+1552 A cone has a base radius of 3 and a height of 6. A cube is inscribed in the cone, so that four of its vertices lie in the circular base, and the other four vertices touch the lateral surface. Find the side length of the cube.
+130031 
Looking at a cross-section along the diagonal of the cube
Let the side of the cube = S = EF
The length of the bottom diagona of the cube = sqrt 2 *S = HF
We can construct similar triangles AFE and ADC
AF / EF = AD / DC
DF = sqrt 2*S / 2 = S/sqrt 2
So AF = 3 - S/sqrt 2
So we have
[ 3 -S/sqrt 2 ] / S = 3/6
[ 3 - S/sqrt 2 ] / S = 1/2
2 [ 3 - S/sqrt 2] = S
6 - sqrt 2 * S = S
6 = S + sqrt 2*S
6 = S ( sqrt 2 + 1)
S = 6 / ( sqrt 2 + 1) = 6 (sqrt 2 - 1) = side of the cube
