Calculate the circumference of a circle whose area is equal to the sum of the area of the circles with diameter 24 cm, 32 cm and 96 cm

Answer

To solve the problem of calculating the circumference of a circle whose area is equal to the sum of the areas of circles with diameters 24 cm, 32 cm, and 96 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the diameters of the circles:**
- Let \( d_1 = 24 \) cm, \( d_2 = 32 \) cm, and \( d_3 = 96 \) cm.

2. **Calculate the areas of the circles:**
- The area \( A \) of a circle is given by the formula:
\[ A = \pi \left( \frac{d}{2} \right)^2 = \frac{\pi d^2}{4} \]
- Therefore, the areas of the three circles are:
- For \( d_1 \):
\[ A_1 = \frac{\pi (24)^2}{4} = \frac{\pi \cdot 576}{4} = 144\pi \text{ cm}^2 \]
- For \( d_2 \):
\[ A_2 = \frac{\pi (32)^2}{4} = \frac{\pi \cdot 1024}{4} = 256\pi \text{ cm}^2 \]
- For \( d_3 \):
\[ A_3 = \frac{\pi (96)^2}{4} = \frac{\pi \cdot 9216}{4} = 2304\pi \text{ cm}^2 \]

3. **Sum the areas of the three circles:**
\[ A_{total} = A_1 + A_2 + A_3 = 144\pi + 256\pi + 2304\pi = 2704\pi \text{ cm}^2 \]

4. **Set the area of the new circle equal to the total area:**
- Let \( d \) be the diameter of the new circle. The area of this circle is:
\[ A = \frac{\pi d^2}{4} \]
- Setting the areas equal gives:
\[ \frac{\pi d^2}{4} = 2704\pi \]

5. **Cancel \( \pi \) from both sides:**
\[ \frac{d^2}{4} = 2704 \]

6. **Multiply both sides by 4:**
\[ d^2 = 2704 \times 4 = 10816 \]

7. **Take the square root to find \( d \):**
\[ d = \sqrt{10816} = 104 \text{ cm} \]

8. **Calculate the circumference of the new circle:**
- The circumference \( C \) is given by:
\[ C = \pi d \]
- Substituting the value of \( d \):
\[ C = \pi \times 104 \]
- Using \( \pi \approx \frac{22}{7} \):
\[ C \approx \frac{22}{7} \times 104 \approx 326.86 \text{ cm} \]

### Final Answer:
The circumference of the circle is approximately **326.86 cm**.