Apothem

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The apothem of a regular polygon is the line segment drawn from the center of the polygon perpendicular to one of its sides. It is also the radius of the inscribed circle.

Formulas

Given the amount of sides, $n$ , and side length, $s$ , the apothem is $\frac{s}{2\tan\left(\frac{\pi}{n}\right)}$ .

Given the amount of sides, $n$ , and the radius of the circumscribed circle, $R$ , the apothem is $R\cos\left(\frac{\pi}{n}\right)$ .

Given the apothem, $a$ , and perimeter, $p$ , the area of the polygon is $\frac{ap}{2}$ , or $as$ , where $s$ is the semiperimeter.