if a regular 9-sided polygon was circumscribed around a circle, what is the difference between the areas of the polygons
So, imagine the polygon made up of nine isosceles triangles, with a common vertex at the centre of the circle. Then cut one of these triangles in half by a line from the centre to the point where the side of the polygon touches the circle. This line is units long, and the angle at the centre of the circle in this triangle is . From this, you should be able to see that the lines joining the centre of the circle to the vertices of the polygon have length .
Using the formula for the area of a triangle , this gives the area of one of the nine original triangles as .
Multiply this by , and subtract , and you're done. OK?