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Geometry/Quick Word Problem

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Question
A circle of radius 6 has chord  of length 6. If point C is selected randomly on the circle, what is the probability that ABC is obtuse?

Answer
To get a right triangle, the other point must involve a side going through the center of the circle.  The length of the side going through the center is the length of the diameter, and that is twice the length of the radius.  This means the length of that side is 12, and that is the hypotenuse of the triangle.  If one side is 6, and the hypotenuse is 12, that means the other side is 6*square-root(3).

The other point, then, could occur anywhere closer than 6*square-root(3) from either side.
That gives the 2*6*root(3) + 6 as where the point could not appear.
Using Excel, the circumference is C = 2*pi*r = 37.69911184.  Now A = 6 + 12*root(3) = 26.78460969.  The region that gives an obtuse triangle is A/C = 0.710483839.
That is around 71% of the time, the triangle will be obtuse.  

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