Find the diameter of the circle whose circumference is equal to the sum of the circumference of circles with radii 5 cm, 8 cm and 10 cm.

Answer

To find the diameter of the circle whose circumference is equal to the sum of the circumferences of circles with radii 5 cm, 8 cm, and 10 cm, we can follow these steps:

### Step 1: Calculate the Circumference of Each Circle
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

1. **For the first circle (radius = 5 cm):**
\[ C_1 = 2\pi \times 5 = 10\pi \text{ cm} \]

2. **For the second circle (radius = 8 cm):**
\[ C_2 = 2\pi \times 8 = 16\pi \text{ cm} \]

3. **For the third circle (radius = 10 cm):**
\[ C_3 = 2\pi \times 10 = 20\pi \text{ cm} \]

### Step 2: Sum the Circumferences
Now, we can find the total circumference of the three circles:
\[ C_{\text{total}} = C_1 + C_2 + C_3 = 10\pi + 16\pi + 20\pi \]
\[ C_{\text{total}} = (10 + 16 + 20)\pi = 46\pi \text{ cm} \]

### Step 3: Set the Total Circumference Equal to the Circumference of the Larger Circle
Let \( C_{\text{big}} \) be the circumference of the larger circle. According to the problem, we have:
\[ C_{\text{big}} = 46\pi \text{ cm} \]

### Step 4: Use the Circumference Formula to Find the Radius of the Larger Circle
Using the circumference formula for the larger circle:
\[ C_{\text{big}} = 2\pi R \]
where \( R \) is the radius of the larger circle. Setting this equal to our previous result:
\[ 2\pi R = 46\pi \]

### Step 5: Solve for the Radius \( R \)
To find \( R \), divide both sides by \( 2\pi \):
\[ R = \frac{46\pi}{2\pi} = \frac{46}{2} = 23 \text{ cm} \]

### Step 6: Calculate the Diameter
The diameter \( D \) of the circle is given by:
\[ D = 2R \]
Substituting the value of \( R \):
\[ D = 2 \times 23 = 46 \text{ cm} \]

### Final Answer
The diameter of the circle is \( 46 \text{ cm} \).

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