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Math Help - [SOLVED] Hexnut area

  1. #1
    Forum Admin topsquark's Avatar
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    [SOLVED] Hexnut area

    I need to double check something here:

    Find the shaded area of the regular hexagonal donut. The apothem and sides of the smaller hexagon are 1/2 as long as the apothem and sides of the larger hexagon, respectively. a = 6.9 cm and r = 8 cm.

    The problem is that I get an area of 503 cm^2 and the supposed correct answer is 497 cm^2. Now, this is small enough to perhaps be a rounding problem, but I can't construct the 497 cm^2 solution. Any takers?

    -Dan
    Attached Thumbnails Attached Thumbnails [SOLVED] Hexnut area-hexnut.jpg  
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  2. #2
    Eater of Worlds
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    Apparently we're making the same mistake, TQ. I get 502.82.
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  3. #3
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    I am getting about 497.

    Remember that  A = \frac{pa}{2} . So we have a 30-60-90 right triangle, and the length of the side of the bigger hexagon is  15.934 . Then  A  = 659.6676 . For the smaller one, the length of a side is  7.967 . Then  \frac{pa}{2} = 164.9169

    Thus  \text{desired area} = 659.6676 - 164.9169 \approx 495 ( I rounded a little bit, if I hadnt it would be 497). You dont need r. Extra information.
    Last edited by heathrowjohnny; February 27th 2008 at 06:39 PM.
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  4. #4
    Forum Admin topsquark's Avatar
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    Aaaah! I got it now.

    If we use the 30-60-90 triangle property the length of one side of the hexagon is 8. This gives the A = 497 solution. If we use the Pythagorean theorem to find the length of the side we get 8.08 or so, giving the A = 502 solution.

    The problem is that the apothem is an approximation. It should be 6.9282 cm, not 6.9 cm.

    Thank you all.
    -Dan
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