The sum of the circumference and daimeter of a circle is 116 cm. Find its radius.

Answer

To solve the problem where the sum of the circumference and diameter of a circle is 116 cm, and we need to find its radius, follow these steps:

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The circumference (C) of a circle is given by the formula:
\[ C = 2\pi r \]
or
\[ C = \pi d \]
where \( d \) is the diameter and \( r \) is the radius.
- The diameter (d) is related to the radius by:
\[ d = 2r \]

2. **Set Up the Equation**:
- According to the problem, the sum of the circumference and diameter is given as:
\[ C + d = 116 \]
- Substituting the formulas for circumference and diameter:
\[ 2\pi r + d = 116 \]

3. **Substitute Diameter**:
- Replace \( d \) with \( 2r \):
\[ 2\pi r + 2r = 116 \]

4. **Factor Out Common Terms**:
- Factor out \( 2r \) from the left side:
\[ 2r(\pi + 1) = 116 \]

5. **Isolate \( r \)**:
- Divide both sides by \( 2(\pi + 1) \):
\[ r = \frac{116}{2(\pi + 1)} \]

6. **Calculate \( \pi \)**:
- Use \( \pi \approx \frac{22}{7} \):
\[ r = \frac{116}{2\left(\frac{22}{7} + 1\right)} \]

7. **Simplify the Denominator**:
- Convert 1 into a fraction with a common denominator:
\[ 1 = \frac{7}{7} \]
Therefore,
\[ \pi + 1 = \frac{22}{7} + \frac{7}{7} = \frac{29}{7} \]

8. **Substitute Back**:
- Now substitute back into the equation for \( r \):
\[ r = \frac{116}{2 \cdot \frac{29}{7}} \]

9. **Multiply by the Reciprocal**:
- Simplify:
\[ r = \frac{116 \cdot 7}{2 \cdot 29} \]

10. **Calculate**:
- First, calculate \( 116 \div 2 = 58 \):
\[ r = \frac{58 \cdot 7}{29} \]
- Now, \( 58 \div 29 = 2 \):
\[ r = 2 \cdot 7 = 14 \]

11. **Final Answer**:
- The radius \( r \) is:
\[ r = 14 \text{ cm} \]