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Thread: geometry help

  1. #1
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    geometry help

    An equilateral triangle is inscribed in a regular hexagon . This triangle's vertices touch the midpoints of three sides of a regular hexagon . The area of the triangle is 12cm^2 . What is the area of the regular hexagon in cm^2 ?
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  2. #2
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    Hello, mathaddict!

    I made a sketch . . . and the answer jumped out at me!


    An equilateral triangle is inscribed in a regular hexagon.
    This triangle's vertices touch the midpoints of three sides of the regular hexagon.
    The area of the triangle is 12 cm².
    What is the area of the regular hexagon in cm² ?

    The diagram looks like this:
    Code:
                    *---*---*
                   /   / \   \
                  /   /   \   \
                 /   /     \   \
                *   /       \   *
                 \ /         \ /
                  *-----------*
                   \         /
                    *-------*

    Divide the hexagon into "unit triangles",
    . . and we have this diagram:
    Code:
                    *---*---*
                   / \ /x\ / \
                  *---*---*---*
                 / \ /x\x/x\ / \
                *---*---*---*---*
                 \ /x\x/x\x/x\ /
                  *---*---*---*
                   \ / \ / \ /
                    *---*---*

    We see that the triangle occupies \frac{9}{24} = \frac{3}{8} of the hexagon.

    Therefore, the hexagon has: . \frac{8}{3}\times12 \:=\:32\text{ cm}^2

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