Test Item #: Sample Item 3

The perimeter of a rectangular field is 74 yards. The length of the field is 27 yards. What is the width, in yards, of the field?

Related Courses

Related access points, related resources, formative assessments.

Students are asked to find the perimeter of a hexagon in which the lengths of two sides are not given but can be found.

Type: Formative Assessment

Students are asked to find the length of a missing side on two polygons given the perimeter of each and the lengths of the other sides.

Students are asked to find the whole number dimensions of every rectangle with a given perimeter and then find the area of each rectangle.

Students are asked to find the perimeters of three different polygons.

Students are asked to find the whole number dimensions of every rectangle with a given area and then find the perimeter of each rectangle.

Lesson Plans

In this lesson, students will explore a real world problem based on the Marilyn Burns book Spaghetti and Meatballs for All!. The problem and further practice finding the distance around rectangles will lead them to discover efficient strategies and formulas for solving perimeter.

Type: Lesson Plan

In this culminating activity, students will use their knowledge of area and perimeter to create a "Mighty Monster" following specific criteria. Given a designated area, students will make their monster on centimeter grid paper and calculate both the area and perimeter of each body part, as well as the combined area and perimeter of the entire figure.

This STEM challenge will engage the students in the ways to create different rectangles that have the same area, but different perimeters. They will also explore how to use the scientific method to test their designs with hypothesis, records, data, and a conclusion. This STEM challenge combines architectural engineering with life science and measurement skills for math.

The students will plan a vegetable garden, deciding which kinds of vegetables to plant, how many plants of each kind will fit, and where each plant will be planted in a fixed-area garden design. Then they will revise their design based on new garden dimensions and additional plant options.  Students will explore the concept of area to plan their garden and they will practice solving 1 and 2-step real-world problems using the four operations to develop their ideas.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

In this lesson, students will review finding perimeter of polygons and apply their knowledge of finding perimeter and area to compute unknown side lengths. Students draw rectangles with a specific perimeter and draw rectangles with a specific area but different perimeters.

In this lesson, students are presented with a problem that requires them to create rectangles with the same perimeter but different areas.  Students also search for relationships among the perimeters and areas of different rectangles and find which characteristics produce a rectangle with the greatest area.

This lesson reinforces perimeter as students arrange tables to decorate.

You are a builder who needs to find out what to charge people for rent based on the needs of the different clients and what they might need in an apartment.

In this lesson, students will use their knowledge of area and perimeter to create a "Mighty Monster”. Given specific criteria related to area and perimeter, students will make their monster on centimeter grid paper and calculate both the area and perimeter of each body part to explore the differences between the two types of measurement.

In this lesson, the students are employees of a fencing company. They are working with a customer to try and get the best deal and design of a fence that will fit the customer's area needs. Students will have to use reasoning skills in order to fill in missing information. Students will also discuss whether or not their designs have met the needs of the customer.

In this lesson, students will explore a real world problem based on the Marilyn Burns book Spaghetti and Meatballs for All!. The problem and further practice finding the distance around rectangles will lead them to conceptually understand finding the perimeter of rectangles.

Students will determine the validity of the statement, "All rectangles with the same area will have the same perimeter" through two investigations.

In this lesson, students are tasked with drawing a house based on given directions. The directions include the area and perimeter of particular features of the house. This resource is recommended as a review of perimeter and area.

Use visuals and formulas to find the perimeter and help Penelope as she creates a rectangular herb garden. Find the perimeter of rectangles using visuals and formulas in this student tutorial. 

Type: Original Student Tutorial

Plan some gardens by applying what you learn about perimeter in this interactive tutorial. 

This Khan Academy video presents finding perimeter by adding side-lengths of various polygons.

Type: Tutorial

Virtual Manipulatives

This activity allows the user to test his or her skill at calculating the perimeter of a random shape. The user is given a random shape and asked to enter a value for the perimeter. The applet then informs the user whether or not the value is correct. The user may continue trying until he or she gets the correct answer.

This activity would work well in mixed ability groups of two or three for about 25 minutes if you use the exploration questions, and 10-15 minutes otherwise.

Type: Virtual Manipulative

This activity operates in one of two modes: auto draw and create shape mode, allowing you to explore relationships between area and perimeter. Shape Builder is one of the Interactivate assessment explorers.

STEM Lessons - Model Eliciting Activity

Mfas formative assessments, original student tutorials mathematics - grades k-5, student resources, parent resources.

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REAL WORLD PROBLEMS INVOLVING AREA AND PERIMETER

Problem 1 :

The diagram shows the shape and  dimensions of Teresa’s rose garden.

(a) Find the area of the garden.

