Triangles and spheres
The sum of the volumes of all spheres with different integer radii less than 7 and greater than 0 which circumscribe a triangle with side lengths 1, cos (15), and cos (75) can be written as A*pi. Find A.
Note: the sphere doesn't necessarily have to circumscribe the triangle at its great circle.
The answer is 588.
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1 solution
1, cos(75) and cos(15) forms a right triangle.
The smallest sphere circumscribing it at great circle will have a radius=hypotenuse/2= 1/2
So all spheres of radii 1,2,3,...6 can circumscribe the triangle
Sum of volumes =4π/3x(1³+2³+...6³)
=4π/3x441=588π