Chapter 11 Class Notes

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Areas of Regular Polygons - Lesson 11-5

Today we started by deriving a formula for the area of an equilateral (regular) triangle. If you start with an equilateral triangle with side length s:

Derivation of the Area of an Equilateral Triangle - Step 1

And then do the standard procedure for finding the length of the altitude (love those 30-60-90 triangles!),

Derivation of the Area of an Equilateral Triangle - Step 2

you are then able to use the formula for the area of a triangle to derive a new formula

Derivation of the Area of an Equilateral Triangle - Step 3

which can be summarized as follows:

Theorem 104 - the Area of an Equilateral Triangle

Next, we can generalize an area formula for all regular polygons. To start, we need to define the radius and the apothem of a regular polygon:

Definition of an Apothem and Radius of a Regular Polygon

If we look at a pentagon, you should be able to see how the formula for its area would be as shown in the table below:

Area of a Regular Pentagon

Table Step 1

The same can be said for an octagon

Regular Octagon Area

Table Step 2

and a dodecagon.

Area of a Regular Dodecagon

Table Step 3

In conclusion, we can derive the general formula to find the area of any given regular polygon:

Area of a Regular Polygon