Find the area of the circle, length of whose circumference is equal to the sum of the lengths of the circumferences of circles with radii 15 cm and 13 cm.

Answer

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

### Step 1: Find the circumferences of the two circles.
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

For the first circle with a radius of 15 cm:
\[ C_1 = 2\pi \times 15 = 30\pi \text{ cm} \]

For the second circle with a radius of 13 cm:
\[ C_2 = 2\pi \times 13 = 26\pi \text{ cm} \]

### Step 2: Calculate the total circumference.
Now, we can find the total circumference of the two circles:
\[ C_{\text{total}} = C_1 + C_2 = 30\pi + 26\pi = 56\pi \text{ cm} \]

### Step 3: Set the total circumference equal to the circumference of the third circle.
Let \( R \) be the radius of the third circle. The circumference of the third circle is:
\[ C_3 = 2\pi R \]
Setting the total circumference equal to the circumference of the third circle gives us:
\[ 2\pi R = 56\pi \]

### Step 4: Solve for \( R \).
We can divide both sides by \( 2\pi \):
\[ R = \frac{56\pi}{2\pi} = 28 \text{ cm} \]

### Step 5: Calculate the area of the third circle.
The area \( A \) of a circle is given by the formula:
\[ A = \pi R^2 \]
Substituting \( R = 28 \) cm:
\[ A = \pi \times (28)^2 = \pi \times 784 \]

### Step 6: Substitute the value of \( \pi \) and calculate the area.
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times 784 \]
Calculating this gives:
\[ A = \frac{22 \times 784}{7} = 22 \times 112 = 2464 \text{ cm}^2 \]

### Final Answer:
The area of the circle is \( 2464 \text{ cm}^2 \).

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