Find the ratio of the volume of a cube to that of the sphere which fits inside the cube.

Answer

To find the ratio of the volume of a cube to that of a sphere that fits inside the cube, we can follow these steps:

### Step 1: Define the side length of the cube
Let the side length of the cube be \( a \).

### Step 2: Calculate the volume of the cube
The volume \( V_{\text{cube}} \) of a cube is given by the formula:
\[ V_{\text{cube}} = a^3 \]

### Step 3: Determine the radius of the sphere
The sphere that fits inside the cube will have a diameter equal to the side length of the cube. Therefore, the radius \( r \) of the sphere is:
\[ r = \frac{a}{2} \]

### Step 4: Calculate the volume of the sphere
The volume \( V_{\text{sphere}} \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Substituting the value of \( r \):
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 \]
Calculating \( \left(\frac{a}{2}\right)^3 \):
\[ \left(\frac{a}{2}\right)^3 = \frac{a^3}{8} \]
Thus, substituting this back into the volume formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{a^3}{8} = \frac{4 \pi a^3}{24} = \frac{\pi a^3}{6} \]

### Step 5: Find the ratio of the volumes
Now we can find the ratio of the volume of the cube to the volume of the sphere:
\[ \text{Ratio} = \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{a^3}{\frac{\pi a^3}{6}} = \frac{a^3 \cdot 6}{\pi a^3} = \frac{6}{\pi} \]

### Final Answer
The ratio of the volume of the cube to that of the sphere which fits inside the cube is:
\[ \frac{6}{\pi} \]