Regular polyhedron resting on a face

Place a regular icosahedron of radius 3 u resting on the PH on one face and work out its height.

Step-by-step solution

Choose one face as the base and rotate the body until its normal is vertical.
The opposite vertex or face is on top: the difference in heights is the height.
Check that every vertex has height ≥ 0.
The height is not the radius: for a tetrahedron it is a√(2/3); for an octahedron, a√6/3.

Result: height = 4.77 u.

Resting on a face changes the plan completely: work out the height before drawing the elevation.