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

Question

Accepted Answer

To determine the radius of a circle when its perimeter and area are numerically equal, we start by using the formulas for the perimeter (circumference) and area of a circle.

1. **Perimeter (Circumference) of a Circle**: The formula for the circumference of a circle is $ C = 2\pi r $, where $ r $ is the radius.

2. **Area of a Circle**: The formula for the area of a circle is $ A = \pi r^2 $.

According to the problem, the perimeter and area are numerically equal, so we set the two expressions equal to each other:

$$ 2\pi r = \pi r^2 $$

To solve for $ r $, we can divide both sides of the equation by $ \pi $ (assuming $ \pi \neq 0 $):

$$ 2r = r^2 $$

Rearranging the equation gives:

$$ r^2 - 2r = 0 $$

Factoring out $ r $ from the equation, we have:

$$ r(r - 2) = 0 $$

This equation gives us two possible solutions for $ r $:

1. $ r = 0 $
2. $ r = 2 $

Since a radius of 0 does not make sense in the context of a circle (as it would imply a degenerate circle with no area or perimeter), we discard $ r = 0 $.

Thus, the only feasible solution is $ r = 2 $.

**Explanation of Why Option 2 is Correct:**

- **Option 2 (2 units)**: As derived above, when the perimeter and area of a circle are numerically equal, the radius must be 2 units. This satisfies the equation $ 2\pi r = \pi r^2 $ when $ r = 2 $.

**Explanation of Why Other Options are Incorrect:**

- **Option 1 (1 unit)**: If $ r = 1 $, then the perimeter would be $ 2\pi \times 1 = 2\pi $ and the area would be $ \pi \times 1^2 = \pi $. These are not numerically equal, so this option is incorrect.

- **Option 3 ($\pi$ unit)**: If $ r = \pi $, then the perimeter would be $ 2\pi \times \pi = 2\pi^2 $ and the area would be $ \pi \times \pi^2 = \pi^3 $. These are not numerically equal, so this option is incorrect.

- **Option 4 (7 units)**: If $ r = 7 $, then the perimeter would be $ 2\pi \times 7 = 14\pi $ and the area would be $ \pi \times 7^2 = 49\pi $. These are not numerically equal, so this option is incorrect.

Therefore, the correct answer is Option 2, with a radius of 2 units.