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. \(\theta \) is the angle of the sector cut out for which the volume of the cone is maximized.

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

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