If the perimeter and the area of a circle are numerically equal, then the diameter of the circle is

Options

A. 2 units
B. `pi` units
C. 4 units
D. 7 units

Answer

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

### Step 1: Understand the formulas
The perimeter (circumference) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius.

The area of a circle is given by the formula:
\[ A = \pi r^2 \]

### Step 2: Set up the equation
According to the problem, the perimeter and the area are equal:
\[ 2\pi r = \pi r^2 \]

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

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

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

### Step 6: Solve for \( r \)
Setting each factor to zero gives us:
1. \( r = 0 \) (not a valid solution for a circle)
2. \( r - 2 = 0 \) which gives \( r = 2 \)

### Step 7: Calculate the diameter
The diameter \( d \) of the circle is given by:
\[ d = 2r \]
Substituting the value of \( r \):
\[ d = 2 \times 2 = 4 \]

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

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