Apothem

Properties of apothems

The apothem ${\displaystyle t}$ can be used to find the area ${\displaystyle S}$ of any regular ${\displaystyle n}$-sided polygon of side length ${\displaystyle a}$ according to the following formula, which also states that the area is equal to the apothem multiplied by half the perimeter since ${\displaystyle na=l}$.

${\displaystyle S=\frac{nat}{2}=\frac{lt}{2}.}$

This formula can be derived by partitioning the ${\displaystyle n}$-sided polygon into ${\displaystyle n}$ congruent isosceles triangles, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height.

The following formulations are all equivalent:

${\displaystyle S=\frac{1}{2}nat=\frac{1}{2}lt=\frac{1}{4}n{a}^2\cot \frac{\pi }{n}=n{t}^2\tan \frac{\pi }{n}}$

An apothem of a regular polygon will always be a radius of the inscribed circle. Therefore, for a regular polygon:

${\displaystyle r=t}$

It is also the minimum distance between any side of the polygon and its center.

This property can also be used to easily derive the formula for the area of a circle, because as the number of sides approaches infinity, the regular polygon's area approaches the area of the inscribed circle of radius ${\displaystyle r=t}$.

${\displaystyle S=\frac{lt}{2}=\frac{(2\pi r)r}{2}=\pi r^2}$