6.7: Area of Regular Polygons
Learning Objectives
- Recognize and use the terms involved in developing formulas for regular polygons.
- Calculate the area and perimeter of a regular polygon.
You already know how to find areas and perimeters of some figures – triangles, parallelograms, and other quadrilaterals. Not surprisingly, the new formulas in this lesson will build on those basic figures – in particular, the triangle.
Parts and Terms for Regular Polygons
Do you remember the names of different polygons from Chapter 1?
First of all, “poly-” means “many” and “-gon” refers to the sides of a shape.
A regular polygon is a shape whose many sides are all congruent.
Regular polygons also have congruent interior angles and congruent central angles (which you will learn about on the next page.)
Polygons are classified by how many sides they have.
Here are a few names to review:
- A pentagon has 5 sides.
- A hexagon has 6 sides.
- A heptagon has 7 sides.
- An octagon has ______ sides. (Hint: how many legs does an octopus have?)
- A nonagon has 9 sides.
- A decagon has ______ sides. (Hint: how many years are in a decade?)
Let’s start with some background on regular polygons.
Here is a general regular polygon with \begin{align*}n\end{align*} sides, where \begin{align*}n\end{align*} stands for some number. Some of its sides are shown in the diagram:
In the diagram, here is what each variable represents:
- \begin{align*}s\end{align*} is the length of each side of the polygon.
- \begin{align*}r\end{align*} is the length of a “radius” of the polygon, which is a segment from the center of the polygon to a vertex (or corner).
- \begin{align*}x\end{align*} is the length of one-half of a side of the polygon (so \begin{align*}x = \frac{1}{2} \ s\end{align*} or \begin{align*}2x = s\end{align*}).
- \begin{align*}a\end{align*} is the length of a segment called the apothem — a segment from the center to a side of the polygon, perpendicular to the side. (Notice that \begin{align*}a\end{align*} is the altitude of each of the triangles formed by two radii and a side.)
Think about it:
A triangle would have \begin{align*}n = \underline{\;\;\;\;\;\;}\end{align*} sides.
A square would have \begin{align*}n = \underline{\;\;\;\;\;\;}\end{align*} sides.
An octagon would have \begin{align*}n = \underline{\;\;\;\;\;\;}\end{align*} sides.
The angle between two consecutive radii measures \begin{align*}\frac{360^\circ}{n}\end{align*} because \begin{align*}n\end{align*} congruent central angles are formed by the radii from the center to each of the \begin{align*}n\end{align*} vertices of the polygon.
We can figure this out because an entire circle is \begin{align*}360^\circ\end{align*}, and you can think of the center of the polygon as having a circle of angles around it. If there are \begin{align*}n\end{align*} central angles (all equivalent), each central angle between each radius is \begin{align*}\frac{360^\circ}{n}\end{align*}.
An apothem divides each of these central angles into two congruent halves; each of these half angles measures \begin{align*}\frac{1}{2} \cdot \frac{360^\circ}{n} = \frac{360^\circ}{2n} = \frac{180^\circ}{n}\end{align*}.
Perimeter of a Regular Polygon
We continue with the regular polygon diagrammed on the previous page. Let \begin{align*}P\end{align*} be the perimeter. Remember that _____ is the number of sides in the polygon and _____ is the length of each side. In simplest terms,
\begin{align*}P = ns\end{align*}
We know this because the perimeter of a shape is the sum of ______________________. Another way to express perimeter is the number of sides times the length of each side.
Example 1
A square has a radius of 6 inches. What is the perimeter of the square?
Notice that a side and two radii make an isosceles right triangle:
- The triangle is isosceles because the legs of the triangle are each a radius of the square. Each radius is _____ inches long and both are the same length, so the triangle is isosceles because its legs are congruent.
- The triangle has a right angle because the central angle is \begin{align*}\frac{360^\circ}{n}\end{align*} and the square has 4 sides (which means \begin{align*}n = 4\end{align*}) so each central angle is \begin{align*}\frac{360^\circ}{4} = 90^\circ\end{align*}.
Not only is this an isosceles right triangle, but it is also a 45–45–90 triangle!
You may remember that if the legs are each _____ inches long, then the hypotenuse of the triangle is \begin{align*}6 \sqrt{2}\end{align*} inches long. Notice that the hypotenuse is also a side of the square.
To find the perimeter, use the formula \begin{align*}P = ns\end{align*}. We know \begin{align*}n = 4\end{align*} and \begin{align*}s = 6 \sqrt{2}\end{align*}.
\begin{align*}P & = ns\\ & = 4 \cdot 6 \sqrt{2} = 24 \sqrt{2} \ \text{inches \ (on a calculator, this length} \approx 33.9 \ \text{inches})\end{align*}
The perimeter of the square is \begin{align*}24 \sqrt{2}\end{align*} inches.
Area of a Regular Polygon
The next logical step is to complete our study of regular polygons by developing area formulas.
Take another look at the regular polygon figure below (it is the same one you saw earlier in this lesson.) Here’s how we can find its area, \begin{align*}A\end{align*}.
Two radii and a side make a triangle with base \begin{align*}s\end{align*} and altitude \begin{align*}a\end{align*}:
There are \begin{align*}n\end{align*} of these triangles in the polygon.
The area of each triangle is: \begin{align*}\frac{1}{2} base \cdot height = \frac{1}{2} sa\end{align*}
The entire area of the polygon is:
\begin{align*}A &= \text{number of triangles} \cdot \text{area of each triangle}\\ A &= n \left( \frac{1}{2} sa \right ) = \frac{1}{2} (ns)a = \frac{1}{2}(Pa) \ \text{because perimeter} \ P = ns\end{align*}
Therefore, the Area of a regular polygon with perimeter \begin{align*}P\end{align*} and apothem \begin{align*}a\end{align*}:
\begin{align*}A = \frac{1}{2} \ Pa\end{align*}
Reading Check
1. How many sides does a pentagon have?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. True or false: All sides of a regular polygon are the same length.
3. If you know the length of one side of a regular pentagon, can you find its perimeter? How? Explain the steps you would use.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
4. True or false: A regular polygon that has \begin{align*}n\end{align*} sides also has \begin{align*}n\end{align*} vertices.
5. In the figure below (you have seen it a few times already!),
a. What does \begin{align*}a\end{align*} stand for? ___________________________
Describe what this is:
b. What does \begin{align*}r\end{align*} stand for? ___________________________
c. What does \begin{align*}s\end{align*} stand for? ___________________________
Graphic Organizer for Lessons 2 – 6: Area
Shape | Draw a Picture | Area Formula | What does each letter in the Area Formula stand for? |
---|---|---|---|
Parallelogram | |||
Triangle | |||
Trapezoid | |||
Rhombus | |||
Kite | |||
Regular | |||
Polygon |
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