How this instrument works
A right cylinder is two parallel circles of equal radius joined by a straight vertical wall — the axis meets each base at a square angle, so the shape never leans. Because every horizontal slice through it is the same circle, its volume is just that circle's area repeated h times: V = πr²h, the same base-times-height rule that gives a volume for a box or any other prism.
The lateral area follows from a simpler trick than it looks. Slit the curved wall along one vertical line and flatten it out: it unrolls into a flat rectangle whose width is the circle's own circumference, 2πr, and whose height is h. Multiply those two sides and you get A_L = 2πrh — a curved surface turned into ordinary rectangle math. Total surface area then adds the two flat caps back on, A = A_L + 2πr², one πr² for the top and one for the bottom.
A case worth noticing: let the height shrink to zero and the cylinder flattens into a disk. Volume and lateral area both vanish, since there is no room inside and no wall left to unroll — but total surface area does not drop to zero. It settles at 2πr², because the calculator is still counting a top face and a bottom face even once they sit directly on top of each other.
- Enter the cylinder's radius into the Radius field — the distance from the central axis out to the curved wall.
- Enter the straight-line gap between the two circular ends into the Height field.
- Read Volume for how much the cylinder holds, in cubic units.
- Read Lateral area for the curved wall alone — the size of a label wrapped around a can with no top or bottom.
- Read Total surface area for the lateral area plus both circular ends combined, the full sheet of material needed to build the solid.
Worked example — a radius-3, height-4 drum
Take a squat storage drum with radius r = 3 m and height h = 4 m. Volume: V = π × 3² × 4 = 36π ≈ 113.097336 cubic metres, the capacity an engineer needs before sizing a pump or checking it against a delivery truck's tank.
The same two numbers give the metal. Lateral area — the curved wall alone, which would unroll into a flat rectangle about 18.850 m wide (the circumference, 2π × 3) and 4 m tall — comes to A_L = 2π × 3 × 4 = 24π ≈ 75.398224 square metres. Add the top and bottom disks, 2π × 3² = 18π ≈ 56.548668 square metres, and the total surface area is A = 24π + 18π = 42π ≈ 131.946891 square metres, the full sheet a fabricator would need to cut and weld.
Questions
How is the volume formula V = πr²h derived?
It is the same rule that gives the volume of any prism: base area times height. Every horizontal cross-section of a right cylinder is a congruent circle of area πr², so stacking h units of that identical circle straight up gives V = πr²h with no calculus required.
Why does lateral area use the circumference instead of the radius directly?
Cut the curved wall along one vertical line and flatten it: it unrolls into a plain rectangle. One side is the height h, the other is the circle's own circumference, 2πr, so the area is base times height, A_L = 2πrh — the same rectangle formula, just applied to an unrolled surface.
What is the most common mistake when finding total surface area?
Adding only one circular base instead of two. Total area is the lateral area plus both flat ends, A = 2πrh + 2πr², so counting a single πr² undercounts by a full base — an error that grows large relative to the total on short, wide shapes like coins or hockey pucks.
What happens to volume and surface area when the height is zero?
Volume and lateral area both fall to zero, since a height-zero cylinder is a flat disk with no room inside and no curved wall to unroll. Total surface area holds at 2πr², though, because the top and bottom circles are still both counted even after they come to sit on top of each other.
How does a right cylinder relate to a sphere of the same radius?
Archimedes showed that a sphere of radius r fits snugly inside a cylinder of radius r and height 2r, and that the cylinder's lateral area then equals the sphere's full surface area exactly, both 4πr². He rated this his finest result and asked for the figure to be cut into his tombstone.