Volume Calculator

Volume formulas

Every formula below is the one this calculator actually applies — the table is generated from the engine at build time, so it cannot drift from the working shown in a result. Choose a row to open that shape with the formula already loaded.

π is the ratio of a circle’s circumference to its diameter, about 3.14159. Where a result is a clean multiple of π the calculator shows the exact form (for example 20π m³) alongside the decimal.

How to calculate volume

1. Identify the shape. Decide which solid your object actually is. A tin is a cylinder; a shoebox is a rectangular prism; a party hat is a cone; a plant pot is usually a conical frustum, not a cylinder. Getting this right matters more than anything that follows.
2. Measure the dimensions the formula needs. A cylinder needs a radius (or a diameter, or a circumference) and a height — nothing else. Heights are measured perpendicular to the base, never along a slanted side. Write down the unit with every number.
3. Apply the formula and report the answer in cubic units. Because volume multiplies three lengths, the unit is cubed. Measure in centimetres and the answer is in cubic centimetres; measure in feet and it is cubic feet.

Composite objects can be split into simpler solids and their volumes added or subtracted. A silo is a cylinder plus a hemisphere. An L-shaped room is two boxes. A washer is a disc with a smaller disc taken out of it, which is exactly what the tube formula does. Calculate each part on its own, then add the parts that are there and subtract the parts that are not — the arithmetic is easier and the mistakes are easier to spot.

Cylinder volume

The volume of a cylinder is V = πr²h.

The reason that formula looks the way it does is worth understanding, because it makes the whole family of prisms obvious: volume = base area × height. A cylinder is a circle carried straight up. The base area = πr², and multiplying it by the height h stacks that circle all the way to the top. The same reasoning gives l × w × h for a box and A × h for any prism at all.

Radius, diameter or circumference

You rarely measure a radius directly — the centre of a drum is the one place a tape measure will not stay. This calculator therefore accepts whichever you can actually get:

• Radius (r) — used directly.
• Diameter (d) — the width straight across, through the centre. The calculator uses r = d ÷ 2 and shows the derived radius.
• Circumference (C) — the distance a tape measure travels around the outside, which is often the easiest measurement of all on a tank or a pipe. The calculator uses r = C ÷ 2π and shows both the radius and the diameter it implies.

The diameter-as-radius error

If you put a diameter into πr²h without halving it, your answer is four times too big — the radius is squared, so doubling it multiplies the result by four, not two. It is the most common single error in volume work and it is completely silent: the number looks perfectly plausible. Switching the input mode to “Diameter + height” removes the risk entirely, and the receipt states the radius it derived so you can check.

The calculator also solves a cylinder backwards: give it a volume and a height and it returns the radius or diameter; give it a volume and a radius and it returns the height.

Cube volume

A cube has the same length in all three directions, so its volume is V = s³ — the edge length multiplied by itself three times.

“Cubed” is not decoration. It is the reason volume behaves so unlike the dimension you measured: doubling the edge multiplies the volume by eight (2³), and trebling it multiplies the volume by twenty-seven (3³). A storage box that is “only 25% bigger on each side” holds nearly twice as much. This is also why unit conversions get cubed — see the conversions section below.

Volume of boxes and rectangular prisms

A rectangle is two-dimensional and has area, not volume. A rectangular prism or box has volume l × w × h.

The distinction is the whole point. A rectangle is a flat figure drawn on paper: it encloses an area, measured in square units. Give it a third dimension — a thickness, a depth, a height — and it becomes a rectangular prism, also called a cuboid or, in everyday language, a box. Only then is there a volume to calculate. If you searched for “rectangle volume”, the box calculator is what you want; if you have a plan with a length and a width and nothing else, you have an area, and the Area Calculator is the right tool.

The three dimensions do not have to share a unit. A 6 ft × 32 in × 1.4 m box is a perfectly ordinary thing to measure with a mix of tools, and each field here carries its own unit selector.

Does a circle have a volume?

A circle is two-dimensional and has no volume. Like a rectangle, it is a flat figure: it has a circumference and an area (A = πr²), and that is all.

People searching for “circle volume” almost always want one of two solids:

• If the object is a round thing with a height — a tin, a pipe, a tank, a column, a disc — you want the cylinder.
• If the object is a round thing that is round in every direction — a ball, a bearing, a dome, a balloon — you want the sphere.

For a flat circle’s area rather than a solid’s volume, use the Circle Calculator.

