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Math Help - 6 Equal circles in an equilateral triangle

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    6 Equal circles in an equilateral triangle

    6 Equal circles in an equilateral triangle-6-circles-triangle.jpg
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    Re: 6 Equal circles in an equilateral triangle

    We can easily exclude (a), (b) and (e). Let's assume that R = 1. Then the combined area of the circles is 6\pi\approx18.850; (c) gives 24.124 and (d) gives 19.392, which is too close to the circle areas. To be sure that (d) is wrong, it is possible to calculate the area of one of the triangle's angles not covered by a circle (which looks like a small triangle with two straight sides and one curved side). If I am correct, its area is \sqrt{3}-\pi/3\approx0.685. Multiplied by 3, this gives 2.055, which together with 6\pi exceeds (d).
    Last edited by emakarov; December 20th 2011 at 08:06 AM.
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    Re: 6 Equal circles in an equilateral triangle

    (12+7($\sqrt{3}$))R^2
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    MHF Contributor ebaines's Avatar
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    Re: 6 Equal circles in an equilateral triangle

    Quote Originally Posted by swordfish774 View Post
    (12+7($\sqrt{3}$))R^2
    Correct. The base length of the traingle is B=4R+2Rsqrt(3), and the area is (B^2/4)sqrt(3). The rest is algebra.
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