The radii of two circle are 25 cm and 18 cm. Find the radius of the circle which has circumference equal to the sum of curcumferences of these two circles.

Answer

To find the radius of a circle whose circumference is equal to the sum of the circumferences of two given circles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Radii**:
- Let the radius of the first circle, \( r_1 = 25 \) cm.
- Let the radius of the second circle, \( r_2 = 18 \) cm.

2. **Write the Formula for Circumference**:
- The formula for the circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]
- For the first circle, the circumference \( C_1 \) is:
\[ C_1 = 2 \pi r_1 = 2 \pi (25) = 50 \pi \text{ cm} \]
- For the second circle, the circumference \( C_2 \) is:
\[ C_2 = 2 \pi r_2 = 2 \pi (18) = 36 \pi \text{ cm} \]

3. **Calculate the Sum of the Circumferences**:
- The total circumference \( C \) of the new circle is:
\[ C = C_1 + C_2 = 50 \pi + 36 \pi = 86 \pi \text{ cm} \]

4. **Set Up the Equation for the New Circle**:
- Let the radius of the new circle be \( R \). Then, its circumference is:
\[ C = 2 \pi R \]
- We can set this equal to the sum of the circumferences of the two circles:
\[ 2 \pi R = 86 \pi \]

5. **Solve for the Radius \( R \)**:
- Divide both sides by \( 2 \pi \):
\[ R = \frac{86 \pi}{2 \pi} = \frac{86}{2} = 43 \text{ cm} \]

### Final Answer:
The radius of the circle which has a circumference equal to the sum of the circumferences of the two circles is \( R = 43 \) cm.

---