# 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.

**Your answer seems reasonable.**Find out if you're right!

**That seems reasonable.**Find out if you're right!

Already have an account? Log in here.