The radii of two circles are 8 cm and 6 cm respectively. Find the diameter of the circle having area equal to the sum of the areas of the two circles.

Answer

To solve the problem, we need to find the diameter of a circle whose area is equal to the sum of the areas of two given circles with radii 8 cm and 6 cm.

### Step-by-Step Solution:

1. **Calculate the area of the first circle:**
- The formula for the area of a circle is given by:
\[ \text{Area} = \pi r^2 \]
- For the first circle with radius \( r_1 = 8 \) cm:
\[ \text{Area}_1 = \pi (8)^2 = \pi \times 64 = 64\pi \text{ cm}^2 \]

2. **Calculate the area of the second circle:**
- For the second circle with radius \( r_2 = 6 \) cm:
\[ \text{Area}_2 = \pi (6)^2 = \pi \times 36 = 36\pi \text{ cm}^2 \]

3. **Find the sum of the areas of the two circles:**
- Now, we add the areas of both circles:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 64\pi + 36\pi = 100\pi \text{ cm}^2 \]

4. **Set up the equation for the area of the new circle:**
- Let the radius of the new circle be \( R \). The area of this circle can be expressed as:
\[ \text{Area} = \pi R^2 \]
- We set this equal to the total area we found:
\[ \pi R^2 = 100\pi \]

5. **Simplify the equation:**
- We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ R^2 = 100 \]

6. **Solve for \( R \):**
- Taking the square root of both sides gives:
\[ R = \sqrt{100} = 10 \text{ cm} \]

7. **Calculate the diameter of the new circle:**
- The diameter \( D \) is twice the radius:
\[ D = 2R = 2 \times 10 = 20 \text{ cm} \]

### Final Answer:
The diameter of the circle having an area equal to the sum of the areas of the two circles is **20 cm**.