Cylinder Calculator: Volume, Lateral & Total Surface Area

The four cylinder measures

A circular cylinder is defined by its base radius $r$ and its height $h$ — two equal circular bases, parallel and directly stacked, connected by a curved side. Given $r$ and $h$, every measure follows directly:

$V = \pi r^2 h \qquad L = 2\pi r h \qquad B = \pi r^2 \qquad A = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)$

Volume is the base's area ($\pi r^2$, the circle formula) times the height — the "base × height" rule shared by every prism. Lateral area comes from "unrolling" the curved side into a flat rectangle: width $2\pi r$ (the base's circumference) and height $h$.

Worked example: r = 3, h = 10

$V = \pi(9)(10) = 90\pi \approx 282.7433 \qquad L = 2\pi(3)(10) = 60\pi \approx 188.4956$ $B = \pi(9) \approx 28.2743 \qquad A = 2\pi(3)(13) = 78\pi \approx 245.0442$

Because $r = 3$ and $h = 10$ are both exact whole numbers here, volume, lateral area, and total surface area all have an EXACT symbolic form ($90\pi$, $60\pi$, $78\pi$) in addition to the decimal approximation.

Worked example: r = 1, h = 1

$V = \pi(1)(1) = \pi \approx 3.1416 \qquad L = 2\pi(1)(1) = 2\pi \approx 6.2832$ $B = \pi \approx 3.1416 \qquad A = 2\pi(1)(2) = 4\pi \approx 12.5664$

A unit cylinder ($r = h = 1$) makes the base area and volume come out to the same number ($\pi$), which is easy to mistake for a pattern — it's only because $r^2h = 1$ here, not a general rule.

Solving in the other direction

Common mistakes

• Forgetting the two bases. Total surface area is $A = L + 2B$, not just the lateral area $L$.
• Confusing lateral area with total surface area. $L = 2\pi rh$ never includes the top and bottom circles by itself.
• Mixing up radius and diameter. Every formula here uses radius $r$, not diameter — halve a diameter measurement before using these formulas.

Where this shows up

• Cans, pipes, tanks, columns: volume tells you capacity; lateral area tells you how much material (a label, paint, insulation) wraps around the side.
• Can labels: the label wrapped around a can is exactly the lateral surface area, $2\pi rh$ — not the total surface area, since the label doesn't cover the top or bottom.
• Building block for other solids: a sphere capped cylinder and a hollow tube both extend the same base formulas.

A cylinder is the natural 3D extension of the circle along a height — knowing just the radius and height is always enough to determine volume, lateral area, base area, and total surface area, each built directly from the circle's own $\pi r^2$ and $2\pi r$ terms.