If the circumference of a circle is reduced by 50%, then by how much per cent is the area of the circle reduced ?

Options

A. `25% `
B. `50% `
C. `65% `
D. `75% `

Answer

To solve the problem step-by-step, we will follow these steps:

### Step 1: Understand the Circumference of a Circle
The formula for the circumference (C) of a circle is given by:
\[ C = 2\pi r \]
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') becomes:
\[ C' = C - 0.5C = 0.5C \]
Substituting the formula for circumference:
\[ C' = 0.5 \times 2\pi r = \pi r \]

### Step 3: Relate the New Circumference to the New Radius
From the new circumference, we can find the new radius (r'):
\[ C' = 2\pi r' \]
Setting the two expressions for the new circumference equal gives:
\[ \pi r = 2\pi r' \]
Dividing both sides by \( \pi \):
\[ r = 2r' \]
Thus, the new radius is:
\[ r' = \frac{r}{2} \]

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

### Step 5: Calculate the Area of the New Circle
Using the new radius, the area of the new circle (A') is:
\[ A' = \pi (r')^2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{\pi r^2}{4} \]

### Step 6: Determine the Reduction in Area
To find out how much the area has been reduced, we calculate the difference between the original area and the new area:
\[ \text{Reduction in Area} = A - A' = \pi r^2 - \frac{\pi r^2}{4} \]
This simplifies to:
\[ \text{Reduction in Area} = \pi r^2 \left(1 - \frac{1}{4}\right) = \pi r^2 \left(\frac{3}{4}\right) = \frac{3\pi r^2}{4} \]

### Step 7: Calculate the Percentage Reduction in Area
The percentage reduction in area is given by:
\[ \text{Percentage Reduction} = \left(\frac{\text{Reduction in Area}}{A}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Reduction} = \left(\frac{\frac{3\pi r^2}{4}}{\pi r^2}\right) \times 100 \]
This simplifies to:
\[ \text{Percentage Reduction} = \left(\frac{3}{4}\right) \times 100 = 75\% \]

### Conclusion
The area of the circle is reduced by **75%**.

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