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Areas of regular polygons

There are multiple methods to find the area of regular polygons. We begin with the equilateral triangle.

The area of an equilateral triangle is $\frac{\sqrt{3}}{4}s^2$, where $s$ is the side length of the triangle.

The area of a square is simply $s^2$, where $s$ is the side length of the square.

The area of a regular pentagon is $\frac{1}{4}s^2\sqrt{5(5+2\sqrt{5}}$, where $s$ is the side length of the regular pentagon. This area formula is less commonly known.

The area of a regular hexagon is $\frac{3\sqrt{3}}{2}s^2$, where $s$ is the side length of the regular hexagon. Notice that this is just $6$ times the area of an equilateral triangle.

The formulas go on and on. There is also a more general formula.

Suppose you have an $n$-sided regular polygon. The $apothem$ is defined as the perpendicular distance from the center of the polygon to any side length. The area of this polygon will then be $\frac{1}{2}$x $apothem$ x (perimeter of the polygon).

Finding the Area of a Regular Polygon

Equilateral triangles and squares aren't bad and have straightforward formulas. However, what if we wanted to find the area of a general regular $n$-gon? How would we do this?

We divide the $n$-gon into triangles! First, for simplicity, let's start with octagon $ABCDEFGH$ with side length $s$ and see what we can do.

Let $0$ be the center of the octagon. Then, we can divide the octagon into eight isosceles triangles with $O$ as the vertex:

Now, note that all these triangles are congruent. Hence, the area of the octagon is eight times the area of one of the triangles, or: $$[ABCDEFGH] = 8[ABO]$$ To find $[ABO]$, we drop the altitude $OK$ of triangle $ABO$.

This is known as the apothem of the polygon. Denote the length of this by $a$. Substituting this in, we have: $$[ABCDEFGH] = 8*as/2$$ This is nice, but we want to generalize this to arbitrary $n$-gons using as few variables as possible. Let the perimeter be $p$. Seeing that $p=8s$, we have: $$\boxed{[ABCDEFGH] = \dfrac{ap}2}$$ Using the same method for an arbitrary $n$-gon, we see that this is the formula for all regular polygons.

The area of the regular polygon with perimeter $p$ and apothem $a$ is $\frac{ap}2$