Question

a regular hexagon is inscribed in a circle. The radius of the circle is 18 units. What is the area of the region bounded by the inside of the circle and the outside of the hexagon. Round your answer to the nearest hundredth.

Answer 1

draw the diagonals of the hexagon and you will have 6 equilateral triangles.
let's look at one of these.
the height can found by Pythagoras
h^2 + 9^2 = 18^2
h^2 = √243 = 9√3
so the area of one of them = (1/2)(18)(9√3) = 81√3
there are 6 of them ,so total area of hexagon = 486√3
Area of circle = π(18)^2 = 324π

area of difference = 324π - 81√3 or appr. 176.1