Before starting this lesson, you should already be familiar with the following terms and concepts.
A regualr polygon is a closed figure in the plane, each of the sides of which has the same length.
Examples:
Specifically, the definition of a tangent.
The sine, cosine, and tangent of the angle x are given in the above picture. Each is a ratio of side lengths. Side o is opposite x, side a is adjacent to x, and side h is the hypotenuse of the triangle.
For background on basic trigonometry, see this lesson.
The apothem of a regular polygon is the perpendicular line from one of the sides of the polygon to its center.
The formula for the area of a regular polygon is
where p is the perimeter of the polygon and a is the apothem length.
And now we derive it!
In this video the formula for the area of a regular polygon is derived using a triangle decomposition technique.
Source: Colin O'Keefe
Given the number of sides and the side length value, this video shows how to derive the length of a regular polygon's apothem using some basic trigonometry.
Source: Colin O'Keefe