Sphere volume

A sphere’s volume is V = 4⁄3 πr³. Because the radius is cubed, size changes bite hard: doubling the radius multiplies the volume by eight, so a ball bearing twice the diameter weighs eight times as much in the same material.

Two related solids share the formula. A hemisphere is exactly half of it, V = ⅔πr³ — the shape of a domed tank end or a bowl. A spherical cap is the slice a plane cuts off a sphere, V = ⅓πh²(3R − h), which is what you need for a shallow dome or for the liquid sitting in the rounded end of a tank. The calculator will also solve a sphere backwards, returning the radius or diameter from a known volume.

Cone and frustum volume

A cone with the same base and height as a cylinder holds exactly one third as much: V = ⅓πr²h. The same one-third relationship holds for every pyramid against its prism, which is why a square pyramid is ⅓a²h.

Most real “cones” are not cones. A plant pot, a bucket, a paper cup and a hopper are all conical frustums — a cone with its point cut off — and their volume is V = ⅓πh(r₁² + r₁r₂ + r₂²), using both radii. Treating a bucket as a cylinder of its top diameter can overstate its capacity by a third or more. The calculator has the frustum as its own shape, along with truncated square and rectangular pyramids for hoppers and skips.

The truncated rectangular pyramid is a case worth flagging: when the length and the width taper at different rates the familiar h⁄3 (A₁ + A₂ + √(A₁A₂)) shortcut no longer applies, because it assumes the two faces are similar. This calculator uses the prismatoid rule, which is exact for a linear taper in both directions.

Tank capacity and liquid volume

A tank asks two different questions, and the calculator keeps them separate:

• Total capacity — the volume of the whole vessel, from its dimensions.
• Filled volume — how much liquid is in it right now, from a dip reading or a sight glass.

For an upright tank the two are simply proportional: liquid rises uniformly, so half the depth is half the capacity. For a tank lying on its side, that is not true and the difference is large. The wetted end of a horizontal cylinder is a circular segment — narrow at the bottom, widest across the middle, narrow again at the top. A horizontal tank filled to a quarter of its depth holds roughly a fifth of its capacity, and one filled to three-quarters of its depth holds about four-fifths. Only the exact halfway point is a true 50%. This calculator uses the segment geometry, A = r²·acos((r − d)/r) − (r − d)·√(2rd − d²), and never a percentage of depth.

Five tank geometries are covered: rectangular, vertical cylinder, horizontal cylinder, capsule (a cylinder with domed ends, where the two hemispheres together are one sphere) and elliptical or oval-section. Each reports total capacity, filled volume, the remaining space and the fill percentage — and the fill percentage by volume is shown next to the fill percentage by depth, because they are not the same number.

For a pool, the same distinction applies with a twist: a pool with a shallow end and a deep end is calculated from the average depth, which is exact only if the floor falls at one constant slope for the whole length. A real pool with a flat shallow shelf, a hopper bottom or a break in slope holds a different amount, usually less. The calculator returns that as an explicit warning rather than leaving you to discover it when the chemicals do not work.

Pipe volume

A pipe is a cylinder, and the only difficulty is which diameter to use. Pipe is specified and sold by its outside diameter — often by a nominal size that matches neither the inside nor the outside measurement exactly — but only the bore holds anything. So the calculator offers two input modes:

• Inside diameter + length, when you know the bore.
• Outside diameter + wall thickness + length, which is what a spec sheet gives you. The calculator computes dᵢ = D − 2t and shows the inside diameter it derived.

The result is reported in litres, US gallons, Imperial gallons, cubic feet and cubic metres together, because a pipe volume is almost always wanted for a fluid job — flushing, chlorinating, priming, draining or dosing — where the unit depends entirely on who you are talking to. In mode B the volume of the wall material itself is reported too, which is what you need for weight.

Volume units and conversions

Volume is measured in cubic units — cubic millimetres, centimetres, metres, inches, feet and yards — or in capacity units built on top of them: millilitres, litres, fluid ounces, cups, pints, quarts and gallons.

Why unit conversions get cubed

This is the conversion mistake that costs the most, and the reasoning that fixes it is short:

1 m³ = 1000000 cm³, because 1 m = 100 cm, and a cubic metre is a cube 100 cm on every edge — so it contains 100 × 100 × 100 = 100³ = 1000000 cm³.

The same logic applies everywhere: a linear factor of k becomes k³ for a volume. There are 12 inches in a foot but 1728 in³ in a cubic foot (12³). There are 3 feet in a yard but 27 ft³ in a cubic yard (3³). Dividing a volume by the linear factor is the classic way to be wrong by two orders of magnitude, and the calculator removes the risk by normalizing every dimension to metres before the formula runs and converting only the finished volume.

