# Formula for Area Bounded by a Circle/Proof/Polygon Method

## < Formula for Area Bounded by a Circle/Proof

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## Prerequisites

- For any regular polygon, , where is the area of the polygon, is the length of the apothem, is the number of sides, and is the length of each side.
- Given a circle inscribed in a regular polygon, the radius of that circle is equal to the apothem of the polygon.
- is the ratio of a circle's circumference to its diameter.
- Constant multiple rule of limits

## Proof

Construct a circle of radius . Construct an n-sided polygon such that the circle is inscribed in the polygon. Then the apothem of the polygon is equal to . Let represent the area of the circle and represent the area of the polygon. Let represent the perimeter of the polygon and represent the circumference of the circle. Then:

Further, let increase without bound. Then:

Since is the ratio of a circle's circumference to its diameter:

QED