To determine the curved surface area of the cylinder, first calculate its height using the volume formula, then apply the curved surface area formula. The volume \( V \) of a cylinder is defined as:
\[ V = \pi r^2 h \]
where \( r \) denotes the radius and \( h \) denotes the height. Given \( V = 396 \, \text{cm}^3 \) and \( r = 3 \, \text{cm} \), substitute these values to solve for \( h \):
\[ 396 = \pi \times 3^2 \times h \]
\[ 396 = 9\pi h \]
\[ h = \frac{396}{9\pi} \]
Approximating \(\pi\) as 3.14 for simplification:
\[ h = \frac{396}{28.26} \approx 14 \, \text{cm} \]
Next, compute the curved surface area (CSA) using the formula:
\[ \text{CSA} = 2\pi rh \]
Substitute \( r = 3 \, \text{cm} \) and \( h = 14 \, \text{cm} \):
\[ \text{CSA} = 2 \times \pi \times 3 \times 14 \]
\[ \text{CSA} = 84\pi \]
Using the approximation \(\pi \approx 3.14\), calculate the CSA:
\[ \text{CSA} = 84 \times 3.14 = 263.76 \approx 264 \, \text{cm}^2 \]
Consequently, the curved surface area of the cylinder is \( 264 \, \text{cm}^2 \).