Find the radius of a circle whose circumference is equal to the sum of the A circumferences of two circles of radi 12 cm and 16 cm.

Answer

To find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles with radii 12 cm and 16 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Two Circles:**
- Let the radius of the first circle \( R_1 = 12 \) cm.
- Let the radius of the second circle \( R_2 = 16 \) cm.

2. **Calculate the Circumference of Each Circle:**
- The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
- For the first circle:
\[ C_1 = 2\pi R_1 = 2\pi \times 12 = 24\pi \text{ cm} \]
- For the second circle:
\[ C_2 = 2\pi R_2 = 2\pi \times 16 = 32\pi \text{ cm} \]

3. **Find the Sum of the Circumferences of the Two Circles:**
- Now, we add the circumferences of the two circles:
\[ C_{total} = C_1 + C_2 = 24\pi + 32\pi = 56\pi \text{ cm} \]

4. **Set the Circumference of the Required Circle Equal to the Total Circumference:**
- Let \( R \) be the radius of the required circle. The circumference of this circle is:
\[ C = 2\pi R \]
- We set this equal to the total circumference:
\[ 2\pi R = 56\pi \]

5. **Solve for the Radius \( R \):**
- We can divide both sides of the equation by \( 2\pi \):
\[ R = \frac{56\pi}{2\pi} = \frac{56}{2} = 28 \text{ cm} \]

### Final Answer:
The radius of the required circle is \( R = 28 \) cm.

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