Assume that `r` is the radius of a circle inscribed in a regular hexagon. Find an expression that yields the area between the circle and the hexagon.

The expression will be of the form Area(hexagon)-Area(circle). Since the area of the circle is easy to describe, we find the area of the circumscribed hexagon:

First note that sides of the hexagon are tangent to the circle. Thus a radius drawn to one of these points of tangency is the apothem of the hexagon. The area of a regular polygon can be found by `A=1/2ap` where `a` is the length of the apothem, and `p` is the perimeter.

Consider the triangle formed by two consecutive radii of the hexagon and a side of the hexagon. The apothem is the height of this triangle and the side of the hexagon is the base of the triangle. Since this is a regular hexagon, the base angles are `60^@` -- this gives us a side length of `(2rsqrt(3))/3` .

** The triangle formed by 1/2 of the side, the apothem, and the radius of the hexagon is a 30-60-90 right triangle. Since the side opposite the `60^@` angle is `r` , the side opposite the `30^@` angle is `(rsqrt(3))/3` and the hypotenuse, which equals the length of a side, is `(2rsqrt(3))/3` **

Thus we have `a=r, p=6*(2rsqrt(3))/3=4rsqrt(3)` . Then the area of the hexagon is `1/2*r*4rsqrt(3)=2r^2sqrt(3)` .

The area of the circle is `pir^2`

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**The area between the inscribed circle and the hexagon is `2r^2sqrt(3)-pir^2=r^2(2sqrt(3)-pi)` **

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**Further Reading**