If the radius of a circle is increased by 20%, then by how much will its area be increased ?

Question

Accepted Answer

To determine how much the area of a circle increases when its radius is increased by 20%, we can use the formula for the area of a circle, which is $ A = \pi r^2 $, where $ r $ is the radius.

1. **Original Area**: 
   - Let the original radius be $ r $.
   - The original area $ A_1 = \pi r^2 $.

2. **New Radius**:
   - The radius is increased by 20%, so the new radius $ r' = r + 0.2r = 1.2r $.

3. **New Area**:
   - The new area $ A_2 = \pi (1.2r)^2 = \pi \times 1.44r^2 = 1.44\pi r^2 $.

4. **Increase in Area**:
   - The increase in area is $ A_2 - A_1 = 1.44\pi r^2 - \pi r^2 = 0.44\pi r^2 $.

5. **Percentage Increase**:
   - The percentage increase in area is given by:
     $$
     \left(\frac{0.44\pi r^2}{\pi r^2}\right) \times 100\% = 44\%
     $$

Thus, the area of the circle increases by 44% when the radius is increased by 20%. Therefore, Option 3 is correct.

**Explanation for Incorrect Options**:

- **Option 1 (22%)**: This option is incorrect because it underestimates the increase in area. The calculation clearly shows that the increase is 44%, not 22%.

- **Option 2 (33%)**: This option is incorrect because it does not accurately reflect the mathematical increase in area. The correct increase is 44%, as derived from the formula.

- **Option 4 (55%)**: This option overestimates the increase in area. The calculated increase is 44%, not 55%.

By following the mathematical derivation, it is evident that the correct answer is indeed Option 3, which states that the area increases by 44%.