Answer:
(54√3 - 27π) in²
Explanation:
The radius of the inscribed circle is the apothem of the hexagon, so is ...
(6 in)·sin(60°) = 3√3 in
The area of the hexagon is half the product of this and the perimeter of the hexagon (6 times the side length).
A = (1/2)(6·6 in)(3√3 in) = 54√3 in²
The area of the circle is ...
A = π·r² = π(3√3 in)² = 27π in²
Then the region between the hexagon and its inscribed circle will be ...
hexagon area - circle area =
(54√3 - 27π) in²