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A regular polygon, remember, is a polygon whose sides and interior angles are all congruent. To understand the formula for the area of such a polygon, some new vocabulary is necessary.

The center of a regular polygon is the point from which all the vertices are equidistant. The radius of a regular polygon is a segment with one endpoint at the center and the other endpoint at one of the vertices. Thus, there are n radii in an n-sided regular polygon. The center and radius of a regular polygon are the same as the center and radius of a circle circumscribed about that regular polygon.

An apothem of a regular polygon is a segment with one endpoint at the center and the other endpoint at the midpoint of one of the sides. The apothem of a regular polygon is the perpendicular bisector of whichever side on which it has its endpoint. A central angle of a regular polygon is an angle whose vertex is the center and whose rays, or sides, contain the endpoints of a side of the regular polygon. Thus, an n-sided regular polygon has n apothems and n central angles, each of whose measure is 360/n degrees. Every apothem is the angle bisector of the central angle that contains the side to which the apothem extends. Below are pictured these characteristics of a regular polygon.

Figure %: A regular polygon with a center (C), radius (r), apothem (a), and central angle

Once you have mastered these new definitions, the formula for the area of a regular polygon is an easy one. The area of a regular polygon is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter.

When an n-sided polygon is split up into n triangles, its area is equal to the sum of the areas of the triangles. Can you see how 1/2(ap) is equal to the sum of the areas of the triangles that make up a regular polygon? The apothem is equal to the altitude, and the perimeter is equal to the sum of the bases. So 1/2(ap) is only a slightly simpler way to express the sum of the areas of the n triangles that make up an n-sided regular polygon.

Figure %: Two n-sided polygons divided into n triangles
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