If the radius of a circle is decreased by `10%`, then the area of the circle is decreased by

Options

A. `89%`
B. `18%`
C. `19%`
D. `25%`

Answer

To solve the problem of how much the area of a circle decreases when the radius is decreased by 10%, we can follow these steps:

### Step 1: Understand the formula for the area of a circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Define the initial radius
Let's assume the initial radius of the circle is \( r \).

### Step 3: Calculate the new radius after a 10% decrease
If the radius is decreased by 10%, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.10r = 0.90r \]

### Step 4: Calculate the initial area of the circle
Using the initial radius \( r \), the initial area \( A \) is:
\[ A = \pi r^2 \]

### Step 5: Calculate the area of the circle with the new radius
Using the new radius \( r' \), the new area \( A' \) is:
\[ A' = \pi (r')^2 = \pi (0.90r)^2 = \pi (0.81r^2) = 0.81 \pi r^2 \]

### Step 6: Calculate the decrease in area
The decrease in area \( \Delta A \) can be calculated as:
\[ \Delta A = A - A' = \pi r^2 - 0.81 \pi r^2 = (1 - 0.81) \pi r^2 = 0.19 \pi r^2 \]

### Step 7: Calculate the percentage decrease in area
To find the percentage decrease in area, we use the formula:
\[ \text{Percentage decrease} = \left( \frac{\Delta A}{A} \right) \times 100 = \left( \frac{0.19 \pi r^2}{\pi r^2} \right) \times 100 = 19\% \]

### Conclusion
Thus, when the radius of the circle is decreased by 10%, the area of the circle is decreased by **19%**.

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