If the circumference and the area of a circle are numerically equal, then what is the numerical value of the diameter ?

Options

A. 1
B. 2
C. 4
D. `pi`

Answer

To solve the problem where the circumference and area of a circle are numerically equal, we can follow these steps:

### 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:
\[ A = \pi r^2 \]
Here, \( r \) is the radius of the circle.

2. **Set the Circumference Equal to the Area**:
- According to the problem, the circumference and area are numerically equal:
\[ 2 \pi r = \pi r^2 \]

3. **Simplify the Equation**:
- We can simplify the equation by dividing both sides 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 \( r \):
\[ r(r - 2) = 0 \]
- This gives us two possible solutions:
\[ r = 0 \quad \text{or} \quad r = 2 \]

6. **Select the Valid Solution**:
- Since \( r = 0 \) does not make sense in the context of a circle, we take:
\[ r = 2 \]

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

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

---