Peel the label off a soup can and it unrolls into a perfect rectangle. That one observation is the whole secret of cylinder surface area: the curved side of any cylinder is a rectangle in disguise, its width equal to the circumference of the base and its height equal to the cylinder's height. Multiply those two and you have the lateral surface area, 2πrh. Then add the two circular ends, each πr², and you have the total surface area: SA = 2πrh + 2πr². Factor out 2πr and the same formula reads SA = 2πr(r + h), which is the form many people find easiest to remember: two pi times radius times radius-plus-height.

The formula has exactly two parts, and keeping them separate is what keeps you out of trouble. The lateral part, 2πrh, answers questions about labels, wraps, paint on the curved side, and insulation around a pipe. The ends part, 2πr², answers questions about lids, caps, and the metal in the top and bottom of a can. Most real jobs need only one of the two, and most expensive mistakes come from adding both when only one was asked for, or forgetting one when both were needed. A label printer who quotes the total surface area for a label job overcharges by the area of two lids; a painter who quotes only the lateral area for a closed tank runs short on the job.

Start with the measurement that trips up more people than any other: radius versus diameter. Tape measures give you diameter, the full width across the circle. The formula wants radius, half of that. A storage tube 10 inches across and 14 inches tall has a radius of 5 inches, not 10. Run the numbers with 5 and you get a lateral area of 2π × 5 × 14 = 439.82 square inches, ends of 2π × 25 = 157.08 square inches, and a total of 596.90 square inches. Enter 10 as the radius by mistake and every part of the answer quadruples or doubles into nonsense: the lateral area alone would read 879.65 square inches. Since the radius is squared in the ends term, a diameter-as-radius error multiplies the ends by four. Always halve the diameter first, before you touch the calculator.

Now a bigger, dirtier, real-world cylinder: a standard 55-gallon steel drum, 23 inches across and 35 inches tall. Halve the diameter to get a radius of 11.5 inches. The lateral area is 2π × 11.5 × 35 = 2,528.98 square inches, the two ends together are 2π × 132.25 = 830.95 square inches, and the total outside surface is 3,359.93 square inches. Divide by 144 to speak in square feet, the unit paint and powder-coating shops quote in, and the drum is 23.33 square feet of steel to coat. That number is what a refinishing shop uses to price the job, and it is also the number that tells you a single spray can, which covers roughly 25 square feet in the manufacturer's fantasy and far less in practice, will not quite do the whole drum.

Not every cylinder is closed. A planter, a bucket, a grain bin with its top off, or a drinking glass has one end open, and the formula sheds one circle. A cylindrical planter with a radius of 1.5 feet and a height of 2 feet needs lateral area of 2π × 1.5 × 2 = 18.85 square feet plus a single base of π × 2.25 = 7.07 square feet, for 25.92 square feet of material. An open pipe or a section of ducting with both ends open needs the lateral area only. And a hollow pipe with thick walls is really two cylinders, an outer one and an inner one, each with its own lateral area; for thin-walled tubing the two are close enough that one calculation suffices, but for a thick concrete pipe the inner surface can be enough smaller to matter for lining estimates. State which version you are computing before you start, because closed, open-top, and open-both-ends answers for the same cylinder differ by whole circles.

Paint is where most homeowners meet this formula, and it is worth doing once all the way through. Take a cylindrical water tank with a radius of 3 feet and a height of 8 feet. The lateral area is 2π × 3 × 8 = 150.80 square feet, the two ends add 2π × 9 = 56.55 square feet, and the total is 207.35 square feet. Two coats double that to 414.70 square feet. At a typical coverage of 350 square feet per gallon, that is 414.70 ÷ 350 = 1.18 gallons, so you buy 2 gallons, not 1: paint is sold in whole gallons, and the second coat always finds the thin spots the first coat missed. If the tank sits on the ground and its bottom never gets painted, subtract one end, 28.27 square feet, and the two-coat total drops to about 358 square feet, still just over a gallon. The formula tells you the paint; the job site tells you which ends count.

Packaging runs on the same arithmetic at a smaller scale. A candle jar with a radius of 1.75 inches and a height of 3.5 inches has a lateral area of 2π × 1.75 × 3.5 = 38.48 square inches, which is the label size before you add a small overlap for the seam, usually a quarter inch of extra wrap. Its total surface, ends included, is 38.48 + 2π × 3.0625 = 57.73 square inches, the glass area the manufacturer is paying for. Label stock is priced by the square inch in bulk, so a candle line selling a hundred thousand jars cares about that 38.48 number to the penny: a tenth of a square inch per label is ten thousand square inches of stock across the run. The same logic sizes soup-can labels, wine-bottle capsules, and the shrink sleeves on sports-drink bottles.

Scaling is the part that surprises people, and it is worth feeling once. Take a cylinder with a radius of 2 and a height of 6: lateral area 75.40, ends 25.13, total 100.53 square units. Double the radius to 4, keeping the height at 6: the lateral area doubles to 150.80, because it grows with the radius, but the ends quadruple to 100.53, because the radius is squared there, and the total lands at 251.33, two and a half times the original rather than twice. Double the height instead, keeping the radius at 2, and only the lateral part doubles; the ends do not move at all. This is why a tall skinny tank and a short fat tank with the same volume can need very different amounts of paint, and why engineers chasing heat loss or material savings care about shape, not just size. A sphere beats a cylinder on surface per volume, but a cylinder beats a box, which is why tanks are round.

Units deserve one careful paragraph because they are the quiet killer of good answers. If radius and height are both in inches, the answer is in square inches; divide by 144 for square feet. If both are in centimeters, divide by 10,000 for square meters. Mixing is the classic failure: a radius in inches with a height in feet gives a number with no meaning at all. Convert first, calculate second. When a European supplier quotes a tank in meters and your paint is rated in square feet per gallon, convert the dimensions to feet before the formula, not the answer after it, because a unit error inside a squared term does not scale back out cleanly. And round only at the end: carrying 2.52898 through intermediate steps and rounding the final answer to two decimals beats rounding every step, which can drift a paint order by a noticeable margin on a big tank.

The formula also shows up in places that do not look like geometry homework. Plumbers size pipe insulation by lateral area, since insulation wraps the curved side and never covers the ends. Farmers estimate the steel in a grain bin or the liner in a silo by total surface area. HVAC technicians compute the outside area of round ductwork to estimate heat loss, because heat escapes through surface, and a duct's surface per foot of run is simply its circumference, 2πr, times the length. Food engineers use surface area to predict how fast a canned product heats during sterilization, since heat enters through the surface. Even the humble toilet-paper roll is a lateral-area problem: the paper area is the circumference times the width of the sheet, wrapped many times over.

To use this calculator, type the radius in the Radius (r) box and the height in the Height (h) box, then read the Total Surface Area, Lateral Surface Area, and Volume outputs together. Use the lateral output for labels, wraps, and insulation; use the total output for closed containers, paint jobs, and material orders; use the volume output when the same cylinder also needs a capacity check. Sanity-check your answer against the anchors on this page: a drum-sized cylinder should land near 3,360 square inches, a small tank near 207 square feet, a candle jar near 38 square inches of side. If your answer is four times too big, you entered the diameter. If it is missing a chunk, you forgot an end. And if you need the surface area of a sphere, a cone, or a rectangular box instead, the companion calculators on this site cover those shapes with the same radius-and-height style inputs.

With the pre-filled defaults — Radius (r) 3, Height (h) 5 — the Surface Area of Cylinder returns Total Surface Area 150.7964 m², Lateral Surface Area 94.2478 m², Volume 141.3717 m³. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.