If the error committed in measuring the radius of a circle is `0.01%`, find the corresponding error in calculating the area.

Answer

To find the corresponding error in calculating the area of a circle when the error in measuring the radius is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We are given that the error in measuring the radius \( r \) of a circle is \( 0.01\% \). We need to find the corresponding error in calculating the area \( A \) of the circle.

2. **Formula for Area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

3. **Percentage Error in Radius**:
The percentage error in measuring the radius can be expressed as:
\[ \frac{\Delta r}{r} \times 100 = 0.01 \]
where \( \Delta r \) is the absolute error in the radius.

4. **Differentiate the Area Formula**:
To find the error in area, we differentiate the area with respect to the radius:
\[ dA = \frac{dA}{dr} \cdot dr \]
From the area formula, we have:
\[ \frac{dA}{dr} = 2\pi r \]
Therefore,
\[ dA = 2\pi r \cdot dr \]

5. **Expressing the Error in Area**:
The relative error in area can be expressed as:
\[ \frac{\Delta A}{A} = \frac{dA}{A} \]
Substituting the expression for \( dA \):
\[ \frac{\Delta A}{A} = \frac{2\pi r \cdot \Delta r}{\pi r^2} \]
Simplifying this gives:
\[ \frac{\Delta A}{A} = \frac{2 \Delta r}{r} \]

6. **Calculating the Percentage Error in Area**:
Now, we can express the percentage error in area:
\[ \frac{\Delta A}{A} \times 100 = 2 \times \frac{\Delta r}{r} \times 100 \]
We know that \( \frac{\Delta r}{r} \times 100 = 0.01 \), so:
\[ \frac{\Delta A}{A} \times 100 = 2 \times 0.01 = 0.02 \]

7. **Final Result**:
Thus, the corresponding error in calculating the area of the circle is:
\[ \Delta A \text{ (percentage)} = 0.02\% \]