If the area of a circle decreases by 36%, then the radius of a circle decreases by

Options

A. 20%
B. 18%
C. 36%
D. 64%

Answer

To solve the problem, we need to find out how much the radius of a circle decreases when the area decreases by 36%. Let's break it down step by step.

### Step 1: Define the initial radius and area
Let the initial 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 decrease in area
The problem states that the area decreases by 36%. Therefore, the remaining area after the decrease can be calculated as:
\[ \text{Remaining Area} = A - 0.36A = 0.64A \]
Substituting the area formula:
\[ \text{Remaining Area} = 0.64 \cdot \pi r^2 \]

### Step 3: Relate the remaining area to the new radius
Let the new radius be \( r' \). The area of the new circle can also be expressed as:
\[ \text{New Area} = \pi (r')^2 \]
Since we know the remaining area is \( 0.64 \pi r^2 \), we can set the two expressions for area equal to each other:
\[ \pi (r')^2 = 0.64 \pi r^2 \]

### Step 4: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ (r')^2 = 0.64 r^2 \]

### Step 5: Solve for the new radius
Taking the square root of both sides gives:
\[ r' = \sqrt{0.64} \cdot r \]
Calculating the square root:
\[ r' = 0.8r \]

### Step 6: Calculate the decrease in radius
The decrease in radius is:
\[ \text{Decrease} = r - r' = r - 0.8r = 0.2r \]

### Step 7: Calculate the percentage decrease in radius
To find the percentage decrease in radius, we use the formula:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{\text{Initial Radius}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Decrease} = \left(\frac{0.2r}{r}\right) \times 100 = 20\% \]

### Conclusion
Therefore, the radius of the circle decreases by **20%**.

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