Capacity units

The metric capacity units are exact by definition: 1 millilitre is exactly 1 cm³, and 1 litre is exactly 1000 cm³ — that is, 0.001 m³, so 1 m³ = 1000 L. The customary units each descend from a single definition: a US liquid gallon is exactly 231 cubic inches, which makes 1 ft³ = 7.4805195 US gal; an Imperial gallon is exactly 4.54609 litres, which makes 1 ft³ = 6.2288355 imp gal. The two gallons differ by about 20%, so a “gallon” with no country attached is not a usable figure — every result on this page names which one it means.

Volume vs capacity

The two words are used interchangeably in conversation and mean slightly different things in practice.

Volume is how much space something occupies — including its own walls. Capacity is how much it will hold — the space inside it. For a sheet of paper or a solid block the distinction is meaningless, but for anything with a wall it matters: a 200-litre steel drum has a capacity of 200 litres and an external volume noticeably larger, and a glass aquarium quoted at 200 litres is quoting the water it holds, not the space it takes up on the floor.

Two practical rules follow. First, measure inside when you want capacity — internal length, internal width, internal depth — or give the calculator an outside dimension and a wall thickness where it supports one, as it does for pipes and tubes. Second, a quoted capacity is usually brim-full and nothing is ever filled to the brim: tanks need ullage, aquariums lose volume to substrate and rock, and a pool is filled to the skimmer, not to the coping.

Worked examples

Each example below was computed by the same engine that powers the calculator, at build time — the givens, the substitution and the answer are the engine’s own output, not numbers typed into the page.

A box measured in three different units

Length

6 ft

Width

32 in

Height

1.4 m

Formula V = l × w × h

Substitution V = 1.8288 × 0.8128 × 1.4 = 2.08103

Answer 2.0810281 m³

Also: 73.490814 ft³ · 2.721882 yd³ · 2081.0281 L · 549.74946 US gal · 457.76219 imp gal

Nothing had to be converted by hand. Each dimension was normalized to metres on its own — 6 ft = 1.8288 m, 32 in = 0.8128 m, 1.4 m = 1.4 m — and only then multiplied. Because the three inputs are not in one system, the answer is reported in the canonical unit, cubic metres.

Load this example →

A cube from one edge

Edge length

3 m

Formula V = a³

Substitution V = 3³ = 27

Answer 27 m³ (exact: 27 m³)

Also: 953.496 ft³ · 35.314667 yd³ · 27000 L · 7132.6454 US gal · 5939.1697 imp gal

Cubing an edge is the whole calculation: three metres in every direction. Trebling the edge would multiply the volume by twenty-seven, not by three — that is what "cubed" means, and it is why volume grows so much faster than a dimension does.

Load this example →

A cylinder from its radius

Radius

2 m

Height

5 m

Formula V = πr²h

Substitution V = π × 2² × 5 = 62.8319

Answer 62.831853 m³ (exact: 20π m³)

Also: 2218.886 ft³ · 82.180961 yd³ · 62831.853 L · 16598.42 US gal · 13821.075 imp gal

The base is a circle of area πr², and the volume is that base area carried up the height. The exact answer is 20π m³; 62.831853 m³ is the same number written as a decimal.

Load this example →

The same cylinder measured across, not from the centre

Diameter

4 m

Height

5 m

Formula V = πr²h

Substitution V = π × 2² × 5 = 62.8319

Answer 62.831853 m³ (exact: 20π m³)

Also: 2218.886 ft³ · 82.180961 yd³ · 62831.853 L · 16598.42 US gal · 13821.075 imp gal

A tape measure gives the diameter, not the radius, so the calculator halves it first and shows the derived radius in the receipt. Putting the diameter straight into πr²h would have given four times the correct answer — the single most common volume mistake there is.

Load this example →

A horizontal cylindrical tank, part full

Diameter

4 ft

Tank length

10 ft

Fill depth

1 ft

Formula V = πr²L; filled = [r²·acos((r−d)/r) − (r−d)√(2rd − d²)] × L

Substitution V = π × 2² × 10 = 125.664

Answer 24.567394 ft³

Also: 0.69567113 m³ · 0.90990348 yd³ · 695.67113 L · 183.77687 US gal · 153.02625 imp gal

The dipstick reads a quarter of the tank's depth, but the tank is not a quarter full: the wetted end of a horizontal cylinder is a circular segment, so it holds 19.550111 % of its 125.66371 ft³ capacity. Reading the depth as a percentage would have overstated the contents by about half as much again.

