If the difference between the circumference and diameter of a circle is 30 cm, then what is the radius of the circle ?

Options

A. 14 cm
B. 3.5 cm
C. 7 cm
D. 6 cm

Answer

To solve the problem step by step, we will follow the mathematical relationships between the circumference, diameter, and radius of a circle.

### Step-by-Step Solution:

1. **Understand the given information**:
We know that the difference between the circumference and the diameter of a circle is 30 cm.

2. **Write the formulas**:
- The formula for the circumference (C) of a circle is:
\[ C = 2\pi r \]
- The formula for the diameter (D) of a circle is:
\[ D = 2r \]

3. **Set up the equation**:
According to the problem, we can write the equation as:
\[ C - D = 30 \text{ cm} \]
Substituting the formulas for circumference and diameter:
\[ 2\pi r - 2r = 30 \]

4. **Factor out common terms**:
We can factor out \(2r\) from the left side of the equation:
\[ 2r(\pi - 1) = 30 \]

5. **Isolate \(r\)**:
To find \(r\), we first divide both sides by \(2(\pi - 1)\):
\[ r = \frac{30}{2(\pi - 1)} \]

6. **Simplify the expression**:
Simplifying the right side gives:
\[ r = \frac{30}{2} \cdot \frac{1}{(\pi - 1)} = \frac{15}{\pi - 1} \]

7. **Substitute the value of \(\pi\)**:
Using \(\pi \approx 3.14\):
\[ r = \frac{15}{3.14 - 1} = \frac{15}{2.14} \]

8. **Calculate the radius**:
Now, we perform the division:
\[ r \approx 7 \text{ cm} \]

### Conclusion:
The radius of the circle is approximately **7 cm**.