Geometry question

Question

A regular hexagon with side lengths of 18 is inscribed in a circle. Everything outside the hexagon is the shaded region. How to find area of shaded region?

Answer 1

If we want to solve this problem without taking any shortcuts. note that the area of the hexagon can be considered as the area of 6 triangles. As there are 360 degrees in a circle, the central angle for each triangle is 360/6 = 60 degrees. Thus, each of these triangles will be an equilateral triangle, as the other two angles are obviously equal, and (180 - 60)/2 = 60.

Than, for an equilateral triangle with lengths of 18, we can split the triangle at the middle of a side and get two 30-60-90 triangles with hypotenuse of the side length. Then, the other 2 sides are 9 and 9√3, and the area of each of the 2 triangles is 1/2 * 9 * 9√3, so the combined area is 9 * 9√3. Now, you could have just looked up that it is s² √3/4

Then, the area of 6 triangles is 6 * 9 * 9√3 = 486√3

Note that the radius of the circle is also 18 (recall the equilateral triangles; the other two sides are radii).

Then, the area of the circle is πr² = π*18² = 324π

Thus, the area outside of the hexagons is 324π - 486√3 = 176.10

As a sanity check, calculate the percent of the area of the circle that is in the hexagon. It will be 486√3/324π = .827