If the radius of a circle is decreased to 25% of its original value, calculate the percentage decrease in the area of the circle.

Options

A. `25%`
B. `43.75%`
C. `50%`
D. `93.75%`

Answer

To solve the problem of calculating the percentage decrease in the area of a circle when the radius is decreased to 25% of its original value, follow these steps:

### Step 1: Understand the relationship between radius and area
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: Determine the new radius
If the radius is decreased to 25% of its original value, we can express the new radius \( r' \) as:
\[ r' = 0.25r \]

### Step 3: Calculate the new area
Using the new radius, the new area \( A' \) can be calculated as follows:
\[ A' = \pi (r')^2 = \pi (0.25r)^2 = \pi (0.0625r^2) = 0.0625 \pi r^2 \]

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

### Step 5: Find the decrease in area
The decrease in area can be calculated by subtracting the new area from the original area:
\[ \text{Decrease in Area} = A - A' = \pi r^2 - 0.0625 \pi r^2 = \pi r^2 (1 - 0.0625) = \pi r^2 (0.9375) \]

### Step 6: Calculate the percentage decrease in area
To find the percentage decrease, we use the formula:
\[ \text{Percentage Decrease} = \left( \frac{\text{Decrease in Area}}{\text{Original Area}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Decrease} = \left( \frac{\pi r^2 (0.9375)}{\pi r^2} \right) \times 100 = 0.9375 \times 100 = 93.75\% \]

### Final Answer
The percentage decrease in the area of the circle is **93.75%**.

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