If the number of square centimetres on the surface of a sphere is equal to the number of cubic centimetres in its volume, what is the diameter of the sphere?

Answer

To solve the problem, we need to find the diameter of a sphere given that its surface area in square centimeters is equal to its volume in cubic centimeters.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
- The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]

2. **Set Up the Equation**:
- According to the problem, the surface area is equal to the volume:
\[ 4\pi r^2 = \frac{4}{3}\pi r^3 \]

3. **Simplify the Equation**:
- We can cancel \( 4\pi \) from both sides (as long as \( r \neq 0 \)):
\[ r^2 = \frac{1}{3} r^3 \]

4. **Rearrange the Equation**:
- Multiply both sides by 3 to eliminate the fraction:
\[ 3r^2 = r^3 \]
- Rearranging gives:
\[ r^3 - 3r^2 = 0 \]

5. **Factor the Equation**:
- Factor out \( r^2 \):
\[ r^2(r - 3) = 0 \]

6. **Solve for \( r \)**:
- This gives us two solutions:
\[ r^2 = 0 \quad \text{or} \quad r - 3 = 0 \]
- The first solution \( r^2 = 0 \) gives \( r = 0 \), which is not valid for a sphere.
- The second solution gives \( r = 3 \).

7. **Find the Diameter**:
- The diameter \( d \) of the sphere is given by:
\[ d = 2r \]
- Substituting the value of \( r \):
\[ d = 2 \times 3 = 6 \text{ cm} \]

### Final Answer:
The diameter of the sphere is **6 cm**.