Surface Area of a Cylinder

The surface area of a cylinder is the total area of its two circular bases plus its curved side (the lateral surface). For a cylinder with radius r and height h, the formula is SA = 2πr² + 2πrh — the two circles (2πr²) plus the rectangle that wraps around the side (2πrh).

Each circular base has area πr², so two bases give 2πr². The curved side, when unrolled, becomes a rectangle: its height is the cylinder's height h, and its width is the base circle's circumference, 2πr. That rectangle's area is 2πr×h = 2πrh. Adding the two parts gives the full formula.

The volume of a cylinder is V = πr²h — the area of the circular base times the height, the same base-times-height idea used for a pyramid's volume, just without the one-third factor since a cylinder's cross-section stays constant all the way up.

Sometimes a problem asks only for the lateral surface area — the curved side without the two circles — which is simply 2πrh. This comes up for objects open at both ends, like a pipe or a tube.

For a cylinder with radius r = 3 and height h = 10: SA = 2π(3²) + 2π(3)(10) = 18π + 60π = 78π ≈ 245.0 square units. Its volume is V = π(3²)(10) = 90π ≈ 282.7 cubic units.

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