The circumference of a circle is 22 cm. Find the area of its quadrant (in `cm^2`).

Options

A. `77/8`
B. `22/7`
C. 78
D. 156

Answer

To find the area of a quadrant of a circle given its circumference, we can follow these steps:

### Step 1: Understand the relationship between circumference and radius
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
We know that the circumference \( C \) is 22 cm. Therefore, we can set up the equation:
\[ 22 = 2\pi r \]

### Step 3: Solve for the radius \( r \)
To isolate \( r \), we can rearrange the equation:
\[ r = \frac{22}{2\pi} \]
This simplifies to:
\[ r = \frac{11}{\pi} \]

### Step 4: Use the approximate value of \( \pi \)
For calculations, we can use the approximate value of \( \pi \) as \( \frac{22}{7} \):
\[ r = \frac{11}{\frac{22}{7}} = \frac{11 \times 7}{22} = \frac{77}{22} = \frac{7}{2} \text{ cm} \]

### Step 5: Calculate the area of the full circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \left(\frac{7}{2}\right)^2 = \pi \times \frac{49}{4} \]

### Step 6: Substitute \( \pi \) with \( \frac{22}{7} \)
Now substituting \( \pi \):
\[ A = \frac{22}{7} \times \frac{49}{4} \]

### Step 7: Simplify the area calculation
Calculating the area:
\[ A = \frac{22 \times 49}{7 \times 4} = \frac{1078}{28} \]
Now simplifying \( \frac{1078}{28} \):
\[ A = \frac{539}{14} \text{ cm}^2 \]

### Step 8: Calculate the area of the quadrant
Since a quadrant is one-fourth of the circle, we divide the area of the circle by 4:
\[ \text{Area of the quadrant} = \frac{539}{14} \times \frac{1}{4} = \frac{539}{56} \text{ cm}^2 \]

### Final Answer
Thus, the area of the quadrant is:
\[ \text{Area of the quadrant} = \frac{539}{56} \text{ cm}^2 \approx 9.64 \text{ cm}^2 \]