The volume of a cylinder is a circle's area stretched along a straight line. First you find the area of the circular base, πr², then you multiply by the height, and the circle sweeps out the whole solid: V = πr²h. That is the entire formula, and it never changes whether the cylinder is a coffee mug, a rain barrel, a concrete form, or a car engine's cylinder. Square the radius, multiply by pi, multiply by height, in that order, and the only decisions left are units and conversions.

Start with something you can hold: a coffee mug with an inside radius of 1.6 inches and an inside height of 3.75 inches. Square the radius to get 2.56, multiply by π to get a base area of 8.04 square inches, and multiply by the height to get 30.16 cubic inches. Cubic inches mean little to a coffee drinker, so convert: one US fluid ounce is 1.8047 cubic inches, and 30.16 ÷ 1.8047 = 16.71 fluid ounces. That is a 16-ounce mug with a little headroom, exactly what the mug was designed to be. Notice the step that matters: measure the inside dimensions when capacity is the question. The outside of that mug might be 1.8 inches in radius, and the extra two-tenths of an inch of ceramic would inflate the volume by more than 25 percent. Wall thickness is invisible until it ruins your answer.

Scale up to the backyard. A rain barrel with an inside radius of 9 inches and a height of 24 inches holds π × 81 × 24 = 6,107.26 cubic inches. Divide by 231, the number of cubic inches in a US gallon, and the barrel holds 26.44 gallons. That number then turns into weight, which is the number that actually constrains where the barrel can sit: fresh water weighs 8.34 pounds per gallon, so a full barrel weighs 26.44 × 8.34 = 220.5 pounds before you count the barrel itself. A deck, a stand, or a dolly rated for 150 pounds will not hold it. Volume problems are rarely just volume problems; the answer usually feeds a weight, a cost, or a structural limit, and this calculator's volume output is the first link in that chain.

Concrete work shows the same chain in a different trade. A Sonotube form 12 inches across and 48 inches tall, the kind used for deck footings and fence posts, has a radius of 6 inches. Its volume is π × 36 × 48 = 5,428.67 cubic inches. Divide by 1,728, the cubic inches in a cubic foot, and the footing needs 3.14 cubic feet of concrete. Bagged concrete mix is usually sold by the cubic foot, often 0.6 cubic feet per 80-pound bag, so 3.14 ÷ 0.6 = 5.24 bags, and you buy 6, because a half-mixed footing cannot wait while you drive back to the store. Rounding up on materials is not sloppy; it is the correct answer to a different question, how much to buy, which is almost always the question actually being asked.

Car engines are cylinders too, and their size is quoted in liters or cubic centimeters for exactly this reason. Take a four-cylinder engine with a bore, the cylinder's diameter, of 3.4 inches and a stroke, the piston's travel distance, of 3.6 inches. The radius is half the bore, 1.7 inches. One cylinder's volume is π × 2.89 × 3.6 = 32.69 cubic inches. Four cylinders give 130.74 cubic inches. Multiply by 16.3871 cubic centimeters per cubic inch and the engine displaces 2,142 cubic centimeters, which the badge rounds to 2.1 liters. When a review says a car has a 2.0-liter engine, it is reporting this exact calculation summed across its cylinders. The same formula sizes motorcycle engines in cc, boat engines in cubic inches, and industrial compressors in displacement per revolution.

Conversions are where cylinder volume problems are won or lost, so here is the full set in one place. Cubic inches to US gallons: divide by 231. Cubic feet to US gallons: multiply by 7.48052. Cubic inches to liters: multiply by 0.0163871. Cubic centimeters to cubic inches: divide by 16.3871. Cubic feet to cubic inches: multiply by 1,728. And the imperial gallon, used in the UK, is 277.42 cubic inches, about 20 percent larger than the US gallon, so a tank rated at 26 imperial gallons holds about 31 US gallons. If a specification came from a British supplier, check which gallon it means before ordering a pump or a heater sized to it. The formula itself never changes; only the final conversion does, and doing the conversion last, after the volume is computed in one consistent unit, keeps the arithmetic clean.

