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To derive a formula for the area of a regular polygon in terms of its apothem length and perimeter. Show the technique of decomposing a regular polygon into triangles in order to determine its area.

First background is presented along with some review material, before the formula is introduced. The formula for the area of a regular polygon is then derived in a video. Finally, another video presents a method for determining the apothem length given only the length of a side.

Tutorial

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