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