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

Answer

To solve the problem step by step, we will follow the mathematical reasoning provided in the video transcript.

### Step 1: Understand the given information
We know that the difference between the circumference and the radius of a circle is 37 cm.

### Step 2: Write 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 3: Set up the equation based on the problem statement
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:
\[ 2 \pi r - r = 37 \]

### Step 4: Factor out the radius
We can factor out \( r \) from the left-hand side:
\[ r(2 \pi - 1) = 37 \]

### Step 5: Solve for the radius \( r \)
To find \( r \), we can rearrange the equation:
\[ r = \frac{37}{2 \pi - 1} \]

### Step 6: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 2 \pi = 2 \times \frac{22}{7} = \frac{44}{7} \]
Now substitute this back into the equation:
\[ r = \frac{37}{\frac{44}{7} - 1} \]

### Step 7: Simplify the denominator
Convert 1 into a fraction with a common denominator:
\[ 1 = \frac{7}{7} \]
Thus, the denominator becomes:
\[ \frac{44}{7} - \frac{7}{7} = \frac{44 - 7}{7} = \frac{37}{7} \]
Now substitute this back into the equation for \( r \):
\[ r = \frac{37}{\frac{37}{7}} = 37 \times \frac{7}{37} = 7 \text{ cm} \]

### Step 8: Calculate the area of the circle
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (7)^2 = \pi \times 49 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times 49 \]
Calculating this gives:
\[ A = 22 \times 7 = 154 \text{ cm}^2 \]

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

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