What Is a Regular Polygon?

A regular polygon is a polygon that has all equal angles and all equal sides. They have many useful properties.

A regular hexagon is shown below:

Notice that the center, O, divides the hexagon into six regions that are congruent to triangle OAB, if we were to draw OA, OB, OC, ... OF.

We proved that there were $180(n-2)$ degrees in the interior angles of a polygon, so a hexagon would have $180(6-2)=720$ degrees in its interior angles. Since all angles are equal, each angle of a hexagon has $\frac{1}{6} \cdot 720 = 120$ degrees.

In general, the degree measure of a regular n-gon is $\frac{180(n-2)}{n}$

Let's go back to our hexagon example:

Notice that $OG$ is perpendicular to $AB$. ($G$ is the midpoint of $AB$). This segment $OG$ is called the apothem of the hexagon.

The most useful property of the apothem is that in a regular n-gon: $area = apothem * semiperimeter$.

We can derive this formula by splitting the n-gon into $n$ triangles. Each one has area $0.5 * base * height$ (where the height is the apothem, and the length of one side is the base). We take $n$ of these for the full area, or $n * 0.5 * base * height$ which is equivalent to $apothem * semiperimeter$ (where semiperimeter equals $\frac{1}{2}$ of the perimeter).

More terms: the apothem is also the inradius of the n-gon, and $OA$ is the circumradius of the hexagon.

Explanation:

We know that regular polygons have equal interior angles, and we know that each interior angle in a regular hexagon is 120 degrees. The center O divides the hexagon into six congruent triangles, which are necessarily equilateral: we know that angle BOA is 60 degrees (360 / 6), and we know that angle ABO is 60 degrees (120 / 2). Triangle AOB is therefore an equilateral triangle with side = 1.

Remembering that the area of a regular n-gon is apothem * semiperimeter, we now know that the semiperimeter is 3 and the apothem is $\frac{\sqrt{3}}{2}$. The area therefore equals $3 \cdot \frac{\sqrt{3}}{2}$