Area of Regular Polygons

Area of Regular Polygons

Author: c o

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.

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Background and Review

Before starting this lesson, you should already be familiar with the following terms and concepts.

Regular Polygons  

A regualr polygon is a closed figure in the plane, each of the sides of which has the same length.  


The Area of a Triangle


Basic Trigonometry

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 and The Area Formula

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!

Deriving the Area Formula

In this video the formula for the area of a regular polygon is derived using a triangle decomposition technique.

Source: Colin O'Keefe

Finding The Apothem Of A Regular Polygon

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