10.12. Area of Regular Polygons http://www.ck12.org
10.12 Area of Regular Polygons
Here you’ll learn how to calculate the area and perimeter of a regular polygon.
What if you were asked to find the distance across The Pentagon in Arlington, VA? The Pentagon, which also
houses the Department of Defense, is composed of 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 921 feet. What is the total distance across the pentagon? Round your answer to the nearest
hundredth. After completing this Concept, you’ll be able to answer questions like this.
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Guidance
A regular polygon is a polygon with congruent sides and angles. 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 issand there arensides in a regular polygon, then the
perimeter isP=ns.
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 acenterandradius, 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^6 BAD. Also, notice that 4 BADis an isosceles triangle. Every regular polygon
withnsides is formed bynisosceles triangles. The height of these isosceles triangles is called theapothem.
The area of each triangle isA 4 =^12 bh=^12 sa, wheresis the length of a side andais the apothem. If there arensides
in the regular polygon, then it is made up ofncongruent triangles.
Area of a Regular Polygon: If there arensides with lengthsin a regular polygon andais the apothem, then
A=^12 asnorA=^12 aP, wherePis the perimeter.