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Hexagonal Dilemma (Posted on 2004-06-02) Difficulty: 4 of 5
A hexagon with sides of length 2, 7, 2, 11, 7, 11 is inscribed in a circle. Find the radius of the circle.

As suggested, *if* it matters, you may assume that the sides listed are given in order

No Solution Yet Submitted by SilverKnight    
Rating: 4.0000 (5 votes)

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Solution Excel solution | Comment 4 of 18 |

If the hexagon is divided into 6 triangles, with one side of the hexagon serving as a side of each triangle, and one of the vertices of each triangle at the center of the circle, the angles at the center need to add up to 360 degrees.  Each of the angles is equal to 2*arcsin(s/(2*r)), as the side of the hexagon is a chord of the circle.  So set up this spreadsheet:

A                B                   C                    D
r                6.99999999999989
                 2                   =2*ASIN(B2/2/B$1)
                 7                   =2*ASIN(B3/2/B$1)
                 2                   =2*ASIN(B4/2/B$1)
                 11                  =2*ASIN(B5/2/B$1)
                 7                   =2*ASIN(B6/2/B$1)
                 11                  =2*ASIN(B7/2/B$1)
                                     =SUM(C2:C7)          =2*PI()

where, in B1 you start out only with a guess, and then use the solver capability to vary B1 so that C8 equals D8.  An initial guess outside a certain range of values will fail to provide a solution, but if it's close enough to what turns out to be the answer, 7, it will converge to that value.

The answer is 7.

  Posted by Charlie on 2004-06-02 15:44:35
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