Ball of chi in a pyramid

Geometry Level 5

\(ABCD\) is a convex quadrilateral such that \( [ABD] = [CDB] \), \( |AB| = 1 \) and \( |BC|=|CD| \). \(S\) is a point in space such that \( |AS| + |DS| = \sqrt{2} \) and the volume of the pyramid \(SABCD\) is equal to \( \frac{1}{6} \).

The surface area of the largest ball that can fit inside such a pyramid can be expressed as \(\frac{a-\sqrt{b}}{c} \pi ,\) where \(a,b,c\) are positive integers, with \(c\) the smallest possible. What is \(a+b+c?\)

Details and assumptions

The ball need not touch all 5 faces of the pyramid.


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