Scaling behaves differently for volume than it does for surface area, and the difference is worth knowing. A cylinder with a radius of 2 and a height of 10 holds π × 4 × 10 = 125.66 cubic units. Double the radius to 4, keeping the height at 10, and the volume becomes π × 16 × 10 = 502.65 cubic units, four times as much, because the radius is squared. Double the height instead, keeping the radius at 2, and the volume merely doubles to 251.33. This is why a slightly wider tank holds dramatically more than a slightly taller one, why a big-bore engine gains displacement faster than a long-stroke engine, and why a drinking glass that looks only a little wider can hold a lot more than you expect. It is also why the radius deserves your most careful measurement: an error in the radius is squared, while an error in the height is not.

Two traps deserve explicit warnings. The first is the diameter trap, the same one that haunts surface area: entering the diameter as the radius quadruples the volume. A 12-inch-diameter form entered as radius 12 instead of 6 would report four times the concrete, and you would order 24 bags for a 6-bag footing. The second trap is the hollow-versus-solid confusion. A pipe's capacity uses its inside radius; a solid rod's volume uses its outside radius; a pipe's material volume is the difference between the two cylinders, outer minus inner. If you are sizing how much water a pipe holds, measure inside. If you are weighing the pipe itself, compute both and subtract. And a horizontal tank that is only partly full is a different problem entirely: the filled cross-section is a circular segment, not a full circle, so the full-cylinder formula gives only the brim-full capacity, and partial fills need the segment formula.

Rounding belongs at the end of the calculation, not in the middle. Carry π at full precision through the squaring and multiplying, convert, and only then round to a sensible number of digits: whole gallons for a tank, tenths of a fluid ounce for a mug, whole cubic feet for concrete. Rounding the radius from 1.7 to 2 before squaring would turn the engine example from 2.14 liters into a fantasy. And when the answer looks wrong, check three things in order: did you halve the diameter, are radius and height in the same unit, and did you convert at the end rather than in the middle. Nine times out of ten, one of those three is the culprit.

One more distinction saves confusion at the store: nominal capacity versus brim-full capacity. A bucket sold as a 5-gallon bucket, about 11 inches across and 15 inches tall, has a brim-full volume of π × 30.25 × 15 = 1,425.50 cubic inches, which is 6.17 gallons. The 5-gallon rating is the safe working fill, not the brim, because a bucket filled to the brim cannot be carried without spilling. The same gap appears at industrial scale: the classic 55-gallon drum, 23 inches across and 35 inches tall, holds π × 132.25 × 35 = 14,541.65 cubic inches brim-full, or 62.95 gallons, while its rated capacity is 55 gallons, leaving headspace for expansion, safe handling, and the liquid's vapor. When a specification says a container holds a certain number of gallons, ask whether that is the brim-full geometry or the rated fill, and use this calculator's volume output as the brim-full truth to compare against.

To use this calculator, type the radius in the Radius (r) box and the height in the Height (h) box, then read the Volume, Total Surface Area, Lateral Surface Area, and Diameter outputs together. The diameter output is your built-in trap detector: if it does not match your tape measure, you entered the wrong radius. Use the volume output for capacity, the surface outputs for paint and material, and the conversions above to speak in gallons, liters, or cubic feet as the job requires. Sanity anchors: a gallon jug is 231 cubic inches, a 16-ounce mug is about 29 cubic inches, a 55-gallon drum is about 12,705 cubic inches. If your answer is nowhere near the anchor for a similar container, recheck the radius before anything else. For spheres, cones, and boxes, the companion calculators on this site run the same radius-and-height style of problem for those shapes.

With the pre-filled defaults — Radius (r) 3, Height (h) 8 — the Volume of Cylinder Calculator returns Volume 226.1947 units³, Total Surface Area 207.3451 units², Lateral Surface Area 150.7964 units², Diameter 6.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.