Find the area of a quadrant of a circle whose circumference is 616 cm.

Answer

To find the area of a quadrant of a circle whose circumference is 616 cm, we can follow these steps:

### Step 1: Use the formula for 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: Substitute the given circumference into the formula.
We know the circumference is 616 cm, so we can set up the equation:
\[ 2\pi r = 616 \]

### Step 3: Solve for the radius \( r \).
To find the radius, we can rearrange the equation:
\[ r = \frac{616}{2\pi} \]
Using \( \pi \approx \frac{22}{7} \), we substitute this value into the equation:
\[ r = \frac{616}{2 \times \frac{22}{7}} \]
\[ r = \frac{616 \times 7}{44} \]
\[ r = \frac{4312}{44} \]
\[ r = 98.5 \text{ cm} \]

### Step 4: Find the area of the whole circle.
The area (A) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (98.5)^2 \]
\[ A = \frac{22}{7} \times (98.5)^2 \]
Calculating \( (98.5)^2 \):
\[ (98.5)^2 = 9702.25 \]
Now substituting this back into the area formula:
\[ A = \frac{22}{7} \times 9702.25 \]

### Step 5: Calculate the area of the quadrant.
Since we need the area of a quadrant (which is one-fourth of the circle), we divide the area of the circle by 4:
\[ \text{Area of Quadrant} = \frac{A}{4} \]
\[ \text{Area of Quadrant} = \frac{\frac{22}{7} \times 9702.25}{4} \]

### Step 6: Simplify the expression.
Calculating this gives:
\[ \text{Area of Quadrant} = \frac{22 \times 9702.25}{28} \]
Calculating \( 22 \times 9702.25 = 213450.5 \):
\[ \text{Area of Quadrant} = \frac{213450.5}{28} \]
\[ \text{Area of Quadrant} = 7616.1 \text{ cm}^2 \]

### Final Answer:
The area of the quadrant of the circle is approximately \( 7616.1 \text{ cm}^2 \).

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