2002 AIME I Problem 15

Polyhedron ABCDEFG has six faces. Face ABCD is a square with AB = 12; face ABFG is a trapezoid with \overline{AB} parallel to \overline{GF}, BF = AG = 8, and GF = 6; and face CDE has CE = DE = 14. The other three faces are ADEG, BCEF, and EFG. The distance from E to face ABCD is 12. Given that EG^2 = p - q\sqrt {r}, where p, q, and r are positive integers and r is not divisible by the square of any prime, find p + q + r.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.