## 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 \[\frac{n}{2} r^2 \times \frac{2 \pi}{n}= \pi r^2\]

.