Find the area of a quadrant of a circle whose circumference is 22 cm.

Explanation

To find the area of a quadrant of a circle whose circumference is 22 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: Set up the equation with the given circumference.
We know from the problem that the circumference is 22 cm. Therefore, we can write:
\[ 2\pi r = 22 \]

### Step 3: Solve for the radius \( r \).
First, we can isolate \( r \) by dividing both sides by \( 2\pi \):
\[ r = \frac{22}{2\pi} \]
Now substituting \( \pi \) with \( \frac{22}{7} \) (a common approximation):
\[ r = \frac{22}{2 \times \frac{22}{7}} = \frac{22 \times 7}{2 \times 22} \]
The \( 22 \) cancels out:
\[ r = \frac{7}{2} \text{ cm} \]

### Step 4: Calculate the area of the quadrant.
The area \( A \) of a quadrant of a circle is one-fourth of the area of the whole circle. The area of a circle is given by:
\[ A = \pi r^2 \]
Thus, the area of the quadrant is:
\[ \text{Area of Quadrant} = \frac{1}{4} \times \pi r^2 \]
Substituting \( r = \frac{7}{2} \):
\[ \text{Area of Quadrant} = \frac{1}{4} \times \pi \left(\frac{7}{2}\right)^2 \]
Calculating \( \left(\frac{7}{2}\right)^2 \):
\[ \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
Now substituting this back into the area formula:
\[ \text{Area of Quadrant} = \frac{1}{4} \times \pi \times \frac{49}{4} = \frac{49\pi}{16} \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area of Quadrant} = \frac{49 \times \frac{22}{7}}{16} \]
Simplifying:
\[ = \frac{49 \times 22}{7 \times 16} = \frac{1078}{112} \]
Dividing \( 1078 \) by \( 112 \):
\[ = 9.625 \text{ cm}^2 \]

### Final Answer:
The area of the quadrant of the circle is \( 9.625 \text{ cm}^2 \).

---