finding the volume of a cube, why doesn't my solution work

Question

Question: the distance from a corner to the centre of a cube is 6. what is the volume of the cube?

Answer: 332.55

I drew a figure of a cube and two lines from a corner to their opposite corner and labeled the middle of the cube e. the triangle aec is right angled because the 2 lines make 4 equal angles when they intersect at e. knowing this and that the distance from a corner to e is 6, I calculated ac to be 8.49 or √72 using the Pythagorean theorem. Since I know ac is 8.49, I calculated x to be 6. 6³ is 216, so the volume of the cube is 216. Please tell me where I went wrong, Thank You.

Answer 1

The angle $\angle AEC$ is NOT $90^{\circ}$, which is why your approach doesn't quite work.

Another approach -- the distance between two opposite corners is the diameter of the sphere, i.e. $12$. This implies the side of the cube is $12/\sqrt{3}$ (can you see why?), which gives the correct answer.

Answer 2

The 2 diagonal lines [through the center of the cube] make 4 angles but the lines are not on a plane.

Let $x$ be the length of an edge of the cube. Then the length of a diagonal of a side is $x\sqrt{2}$. Furthermore, there is a right triangle where the legs are 1. the diagonal of a side [such as say from the lower-right to upper-left corner of the side facing us in your picture], 2. an edge--such as the lower edge of the right side of the cube, and where the hypo is a diagonal across the entire cube.

This gives us $x^2 + 2x^2 = (6+6)^2$ which gives us $x =\sqrt{48}$; and $x^3=48\sqrt{48}$ is the answer.