Question

How do we find the apothem of a regular polygon?

Answer 1

Apothem of a regular polygon with `n` sides and one side as `a` is `a/2cot(pi/n)`.

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 `(2pi)/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 `pi/n`

and if apothem is `x`, we have

`x/(a/2)=cot(pi/n)`

and hence apothem is `a/2cot(pi/n)`

Hence, apothem of a regular polygon with `n` sides and one side as `a` is `a/2cot(pi/n)`.