The difference between the circumference and radius of a circle is 37 cm. Find the area of the circle

Answer

To solve the problem, we need to find the area of a circle given that the difference between its circumference and radius is 37 cm.

### Step-by-Step Solution:

1. **Understand the formulas**:
- The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
- The radius of the circle is denoted as \( r \).

2. **Set up the equation**:
- According to the problem, the difference between the circumference and the radius is 37 cm:
\[ C - r = 37 \]
- Substituting the formula for circumference into this equation gives:
\[ 2\pi r - r = 37 \]

3. **Factor out \( r \)**:
- We can factor \( r \) from the left side:
\[ r(2\pi - 1) = 37 \]

4. **Solve for \( r \)**:
- Rearranging the equation gives:
\[ r = \frac{37}{2\pi - 1} \]
- Now, we will substitute \( \pi \) with \( \frac{22}{7} \):
\[ r = \frac{37}{2 \times \frac{22}{7} - 1} \]
- Calculating \( 2 \times \frac{22}{7} \):
\[ 2 \times \frac{22}{7} = \frac{44}{7} \]
- Thus, we have:
\[ r = \frac{37}{\frac{44}{7} - 1} \]
- Converting 1 to a fraction with a denominator of 7:
\[ 1 = \frac{7}{7} \]
- Therefore:
\[ r = \frac{37}{\frac{44}{7} - \frac{7}{7}} = \frac{37}{\frac{44 - 7}{7}} = \frac{37 \times 7}{37} = 7 \text{ cm} \]

5. **Calculate the area of the circle**:
- The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
- Substituting the values we have:
\[ A = \frac{22}{7} \times (7)^2 \]
- Calculating \( (7)^2 \):
\[ (7)^2 = 49 \]
- Now substituting back:
\[ A = \frac{22}{7} \times 49 \]
- Simplifying:
\[ A = 22 \times 7 = 154 \text{ cm}^2 \]

### Final Answer:
The area of the circle is \( 154 \text{ cm}^2 \).