If the radius of a circle is decreased by 10%, then what will be the percentage decrease in the area of circle?

Options

A. 19
B. 20
C. 21
D. 23

Answer

To find the percentage decrease in the area of a circle when its radius is decreased by 10%, we can follow these steps:

### Step 1: Define the original radius and area
Let the original radius of the circle be \( r \). The area \( A \) of the circle is given by the formula:
\[ A = \pi r^2 \]

### Step 2: Calculate the new radius after a 10% decrease
If the radius is decreased by 10%, the new radius \( r' \) can be calculated as:
\[ r' = r - \frac{10}{100} \times r = r \left(1 - 0.1\right) = r \times 0.9 = \frac{90}{100} r = \frac{9}{10} r \]

### Step 3: Calculate the new area using the new radius
The area \( A' \) of the circle with the new radius \( r' \) is:
\[ A' = \pi (r')^2 = \pi \left(\frac{9}{10} r\right)^2 = \pi \left(\frac{81}{100} r^2\right) = \frac{81}{100} \pi r^2 \]

### Step 4: Find the decrease in area
The decrease in area \( \Delta A \) can be calculated as:
\[ \Delta A = A - A' = \pi r^2 - \frac{81}{100} \pi r^2 = \pi r^2 \left(1 - \frac{81}{100}\right) = \pi r^2 \left(\frac{19}{100}\right) \]

### Step 5: Calculate the percentage decrease in area
The percentage decrease in area is given by:
\[ \text{Percentage decrease} = \frac{\Delta A}{A} \times 100 = \frac{\pi r^2 \left(\frac{19}{100}\right)}{\pi r^2} \times 100 \]
This simplifies to:
\[ \text{Percentage decrease} = \frac{19}{100} \times 100 = 19\% \]

### Final Answer
The percentage decrease in the area of the circle when the radius is decreased by 10% is **19%**.

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