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

< Formula for Area Bounded by a Circle/Proof

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A=\pi r^2

Prerequisites

  • For any regular polygon, A=\frac{lns}{2} , where A is the area of the polygon, l is the length of the apothem, n is the number of sides, and s 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.
  • \pi is the ratio of a circle's circumference to its diameter.
  • Constant multiple rule of limits

Proof

Two circ polygon

A circle of radius r inscribed in a pentagon and a dodecagon (left and right, respectively)

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

A_p=\frac{rns}{2}\approx A_c

P=ns\approx C

Further, let n increase without bound. Then:

\lim_{n\to\infty}\frac{rns}{2}=A_c\ ,\ \lim_{n\to\infty}ns=C

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

\pi=\frac{C}{2r}=\frac{\lim\limits_{n\to\infty}ns}{2r}=\lim_{n\to\infty}\frac{ns}{2r}

A_c=\lim_{n\to\infty}\frac{rns}{2}=\lim_{n\to\infty}\frac{r^2ns}{2r}=r^2\lim_{n\to\infty}\frac{ns}{2r}=\pi r^2

QED

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