TITLE: Area of a circle inscribed in a polygon
QUESTION [10 upvotes]: If a circle is inscribed in a polygon, show that,   
$$\dfrac{\text{(Area of inscribed circle)}}{\text{(Perimeter of inscribed circle)}} = \dfrac{\text{(Area of Polygon)}}{\text{(Perimeter of Polygon)}}$$

REPLY [2 votes]: Divide the polygon into triangles by drawing a line segment from each vertex to the center of the circle.  Each triangle includes one side of the polygon and a sector of the inscribed circle. Letting $r$ = the radius of the circle:
Area of sector = $(r/2)$×(Arc length of sector)
Area of triangle = $(r/2)$×(Length of included polygonal side)
Add up all the triangle and sector areas as above, and find that the area/perimeter ratio for both the polygon and the circle are $r/2$.