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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.enter image description here

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    $\begingroup$ Hint: Review your justification that AEC is a right triangle. $\endgroup$ – Josh B. Jan 22 at 3:24
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    $\begingroup$ You should be able to tell from your diagram that the four angles at the center of the cube are not equal. All four angles are apex angles of isosceles triangles whose equal sides are $6.$ But the triangles for two angles have base length equal to the side of the cube, whereas the base of the other two triangles is $\sqrt2$ times the side of the cube. $\endgroup$ – David K Jan 22 at 3:33
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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.

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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.

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