The Star Medallion

Geometry Level 3

A pentagram is inscribed inside a circle with radius rr as shown in the picture above. The formula that puts the area of the pentagram SS in terms of the circle's radius rr can be expressed as follows:

S=a10abacr2  , \large S = \dfrac{a \sqrt{10a-b\sqrt a}}c r^2 \; ,

where a,ba,b and cc are positive integers with aa square-free and a,ca,c coprime.

Find the value of a+b+ca+b+c.

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