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Challenge 392: Operation Octahedron

Investigate this octahedron...

An octahedron has vertices A, B, C , D, X and Y.

ABCD is a square, with X above the plane containing ABCD and Y below this plane.

All the faces of the octahedron are equilateral triangles.

Given that the volume of the octahedron is 288cm3, find the total length of all its edges in the form a√2 cm, where a is an integer.

What is the ratio of the octahedron's surface area to its volume, in its simplest form?