10.4 Polygons, Perimeter, and Circumference

Learning Objectives

After completing this section, you should be able to:

1. Identify polygons by their sides.
2. Identify polygons by their characteristics.
3. Calculate the perimeter of a polygon.
4. Calculate the sum of the measures of a polygon’s interior angles.
5. Calculate the sum of the measures of a polygon’s exterior angles.
6. Calculate the circumference of a circle.
7. Solve application problems involving perimeter and circumference.

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 ${180}^{\circ }$. 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.

Perimeter

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:

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,

$$\begin{array}{ccc}\hfill 24& \hfill =\hfill & 2l+2(4)\hfill \\ \hfill & \hfill =\hfill & 2l+8\hfill \\ \hfill 24-8& \hfill =\hfill & 2l\hfill \\ \hfill 16& \hfill =\hfill & 2l\hfill \\ \hfill 8& \hfill =\hfill & l\hfill \end{array}$$

The length is 8 units.

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

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.

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

$$\begin{array}{ccc}\hfill S& \hfill =\hfill & (4-2){180}^{\circ }\hfill \\ \hfill & \hfill =\hfill & 2(180)={360}^{\circ }.\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill S& \hfill =\hfill & \left(7-2\right){180}^{\circ }\hfill \\ \hfill & \hfill =\hfill & 5(180)\hfill \\ \hfill & \hfill =\hfill & {900}^{\circ }.\hfill \end{array}$$

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

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

$$a=\frac{(7-2){180}^{\circ }}{7}={128.57}^{\circ }\text{.}$$

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 $n$ sides is found using the following formula:

In Figure 10.71, we have a regular hexagon $\mathit{ABCDEF}$. 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, $b=\frac{{360}^{\circ }}{6}={60}^{\circ }$.

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

Circles and Circumference

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

The radius is ½ of the diameter of a circle. The symbol $\pi =3.141592654\dots$ 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

$$\begin{array}{ccc}\hfill C& \hfill =\hfill & 2\pi (3.5)\hfill \\ \hfill & \hfill =\hfill & 21.99\text{in}.\hfill \end{array}$$

Key Terms

• perimeter
• polygon
• pentagon
• hexagon
• heptagon
• octagon
• quadrilateral
• trapezoid
• parallelogram
• circumference

Key Concepts

• Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides.
• The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by calculating the sum of the lengths of its sides.
• The sum of the interior angles of a regular polygon with $n$ sides is found using the formula $S=\left(n-2\right){180}^{\circ }.$ The measure of a single interior angle of a regular polygon with $n$ sides is determined using the formula $a=\frac{\left(n-2\right){180}^{\circ }}{n}.$
• The sum of the exterior angles of a regular polygon is ${360}^{\circ }.$ The measure of a single exterior angle of a regular polygon with $n$ sides is found using the formula $b=\frac{{360}^{\circ }}{n}.$
• The circumference of a circle is $C=2\pi r,$ where $r$ is the radius, or $C=\pi d,$ and $d$ is the diameter.

Formulas

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

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

$$S=(n-2){180}^{\circ }.$$

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

$$a=\frac{(n-2){180}^{\circ }}{n}\text{.}$$

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

$$b=\frac{{360}^{\circ }}{n}\text{.}$$

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