Find the radius of the circle whose area is equal to the sum of the areas of three smaller circles of radii 3 cm , 4 cm and 12 cm .

Answer

To find the radius of the circle whose area is equal to the sum of the areas of three smaller circles with radii 3 cm, 4 cm, and 12 cm, we can follow these steps:

### Step 1: Calculate the area of each smaller circle.
The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

- For the first circle with radius 3 cm:
\[ A_1 = \pi (3^2) = \pi \times 9 = 9\pi \, \text{cm}^2 \]

- For the second circle with radius 4 cm:
\[ A_2 = \pi (4^2) = \pi \times 16 = 16\pi \, \text{cm}^2 \]

- For the third circle with radius 12 cm:
\[ A_3 = \pi (12^2) = \pi \times 144 = 144\pi \, \text{cm}^2 \]

### Step 2: Sum the areas of the three smaller circles.
Now, we add the areas of the three circles:
\[ \text{Total Area} = A_1 + A_2 + A_3 = 9\pi + 16\pi + 144\pi \]
\[ \text{Total Area} = (9 + 16 + 144)\pi = 169\pi \, \text{cm}^2 \]

### Step 3: Set the area of the larger circle equal to the total area of the smaller circles.
Let the radius of the larger circle be \( R \). The area of the larger circle is:
\[ A = \pi R^2 \]
Setting this equal to the total area we found:
\[ \pi R^2 = 169\pi \]

### Step 4: Simplify the equation.
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ R^2 = 169 \]

### Step 5: Solve for \( R \).
Taking the square root of both sides gives:
\[ R = \sqrt{169} = 13 \, \text{cm} \]

### Conclusion:
The radius of the circle whose area is equal to the sum of the areas of the three smaller circles is \( 13 \, \text{cm} \).

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