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area of regular polygon
Theorem Given a regular $n$ -gon with apothem of length $a$ and perimeter $P$ , its area is
$$A=\frac{1}{2}aP.$$
Proof. Given a regular $n$ -gon $R$ , line segments can be drawn from its center to each of its vertices. This divides $R$ into $n$ congruent triangles. The area of each of these triangles is $\displaystyle \frac{1}{2}as$ , where $s$ is the length of one of the sides of the triangle. Also note that the perimeter of $R$ is $P=ns$ . Thus, the area $A$ of $R$ is
To illustrate what is going on in the proof, a regular hexagon appears below with each line segment from its center to one of its vertices drawn in red and one of its apothems drawn in blue.
(0,-1.732) \psline[lin... ...2) \pspolygon(2,0)(1,1.732)(-1,1.732)(-2,0)(-1,-1.732)(1,-1.732) \end{pspicture}](http://images.planetmath.org/cache/objects/9501/js/img3.png)
area of regular polygon is owned by Warren Buck.
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