The volume of a sphere is `36 pi cm^(3)`. Find the surface area of the sphere.

Answer

To find the surface area of a sphere given its volume, we can follow these steps:

### Step 1: Write down the formula for the volume of a sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Set the volume equal to the given value.
We know the volume of the sphere is \( 36 \pi \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 36 \pi \]

### Step 3: Simplify the equation.
To simplify, we can divide both sides by \( \pi \):
\[ \frac{4}{3} r^3 = 36 \]

### Step 4: Solve for \( r^3 \).
Next, we multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 36 \times \frac{3}{4} \]
Calculating the right side:
\[ r^3 = 27 \]

### Step 5: Find the radius \( r \).
Now, take the cube root of both sides to find \( r \):
\[ r = \sqrt[3]{27} = 3 \, \text{cm} \]

### Step 6: Write down the formula for the surface area of a sphere.
The formula for the surface area \( A \) of a sphere is:
\[ A = 4 \pi r^2 \]

### Step 7: Substitute the value of \( r \) into the surface area formula.
Substituting \( r = 3 \, \text{cm} \):
\[ A = 4 \pi (3)^2 \]
Calculating \( (3)^2 \):
\[ A = 4 \pi \times 9 \]
\[ A = 36 \pi \, \text{cm}^2 \]

### Step 8: If required, use \( \pi \approx \frac{22}{7} \) to find a numerical value.
Using \( \pi \approx \frac{22}{7} \):
\[ A = 36 \times \frac{22}{7} = \frac{792}{7} \, \text{cm}^2 \]
Calculating the division:
\[ A \approx 113.14 \, \text{cm}^2 \]

### Final Answer:
The surface area of the sphere is approximately \( 113.14 \, \text{cm}^2 \).

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