Find the radius of a circle having area equal to the sum of the areas of two circles with radius 20 cm and 15 cm respectively.

Answer

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

### Step 1: Calculate the area of the first circle
The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
For the first circle with a radius of 20 cm:
\[ A_1 = \pi (20)^2 = \pi \times 400 = 400\pi \text{ cm}^2 \]

### Step 2: Calculate the area of the second circle
Now, we calculate the area of the second circle with a radius of 15 cm:
\[ A_2 = \pi (15)^2 = \pi \times 225 = 225\pi \text{ cm}^2 \]

### Step 3: Find the sum of the areas of the two circles
Now, we add the areas of the two circles:
\[ A_{\text{total}} = A_1 + A_2 = 400\pi + 225\pi = 625\pi \text{ cm}^2 \]

### Step 4: Set the area of the new circle equal to the total area
Let \( r \) be the radius of the new circle. The area of this new circle can be expressed as:
\[ A_{\text{new}} = \pi r^2 \]
We set this equal to the total area we calculated:
\[ \pi r^2 = 625\pi \]

### Step 5: Solve for \( r^2 \)
We can divide both sides of the equation by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r^2 = 625 \]

### Step 6: Find the radius \( r \)
To find \( r \), we take the square root of both sides:
\[ r = \sqrt{625} = 25 \text{ cm} \]

### Final Answer
The radius of the circle is \( 25 \) cm.

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