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Dec 9, 2020

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HomeMathematicsGeometryFlexBooksCK-12 Geometry - Second EditionCh106. Area and Perimeter of Regular Polygons

10.6 Area and Perimeter of Regular Polygons

Difficulty Level: Basic | Created by: CK-12

Last Modified: Dec 25, 2014

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Learning Objectives

Calculate the area and perimeter of a regular polygon.

Review Queue

1. What is a regular polygon?

Find the area of the following regular polygons. For the hexagon and octagon, divide the figures into rectangles and/or triangles.

2. ￼

3. ￼

4. Find the length of the sides in Problems 2 and 3.

Know What? The Pentagon in Arlington, VA houses the Department of Defense, is two regular pentagons with the same center. The entire area of the building is 29 acres (40,000 square feet in an acre), with an additional 5 acre courtyard in the center. The length of each outer wall is 921feet. What is the total distance across the pentagon? Round your answer to the nearest hundredth.

acre courtyard in the center. The length of each outer wall is 921feet. What is the total distance across the pentagon? Round your answer to the nearest hundredth.

Perimeter of a Regular Polygon

Recall that a regular polygon is a polygon with congruent sides and angles. In this section, we are only going to deal with regular polygons because they are the only polygons that have a consistent formula for area and perimeter. First, we will discuss the perimeter.

Recall that the perimeter of a square is 4 times the length of a side because each side is congruent. We can extend this concept to any regular polygon.

Perimeter of a Regular Polygon: If the length of a side is s and there are n sides in a regular polygon, then the perimeter is P=ns.

Example 1: What is the perimeter of a regular octagon with 4 inch sides?

Solution: If each side is 4 inches and there are 8 sides, that means the perimeter is 8(4 in) = 32 inches.

Example 2: The perimeter of a regular heptagon is 35 cm. What is the length of each side?

Solution: If P=ns, then 35 cm=7s. Therefore, s=5 cm.

Area of a Regular Polygon

In order to find the area of a regular polygon, we need to define some new terminology. First, all regular polygons can be inscribed in a circle. So, regular polygons have a center and radius, which are the center and radius of the circumscribed circle. Also like a circle, a regular polygon will have a central angle formed. In a regular polygon, however, the central angle is the angle formed by two radii drawn to consecutive vertices of the polygon. In the picture below, the central angle is ∠BAD. Also, notice that △BAD is an isosceles triangle. Every regular polygon with n sides is formed by n isosceles triangles. In a regular hexagon, the triangles are equilateral. The height of these isosceles triangles is called the apothem.

Apothem: A line segment drawn from the center of a regular polygon to the midpoint of one of its sides.

We could have also said that the apothem is perpendicular to the side it is drawn to. By the Isosceles Triangle Theorem, the apothem is the perpendicular bisector of the side of the regular polygon. The apothem is also the height, or altitude of the isosceles triangles.

Example 3: Find the length of the apothem in the regular octagon. Round your answer to the nearest hundredth.

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