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

Options

A. `111cm^(2)`
B. `184cm^(2)`
C. `154cm^(2)`
D. `259cm^(2)`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the relationship between circumference, radius, and the given difference.
We know that the circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. According to the problem, the difference between the circumference and the radius is 37 cm. This can be expressed as:
\[ C - r = 37 \]

### Step 2: Substitute the formula for circumference into the equation.
Substituting the formula for circumference into the equation gives us:
\[ 2\pi r - r = 37 \]

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

### Step 4: Solve for \( r \).
To find \( r \), we can rearrange the equation:
\[ r = \frac{37}{2\pi - 1} \]
Now, we will substitute the value of \( \pi \) (approximately \( \frac{22}{7} \)) into the equation:
\[ r = \frac{37}{2 \times \frac{22}{7} - 1} \]
Calculating \( 2 \times \frac{22}{7} \):
\[ 2 \times \frac{22}{7} = \frac{44}{7} \]
Thus, we have:
\[ r = \frac{37}{\frac{44}{7} - 1} \]
To subtract 1, we convert 1 into a fraction with the same denominator:
\[ 1 = \frac{7}{7} \]
So, we get:
\[ r = \frac{37}{\frac{44}{7} - \frac{7}{7}} = \frac{37}{\frac{44 - 7}{7}} = \frac{37}{\frac{37}{7}} = 37 \times \frac{7}{37} = 7 \text{ cm} \]

### Step 5: Calculate the area of the circle.
Now that we have the radius \( r = 7 \) cm, we can calculate the area \( A \) of the circle using the formula:
\[ A = \pi r^2 \]
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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