If the circumference and area of a circle are numerically equal, then the diameter is equal to :

Options

A. area of the circle
B. `(pi)/(2)`
C. `2pi`
D. 4

Answer

To solve the problem, we need to find the diameter of a circle when its circumference and area are numerically equal.

### Step-by-Step Solution:

1. **Understand the formulas**:
- The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
- The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]

2. **Set the circumference equal to the area**:
Since the problem states that the circumference and area are numerically equal, we can write:
\[ 2\pi r = \pi r^2 \]

3. **Simplify the equation**:
We can divide both sides of the equation by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ 2r = r^2 \]

4. **Rearrange the equation**:
Rearranging gives us:
\[ r^2 - 2r = 0 \]

5. **Factor the equation**:
Factoring out \( r \):
\[ r(r - 2) = 0 \]

6. **Solve for \( r \)**:
This gives us two solutions:
\[ r = 0 \quad \text{or} \quad r = 2 \]
Since a radius of 0 does not represent a valid circle, we take:
\[ r = 2 \]

7. **Find the diameter**:
The diameter \( d \) of a circle is twice the radius:
\[ d = 2r = 2 \times 2 = 4 \]

### Final Answer:
The diameter of the circle is \( 4 \).

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