The diameter of two circles are 32 cm and 24 cm. Find the radius of the circle having its area equal to sum of the areas of the two given circle.

Answer

To solve the problem step by step, we will find the radius of a new circle whose area is equal to the sum of the areas of two given circles with diameters of 32 cm and 24 cm.

### Step 1: Find the radii of the two circles.
- For Circle C1 (diameter D1 = 32 cm):
\[ r_1 = \frac{D_1}{2} = \frac{32 \, \text{cm}}{2} = 16 \, \text{cm} \]

- For Circle C2 (diameter D2 = 24 cm):
\[ r_2 = \frac{D_2}{2} = \frac{24 \, \text{cm}}{2} = 12 \, \text{cm} \]

### Step 2: Calculate the areas of the two circles.
- Area of Circle C1:
\[ \text{Area}_{C1} = \pi r_1^2 = \pi (16 \, \text{cm})^2 = 256\pi \, \text{cm}^2 \]

- Area of Circle C2:
\[ \text{Area}_{C2} = \pi r_2^2 = \pi (12 \, \text{cm})^2 = 144\pi \, \text{cm}^2 \]

### Step 3: Find the sum of the areas of the two circles.
\[ \text{Total Area} = \text{Area}_{C1} + \text{Area}_{C2} = 256\pi + 144\pi = 400\pi \, \text{cm}^2 \]

### Step 4: Set the area of the new circle equal to the total area.
Let the radius of the new circle (C3) be \( r_3 \). The area of Circle C3 is given by:
\[ \text{Area}_{C3} = \pi r_3^2 \]
Setting this equal to the total area:
\[ \pi r_3^2 = 400\pi \]

### Step 5: Solve for \( r_3 \).
Dividing both sides by \( \pi \):
\[ r_3^2 = 400 \]
Taking the square root of both sides:
\[ r_3 = \sqrt{400} = 20 \, \text{cm} \]

### Final Answer:
The radius of the new circle is \( 20 \, \text{cm} \).

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