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Math Help - Area

  1. #1
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    Area

    What might be the dimensions 0f a pentagonal prism if its surface area is 180cm
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  2. #2
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    The surface area of a regular pentagonal prism is given by

    TSA = 2A_{\textrm{pentagon}} + 5lh, where l is one of the side lengths and h is the height.


    The area of a regular polygon is given by

    A = \frac{l^2n}{4\tan{\frac{\pi}{n}}}, where n is the number of sides, so the pentagon's area is


    \frac{5l^2}{4\tan{\frac{\pi}{5}}}

     = \frac{5l^2}{4\sqrt{5-2\sqrt{5}}}.


    Therefore

    TSA = 2\left(\frac{5l^2}{4\sqrt{5-2\sqrt{5}}}\right) + 5lh

     = \frac{5l^2}{2\sqrt{5 - 2\sqrt{5}}} + 5lh.


    Choose some values of l and h which will make this value 180\,\textrm{cm}^2.
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