(b) Teresa wants to buy mulch for her garden. One bag of mulch covers  12 square feet. How many bags will she need?

By drawing a horizontal line, we can divide the given shape into two parts as shown below.

(1) ABCD is a rectangle

(2) CEFG is also a rectangle

Area of the garden

= Area of rectangle ABCD + Area of the rectangle CEFG

Area of rectangle ABCD :

length AB = 15 ft and width BD = 9 ft

= length x width 

 =  15 x 9

=  135 ft ²  ----(1)

Area of rectangle CEFG :

length CE = 24 ft and width CF = AF - AC ==> 18 - 9 = 9 ft

 =  24 x 9

=  216 ft ²  ----(1)

(1) + (2) 

Area of the rose garden = 135 + 216 ==> 351 ft ²

Number of bags that she needed = 351/12 ==> 29.25

So, she will need 30 bags of mulch

Problem 2 :

The length of a rectangle is 4 less than 3 times its width. If its length is 11 cm, then find the perimeter. 

Let w be the width of the rectangle.

Then, its length is (3w - 4).

Given : Length is 11 cm. 

Then, 

Length (l)  =  11

3w - 4  =  11

3w  =  15

w  =  5

So, the perimeter of the rectangle is 

=  2(l + w)

=  2(11 + 5)

=  2(16)

=  32 cm

Problem 3 :

The diagram shows the floor plan of a hotel  lobby. Carpet costs $3 per square foot. How  much will it cost to carpet the lobby?

By observing the above picture, we can find two trapeziums of same size. Since both are having same size. We can find area of one trapezium and multiply the area by 2.

Area of trapezium = (1/2) h (a +  b)

h = 15.5 ft  a = 30 ft  and b = 42 ft

  =  (1/2) x 15.5 x (30 + 42)

  =  (1/2) x 15.5 x 72 ==> 15.5 x 36==> 558 square feet

Area of floor of a hotel  lobby = 2 x 558

 =  1116 square feet

Cost of carper per square feet = $3

=  3 x 1116 ==> $ 3348

Amount spent for carpet =    $ 3348.

Problem 4 :

The cost of fencing a circle shaped garden is $20 per foot. If the radius of the garden is 14 feet, find the total cost of fencing the garden. ( π  =  22/7).  

To know the length of fencing required, find the circumference of the circle shaped garden.

Circumference of the circle shaped garden is 

=  2πr

Substitute 22/7 for π and 14 for r. 

=  2(22/7)(14)

=  88 feet

Total cost of fencing is 

=  88(20)

=  $1760

Problem 5 :

Jess is painting a giant arrow on a playground. Find the area of  the giant arrow. If one can of paint covers 100 square feet,  how many cans should Jess buy?

Now we are going to divide this into three shapes. Two triangles and one rectangle.

Area of rectangle = length x width

   =  18 x 10 ==> 180 square feet

Area of one triangle = (1/2) x b x h

   =  (1/2) x 6 x 10 ==> 30 square feet

Area of two triangles = 2 x 30 = 60 square feet

Total area of the given shape = 180 + 60

=   240 square feet

one can of paint covers 100 square feet

Number of cans needed = 240/100 = 2.4 approximately 3.

So, Jessy has to 3 cans of paint.

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Word Problems Involving Perimeter and Area of Polygons (Carpentry Themed) Worksheets

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Perimeter is the measurement of the total length of the sides of a given shape or polygon. When we want to measure the perimeter of a polygon, all we need to do is to add the length of all its sides. 

While area can be defined as the region covered by a flat shape or the surface of an object.  The area of a figure is the number of unit squares that occupied the surface of a closed figure. 

PERIMETER OF POLYGONS

• Square P = 4s
• Rectangle P = 2L + 2W
• Trapezoid P = a + b + c + d
• Parallelogram P = 2a + 2b
• Triangle P = a + b + c

AREA OF POLYGONS

The area of a triangle is equal to the half of the product of its base and height or A = ½ bh, where b = base of the triangle and h = height/altitude of the triangle

This is a fantastic bundle which includes everything you need to know about Word Problems Involving Perimeter and Area of Polygons across 21 in-depth pages.

Each ready to use worksheet collection includes 10 activities and an answer guide. Not teaching common core standards ? Don’t worry! All our worksheets are completely editable so can be tailored for your curriculum and target audience.

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Worksheets Activities Included

Ages 8-9 (Basic)

• Sawmill Errands
• Drill and Cut
• Constructing Layout
• Carpentry Shop Moments
• Enclosing the Vacant Lot

Ages 9-10 (Advanced)

• Paint the Area
• Floor Tile Project
• The Missing Toolkit
• Scale Drawing
• The Carpenter’s Dream

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