How this instrument works
A cylinder's total surface area is what you get when you cut the shape open and lay every piece flat. Unroll the curved wall along a single vertical seam and it becomes a rectangle exactly 2πr wide — the base circle's own circumference — and h tall, with no stretching or distortion, since a cylinder is one of the developable surfaces that flattens cleanly. The two ends peel away separately as an ordinary pair of circles, each of radius r. Add the rectangle's area, 2πrh, to the pair of circles, 2πr², and every scrap of the cylinder's skin is accounted for once, with nothing hidden and nothing counted twice.
This is a different figure from a cylinder's lateral area alone, which stops at the curved wall and leaves the two circular ends out entirely. The distinction matters in practice, not just on paper: a paper label wrapped around a can only needs the lateral term, but painting the whole can, plating a roller in metal, or working out how much sheet stock a closed drum consumes all need the caps folded back in — the two circles a lateral-only figure quietly leaves off.
Which term dominates depends on proportion, not overall size. A short, wide cylinder — a hockey puck's shape — carries most of its surface in the two caps, while a tall, narrow one — a length of pipe — carries almost all of it in the curved wall, with the caps shrinking toward a negligible sliver as the height stretches on. That tension is also why the calculus problem of minimizing a can's surface for a fixed volume has a clean answer, height equal to twice the radius, that most real cans quietly ignore in favor of standard sheet sizes and easier stamping.
- Enter the cylinder's Radius in the Radius field, using any length unit you like.
- Enter the Height in that same unit — the sheet assumes Radius and Height share one unit system.
- Read Total surface area for the combined figure: curved wall plus both circular caps, reported in your unit squared.
- To check a physical object, measure radius and height with calipers or a tape before typing the numbers in.
Worked example — radius 3, height 4
Take a cylinder with radius r = 3 and height h = 4, roughly the shape of a squat paint can, measured in whatever unit you like. The lateral wall alone is 2π × 3 × 4 = 24π ≈ 75.398224 square units, and the two caps add 2π × 3² = 18π ≈ 56.548668 square units on top of that.
Add the two pieces together and the total exterior is 24π + 18π = 42π ≈ 131.946891 square units — the exact figure this sheet returns for area given those same inputs, carried to full double precision rather than rounded at each step. Leave off the caps and you would understate the material by the full 18π ≈ 56.548668 square units, nearly as large as the curved wall's own contribution.
Questions
What is the formula for the total surface area of a cylinder?
A = 2πrh + 2πr², where r is the base radius and h is the height. The first term, 2πrh, is the curved wall unrolled into a rectangle; the second, 2πr², is the pair of flat circular caps. Add them and every surface of the cylinder is covered exactly once.
How is this different from a cylinder's lateral surface area?
Lateral surface area reports the curved wall alone, 2πrh, and stops there — useful for a label or a rolled sheet of ducting. This total figure adds the two circular caps, 2πr², on top, which is what you need for painting, plating, or otherwise covering a cylinder's entire outside, ends included.
Why does unrolling the cylinder explain the formula?
Cut the curved wall along one vertical line and it flattens into a rectangle whose width is the base's own circumference, 2πr, and whose height is h, giving area 2πrh with no distortion since a cylinder unrolls without stretching. The two ends peel off separately as circles of area πr² each, contributing 2πr² together.
Which real cylinder proportions minimize surface area for a given volume?
Calculus shows the minimum-surface cylinder for a fixed volume has height equal to twice the radius, h = 2r — a genuinely squat shape, not the tall can most people picture. Real cans rarely hit that ratio, since sheet-metal stamping, stacking, and shelf display all pull the proportions elsewhere.
What happens to the total surface area as the height shrinks to zero?
The lateral term 2πrh vanishes and the total collapses to 2πr², the area of the two caps alone — sensible, since a cylinder with zero height is just a flat disk with a front and back but no wall between them. Push the radius to zero instead and the whole surface shrinks to zero, since no wall or cap remains.
Which unit should I use for radius and height?
Any consistent length unit works, since the formula combines areas built from that one length: centimetres in give square centimetres out, inches in give square inches out. Keep radius and height in the same unit before entering them, or the two terms in the formula will disagree with each other.