geometry

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.

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  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

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