Load this example →

A pipe from its outside diameter and wall thickness

Outer diameter

4.5 in

Wall thickness

0.25 in

Pipe length

100 ft

Formula V = π(dᵢ/2)²L

Substitution V = π × (0.333333/2)² × 100 = 8.72665

Answer 8.7266463 ft³ (exact: 25π/9 ft³)

Also: 0.2471111 m³ · 0.32320912 yd³ · 247.1111 L · 65.279847 US gal · 54.356844 imp gal

Pipe is sold by its outside diameter, but water only flows through the bore. Subtracting twice the wall thickness gives an inside diameter of 0.33333333 ft, and that hundred-foot run holds 247.1111 L — 65.279847 US gal, or 54.356844 imp gal. Using the outside diameter instead would have overstated the fill by about a quarter.

Load this example →

Common volume calculation mistakes

1. Putting the diameter where the radius goes. πr²h with a diameter in place of r gives four times the correct volume, because the radius is squared. A tape measure naturally reads across the full width, so this is the single most common error. Use the diameter input mode and let the calculator halve it — the derived radius is shown in the receipt so you can see it happened.
2. Reporting volume in square units. Volume multiplies three lengths, so the unit is cubed: cm³, ft³, m³. An answer labelled ft² is an area, and an unlabelled number is not an answer at all.
3. Mixing centimetres and metres in one formula. A box entered as 200 (cm) × 3 (m) × 1.5 (m) is out by a factor of a hundred. Give every dimension its own unit selector — the calculator normalizes each one independently before it multiplies anything.
4. Converting with a linear factor instead of a cubic one. There are 100 cm in a metre, but 1000000 cm³ in a cubic metre. Dividing a volume in cm³ by 100 to “get metres” is the same mistake as scaling a photo by area when you meant width.
5. Confusing area with volume. A floor plan gives you square feet. Multiplying by a depth is what turns it into a volume — and if the depth is in inches while the area is in square feet, that multiplication needs a conversion too.
6. Treating external size as capacity. A tank’s outside dimensions include its walls. For anything with a meaningful wall thickness — a steel drum, a glass aquarium, a concrete cistern — measure inside, or expect to be over.
7. Using the outside diameter of a pipe. Pipe is specified and sold by its outside diameter, and often by a nominal size that matches neither. Only the bore holds water: subtract twice the wall thickness first. On a 4½-inch pipe with a ¼-inch wall the difference is about a quarter of the volume.
8. Assuming a horizontal tank fills in proportion to its depth. It does not. The wetted end of a horizontal cylinder is a circular segment, so a tank at a quarter of its depth holds roughly a fifth of its capacity, not a quarter. Only the halfway point is a true 50%. This calculator uses the segment geometry.
9. Rounding part-way through. Rounding the radius, then the base area, then the volume compounds the error three times over. Every calculation here runs at full internal precision and rounds once, for display only — which is why the precision selector never changes the underlying number.

Where volume calculations show up

• Boxes & rooms

  Shipping cartons, crates and the air volume of a room for ventilation or heating sizing. Length × width × height, in whatever units the tape measure and the plan happen to use.

• Cylinders

  Drums, silos, columns, buckets and cores. Measure across the top with a tape and let the calculator halve it — the diameter is what you can actually reach.

• Tanks

  Total capacity from the dimensions, and how much is actually in there from a dip reading. Horizontal tanks need the segment geometry, not a percentage of depth.

• Pipes

  How much water a run holds — for flushing, chlorinating, priming or draining. Pipe is sold by its outside diameter; only the bore holds anything.

• Pools

  Water volume for dosing chemicals and sizing a pump. A flat-bottomed pool is exact; a sloping floor is a model, and the calculator says so.

• Aquariums

  Gross internal volume from the glass dimensions, then the water depth for what is really in the tank. Substrate and rock displace more than people expect.

• Concrete

  Slabs, footings, columns and pads, in cubic yards or cubic metres. Volume is the geometry step; ordering also needs a waste allowance the supplier will advise on.

• Gravel

  Loose volume for a driveway or a base layer. Converting that volume into a delivered weight depends on the material’s density — that belongs to a materials calculator.

• Soil & mulch

  Bed volume from area and depth, which is where bag counts start. Depth is the number people get wrong: 2 in over 200 ft² is a lot less than it sounds.

