Radius not given?

Calculus Level 4

A Cone is made from a circular sheet by cutting out a sector and gluing the cut edges of the remaining sheet together, as shown in the above figure. θ\theta is the angle of the sector cut out for which the volume of the cone is maximized.

If the value of θ\theta can be written as a(1bc)π a\left(1-\frac { \sqrt { b } }{ \sqrt { c } } \right)\pi , where a,b,a,b, and cc are positive integers and bb and cc are not divisible by the square of any prime, determine the value of a+b+ca+b+c.

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