## Area of a Regular Polygon in a Circle of Radius r

Suppose a regular polygon with
$n$
sides is inscribed in a circle of radius
$r$
, with the vertices of the polygon touching the circle.
From the centre of the circle lines are drawn to each vertex, and
$n$
triangles are formed. The angle of each triangle at the centre of the circle is
$\frac{360}{n}$
, and the area of each triangle is
$\frac{1}{2}r^2 sin (\frac{360}{n})$
.

There are
$n$
such triangles, so the area of the polygon is
$\frac{n}{2}r^2 sin (\frac{360}{n})$
.
As the number of sides gets larger
$sin (\frac{360}{n})$
tends to
$\frac{2 \pi}{n}$
, so the area of the polygon tends to
$\frac{n}{2} r^2 \times \frac{2 \pi}{n}= \pi r^2$
.