How do we find the apothem of a regular polygon?

Dec 12, 2016

Apothem of a regular polygon with $n$ sides and one side as $a$ is $\frac{a}{2} \cot \left(\frac{\pi}{n}\right)$.

Explanation:

Apothem is the line joining the center of a regular polygon to the middle point any of its side. It is also the radius of incircle of the regular polygon.

Assume there are $n$ sides of the polygon and each side is $a$. Joining center of the polygon to two ends of same side will form an isosceles triangle whose angle at vertex will be $\frac{2 \pi}{n}$ and drawing the perpendicular from vertex to side will form a right angle side (as shown below), with altitude forming apothem and angle at vertex being $\frac{\pi}{n}$

and if apothem is $x$, we have

$\frac{x}{\frac{a}{2}} = \cot \left(\frac{\pi}{n}\right)$

and hence apothem is $\frac{a}{2} \cot \left(\frac{\pi}{n}\right)$

Hence, apothem of a regular polygon with $n$ sides and one side as $a$ is $\frac{a}{2} \cot \left(\frac{\pi}{n}\right)$.