Answer has to be in exact form. Find the area of the region between a regular hexagon with sides of 6" and its inscribed circle.

Answer 1

You can split your hexagon up into 6 equal equilateral triangles with sides of 6
Drawing the height in one of them, let the height be h
h^2 + 3^2 = 6^2
h^2 = 36-9
h = √27 or 3√3
area of one is (1/2)(6)(3√3) = 9√3
so the area of the hexagon is 54√3

the inscribed circle has radius 3√3 or √27
area of that circle = π(√27)^2 = 27π

So area between circle and hexagon
= 54√3 - 27π