Areas and Perimeters of Regular Polygons

Introduction: Area is a measure of the amount of space contained inside a closed figure. Perimeter is a measure of the distance around a closed figure. We examine these concepts for regular polygons.

The Lesson:

The area and perimeter of a regular polygon can involve relatively simple figures such as an equilateral triangle or a square. A diagram below illustrates these concepts. To find the perimeter, we add the lengths of the sides. To find the area, we use for the triangle and s² for the square. The perimeter of the triangle is 3s and of the square 4s.

Because the triangle is equilateral, each angle is 60º. This allows us to calculate h in terms of s. The height h divides the equilateral triangle into two congruent 30-60-90 triangles, each with a hypotenuse of s and legs of h and . We have h . This gives us the area of the equilateral triangle as . If this calculation was unfamiliar, you may want to reference the lesson on the trigonometry of special triangles.

These cases where the number of sides of the regular polygon is 3 or 4 are easy to calculate. In fact, the area of a regular hexagon, in which the number of sides n = 6, is easy to calculate since a hexagon can be decomposed into 6 equilateral triangles. The area is .

To derive a formula for the area of a regular polygon if the number of sides is n requires applying some more trigonometry. We examine a diagram of a (partial) regular polygon of side s and number of sides n. A diagram is shown below. Assume that point O is the center of the regular polygon and r, the distance from the center to a vertex, is called the radius of the polygon. The perimeter is clearly ns. We derive a formula for the area in terms of the radius r.

• Triangle AOB is a central triangle. There are n such triangles in this polygon, one for each side.
• Angle AOB has a measure of , and the indicated angle MOB has a measure of .
• Since AB is a side of measure s, . Using basic right triangle trigonometry, we see that and .
• This gives us the area of triangle AOB = .

Summary: Since there are n such triangles in this regular polygon, we have a formula:

The area of a regular polygon of n sides and radius r is .

Let's Practice:

1. A regular hexagon has a side of 8 feet. What is its area?

We use

Area = .

2. A regular hexagon has a radius of 5 meters. What is the area? What is the perimeter?

We use

Area = .

This gives us

m².

A hexagon is a special case in which each central triangle is equilateral. This tells us that r = s, the length of a side of the hexagon.

The perimeter is 6s = 30 meters.

Because we can easily find that s = 5, we could also have used

Area = .

3. A regular octagon has a radius of 6. What is the area? What is the perimeter?

An octagon has 8 sides so we use

Area = .

To find the perimeter, we need s.

We know from the central triangle that

.

This gives us s = 4.592 and

perimeter = 8s = 36.736.