One-Half the Apothem Times the Perimeter

An apothem of a regular polygon is the length of the segment from the center to the midpoint of a side. The area of a regular polygon can be calculated as one-half the apothem multiplied by the perimeter. This Demonstration explains why that formula works.


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Snapshot 1: each triangle is identical, bounded on the sides by the green circumradii
Snapshot 2: the area is equal to 1/2 the area multiplied by the perimeter
Snapshot 3: for as few or as many sides as an -gon has, this formula works
Imagine a regular -sided polygon divided into a set of identical isosceles triangles. One such triangle is the sector shown in light blue in the Demonstration. Each triangle has a base of side length and a height equal to the apothem. Thus the area of each triangle is . If we add up all the triangle areas, the area of the entire polygon can be written as , where is the perimeter. As , the polygon approaches a circle. An apothem equals the radius , while the perimeter equals , giving the well-known result for the area of a circle: .
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