Regular Polygons & Apothem

Interior angle (n-2)180/n, exterior 360/n, area 12 a P.

The idea

A polygon is regular when all of its sides are equal and all of its angles are equal: the equilateral triangle, the square, the regular pentagon, and so on. For a regular polygon the side count $n$ determines every angle.

Join one vertex to every other vertex and an $n$-gon splits into $n - 2$ triangles, so its interior angles total $(n-2)\,180$ degrees. A regular polygon divides that total equally among its $n$ vertices, so each interior angle is $\dfrac{(n-2)\,180}{n}$. The turns made in one full trip around the boundary total one revolution, so each exterior angle is $\dfrac{360}{n}$; at each vertex the interior and exterior angles lie along a straight line and are supplementary.

The same splitting gives the area. The apothem $a$ is the distance from the center to the midpoint of a side. Join the center to every vertex, and the polygon splits into $n$ triangles, each with one side, of length $s$, as its base and the apothem as its height. The bases total the perimeter $P = ns$, so the areas add to $A = \tfrac12\,a\,P.$

Ways to work on it

• Walkthrough. Hexagon angles and an apothem-area computation.
• Practice. An interior or exterior angle for a random regular polygon.
• Hardest. Recover the number of sides from an exterior angle, or the area from apothem and side.

Not sure where to start? Take the ten-question placement test.