Sphere Sandwich

Geometry Level 4

We begin with a cube $C_1$ .

Inside this cube $C_1$ we fit a sphere $S$ , which touches the cube in the midpoint of each of the six faces.

Inside the sphere $S$ we fit a smaller cube $C_2$ , whose eight vertices touch the sphere.

Express the volume of cube $C_2$ as a percentage of the volume of $C_1$ . Round off to a whole number.

25% 58% 32% 19% 50% 35%

1 solution

Arjen Vreugdenhil
Feb 4, 2016

Suppose the sphere has diameter $d$ .

Then $C_1$ is a cube with sides $d$ , and volume $d^3$ .

As for $C_2$ , if $a$ is its side, then the distance between two opposite vertices is the body diagonal, $\sqrt 3 a$ . This should be equal to the diameter of the sphere. Thus the side of $C_2$ is $a = d/\sqrt 3$ , and its volume is $(d/\sqrt 3)^3 = d^3/(3\sqrt 3)$ .

Therefore the ratio of volumes is $\frac{d^3/(3 \sqrt 3)}{d^3} = \frac1{3\sqrt 3} \approx 0.19245$ so the answer is $\boxed{19\%}$ .

Sphere Sandwich

1 solution

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