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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