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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 = .
1. 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 = .
1. 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.

Examples A hexagon has a side of length 10 cm. What is the area and what is the perimeter? What is your answer?  A STOP sign has a side of 1 foot in length. What is the perimeter and what is the area? What is your answer? M Ransom

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