Find the area of a circle whose circumference is 66 cm .

Answer

To find the area of a circle whose circumference is 66 cm, we will follow these steps:

### Step 1: Use the formula for circumference
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
Given that the circumference is 66 cm, we can set up the equation:
\[ 66 = 2 \pi r \]

### Step 3: Solve for the radius (r)
We will rearrange the equation to solve for \( r \):
\[ r = \frac{66}{2 \pi} \]

### Step 4: Substitute the value of π
We will use \( \pi \approx \frac{22}{7} \):
\[ r = \frac{66}{2 \times \frac{22}{7}} \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ r = \frac{66 \times 7}{2 \times 22} \]

### Step 6: Perform the calculations
Calculating the numerator and denominator:
- The numerator: \( 66 \times 7 = 462 \)
- The denominator: \( 2 \times 22 = 44 \)

Now we have:
\[ r = \frac{462}{44} \]

### Step 7: Further simplify
We can simplify this fraction:
\[ r = \frac{462 \div 22}{44 \div 22} = \frac{21}{2} \text{ cm} \]

### Step 8: Calculate the area of the circle
The area (A) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( \pi \) and \( r \):
\[ A = \frac{22}{7} \left( \frac{21}{2} \right)^2 \]

### Step 9: Calculate \( r^2 \)
Calculating \( \left( \frac{21}{2} \right)^2 \):
\[ \left( \frac{21}{2} \right)^2 = \frac{441}{4} \]

### Step 10: Substitute back into the area formula
Now substituting back:
\[ A = \frac{22}{7} \times \frac{441}{4} \]

### Step 11: Multiply and simplify
Calculating:
\[ A = \frac{22 \times 441}{7 \times 4} = \frac{9702}{28} \]

### Step 12: Final simplification
Now, we simplify \( \frac{9702}{28} \):
\[ A = 346.5 \text{ cm}^2 \]

Thus, the area of the circle is **346.5 cm²**.

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