If the circumference of a circle is reduced by 50%, its area will be reduced by

Options

A. `12.5%`
B. `25%`
C. `50%`
D. `75%`

Answer

To solve the problem, we need to determine how much the area of a circle is reduced when its circumference is reduced by 50%. Let's break it down step by step.

### Step 1: Understand the relationship between circumference and radius
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 2: Calculate the new circumference
If the circumference is reduced by 50%, the new circumference \( C' \) will be:
\[ C' = C - 0.5C = 0.5C \]
Substituting the formula for circumference, we get:
\[ C' = 0.5 \times 2\pi r = \pi r \]

### Step 3: Find the new radius
Using the new circumference, we can find the new radius \( r' \):
\[ C' = 2\pi r' \implies \pi r = 2\pi r' \implies r' = \frac{r}{2} \]

### Step 4: Calculate the original area
The area \( A \) of the original circle is given by:
\[ A = \pi r^2 \]

### Step 5: Calculate the new area
Now, we calculate the area of the new circle with radius \( r' \):
\[ A' = \pi (r')^2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{\pi r^2}{4} \]

### Step 6: Determine the reduction in area
To find the reduction in area, we subtract the new area from the original area:
\[ \text{Reduction in Area} = A - A' = \pi r^2 - \frac{\pi r^2}{4} \]
Finding a common denominator:
\[ = \frac{4\pi r^2}{4} - \frac{\pi r^2}{4} = \frac{3\pi r^2}{4} \]

### Step 7: Calculate the percentage reduction in area
To find the percentage reduction, we use the formula:
\[ \text{Percentage Reduction} = \left(\frac{\text{Reduction in Area}}{\text{Original Area}}\right) \times 100 \]
Substituting the values:
\[ = \left(\frac{\frac{3\pi r^2}{4}}{\pi r^2}\right) \times 100 = \left(\frac{3}{4}\right) \times 100 = 75\% \]

### Final Answer
The area of the circle will be reduced by **75%** when the circumference is reduced by 50%.

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