• Shipping & storage

  Cubic capacity of containers, vans and self-storage units, and whether a stack of cartons will actually fit inside it.

This page answers the geometry question: how much space is inside this shape. When the next question is a purchase — how many tonnes of gravel, how many bags of mulch, how many tiles — a materials calculator is the right tool, because the answer then depends on density, coverage, waste and packaging rather than on geometry alone.

Related calculators

• Area Calculator

  Square footage, land area and 3D surface area — the two-dimensional half of the same measuring job.

• Circle Calculator

  Radius, diameter, circumference and area of a circle, solved from any one of them.

• Triangle Calculator

  Sides, angles and area of a triangle — useful when a prism’s cross-section is not a rectangle.

• Conversion Calculator

  General unit conversion beyond volume: length, mass, temperature, speed and more.

• Gravel Calculator

  Turns a driveway or base-layer volume into tonnes and a delivered order, using material density.

• Mulch Calculator

  Bed volume into bags or cubic yards of mulch, with depth guidance per material.

• Tile Calculator

  Tile counts, layout and waste for a floor or wall — the surface job rather than the solid one.

Methodology, assumptions and limitations

Every result comes from the dedicated volume engine (volume-engine v1.0.0), which applies the standard geometry formula for the chosen solid — s³ for a cube, l×w×h for a box, πr²h for a cylinder, ⅓πr²h for a cone, 4⁄3πr³ for a sphere, ⅓πh(r₁²+r₁r₂+r₂²) for a frustum, and so on for all 27 supported solids. Each dimension carries its own unit and is normalized to metres BEFORE any formula runs, so mixing feet, inches and metres in one calculation can never apply a conversion factor twice; only the finished volume is converted to the output unit. Cubic units are derived from their exact linear definitions (1 in = 0.0254 m exactly, so 1 in³ = 0.0254³ m³), the US gallon from its statutory 231 cubic inches and the Imperial gallon from its statutory 4.54609 litres. Partially filled horizontal cylindrical, capsule and elliptical tanks use the exact circular-segment geometry, never a proportion of fill depth. All arithmetic runs at full internal double precision; the precision selector affects display only and never a stored value. Each result ships a receipt with the formula, the substitution, the numbered steps, the normalized inputs, the derived values, an exact π or rational form where one exists, unit equivalents, and self-verification checks that are reported as passed, failed or not applicable — never silently skipped.

Assumptions

• Every geometric dimension is a strictly positive length; the only fields that may be zero are the ones that mean “none of it”, such as a fill depth on an empty tank.
• Each dimension carries its own unit and is normalized to metres independently before any formula is applied.
• Heights are measured perpendicular to the base, never along a slanted side.
• Tank and container dimensions are taken as internal measurements; a wall thickness is only accounted for where the shape asks for one, as pipes and tubes do.
• A pool entered with a shallow and a deep depth is modelled as one flat floor sloping at a constant rate for the whole length — the calculator states this as a warning on the result.
• A ± tolerance produces the mathematical extremes the geometry reaches at the ends of the entered bands. It is a possible measurement range, not a statistical confidence interval, and carries no probability.

Limitations

• The calculator reports geometry only — no material weights, densities, coverage rates, waste factors, bag counts or prices. A purchase question belongs to a materials calculator.
• Composite objects are not built inside one calculation: split them into simpler solids, calculate each, then add or subtract the parts by hand.
• A quoted capacity is brim-full. Ullage, freeboard, substrate displacement and a pool filled to the skimmer rather than the coping are not modelled.
• Irregular and organic shapes have no closed-form volume here; approximate them with the nearest supported solid, or measure them by displacement.
• The measurement value parser accepts plain decimals, feet-and-inches compounds and prime marks, but not E-notation — “1e6” is rejected with a structured error rather than silently misread.

No independent third-party review. The author is responsible for the methodology. Correctness is enforced inside the repository by 157 golden reference vectors with hand-derived expectations, 36 invariant and property tests, and an independent numerical audit that re-derives every volume by tanh-sinh quadrature of the solid's cross-section — code that shares nothing with the production formulas.

Calculation engine

volume-engine v1.0.0

Shapes and input modes

27 shapes, 48 input modes, 12 inverse solvers

Units

8 length units, 18 volume and capacity units

Author

Ugo Candido

Last reviewed

August 21, 2026

Where the calculation runs

In your browser. Dimensions are never transmitted.

Frequently asked questions