The circumference of a circle exceeds its diameter by 30 cm. Find the radius of the circle.

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the variables
Let the radius of the circle be \( r \) cm.

### Step 2: Find the diameter
The diameter \( d \) of the circle can be expressed in terms of the radius:
\[ d = 2r \text{ cm} \]

### Step 3: Find the circumference
The circumference \( C \) of the circle can be expressed as:
\[ C = 2\pi r \text{ cm} \]

### Step 4: Set up the equation
According to the problem, the circumference exceeds the diameter by 30 cm. This can be written as:
\[ C - d = 30 \]
Substituting the expressions for circumference and diameter, we have:
\[ 2\pi r - 2r = 30 \]

### Step 5: Factor out common terms
We can factor out \( 2r \) from the left side of the equation:
\[ 2r(\pi - 1) = 30 \]

### Step 6: Solve for \( r \)
To isolate \( r \), divide both sides by \( 2(\pi - 1) \):
\[ r = \frac{30}{2(\pi - 1)} \]
This simplifies to:
\[ r = \frac{15}{\pi - 1} \]

### Step 7: Substitute the value of \( \pi \)
Using the approximate value of \( \pi \approx \frac{22}{7} \):
\[ r = \frac{15}{\frac{22}{7} - 1} \]
First, convert \( 1 \) to a fraction with a denominator of \( 7 \):
\[ 1 = \frac{7}{7} \]
Thus:
\[ r = \frac{15}{\frac{22 - 7}{7}} = \frac{15}{\frac{15}{7}} = 15 \times \frac{7}{15} \]
This simplifies to:
\[ r = 7 \text{ cm} \]

### Conclusion
The radius of the circle is \( 7 \) cm.

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