The Apothem and Radius

The apothem and radius are terms defined for regular polygons only, but they carry a great deal of importance.

The apothem of a regular polygon is a segment connecting the center of the polygon to a midpoint of one of the sides, and the radius of a regular polygon is a segment connecting the center of the polygon to one of the vertices.

Now, we will derive some important properties concerning the apothem and radius. Denote the apothem as $a$, the radius as $r$, the side length as $s$, the number of sides as $n$, the center as $O$, and the vertices as $A_1,A_2,...,A_n$.

First, familiarize yourself with the following theorems, which are all fairly trivial:

The apothem is perpendicular to the side it meets.

All apothems are congruent.

All radii are congruent.

Now on to the more interesting stuff!

1. The area of a regular polygon is $\frac{ans}{2}$, or one half the product of the apothem and perimeter.

Calculating the area of $OA_1A_2$, $$[OA_1A_2]=\frac{1}{2}A_1A_2\cdot OM$$ $$[OA_1A_2]=\frac{1}{2}as$$ To find the total area, just multiply by the number of sides! $$[A_1A_2...A_n]=\frac{1}{2}ans$$

1. The area of a regular polygon is $\frac{n}{2}r^2\sin(\frac{2\pi}{n})$.

Calculating the area of $OA_1A_2$, $$[OA_1A_2]=\frac{1}{2}OA_1\cdot OA_2\sin(\frac{2\pi}{n})$$ $$[OA_1A_2]=\frac{1}{2}r^2\sin(\frac{2\pi}{n})$$ Again, to find the total area, just multiply by $n$, the number of sides!

For an interesting observation, those well-versed in basic limits will know the following: $$lim_{n\to 0}{\frac{\sin n}{n}}=1$$ Using this limit here, one may see that as $n$ goes to $\infty$, the area approaches $\pi r^2$, which is exactly what we would expect, since the polygon approaches a circle.

1. The length of the apothem is $a=\frac{s}{2\tan(\pi/n)}$, while the length of the radius is $r=\frac{s}{2\sin(\pi/n)}$.

Draw the triangle $OA_1M$. This is a right triangle, so one may use the basic definitions of the trigonometric ratios to quickly derive the equations above.

Note that $a=r\cos(\frac{\pi}{n})$, which means that $a<r$, but that $a$ and $r$ get closer as $n$ increases.

While we are on the topic of lengths, note that by the Pythagorean Theorem: $$\left(\frac{s}{2}\right)^2+a^2=r^2$$

As a last note, we may use the apothem or radius to find the area purely in terms of the side length, $s$, and the number of sides, $n$: $$[A_1A_2...A_n]=\frac{1}{2}ans$$ $$[A_1A_2...A_n]=\frac{1}{2}\frac{s}{2\tan(\pi/n)}ns$$ $$[A_1A_2...A_n]=\frac{ns^2}{4\tan(\pi/n)}$$

This is more than enough knowledge concerning the apothem and perimeter. Good luck!