Volume Calculator
Ten 3D shapes, in cubic units and in capacity
| Measure | Result | Basis |
|---|
Enter every dimension above to see the result.
One set of dimensions, seven solids
| Solid | Built from | Volume | vs cylinder |
|---|
Why the numbers land where they do
Read the last column and two families jump out. Anything that comes to a point holds a third of the solid that wraps around it, and anything cut in half holds half. Those ratios are exact, not approximations, so they double as a sanity check: if your answer is not roughly a third of the box you could fit the object inside, one of your measurements is wrong.
- ×⅓Cone against its cylinder, square pyramid against its box. Same base, same height, one third the contents.
- ×½Hemisphere against its sphere. Cut the ball, halve the volume, but the surface area does not halve because you have added a flat face.
- ×⅔Sphere against the cylinder that just contains it, when the cylinder height equals the diameter. Archimedes liked this one enough to have it carved on his tomb.
A cone holds exactly one third of the cylinder around it, and the derivation of that third, along with slant height and cone surface area, sits on the cone volume calculator. For tanks, pipes, hollow cylinders and part filled depths, use the cylinder volume calculator.
Which shape is my object?
Every volume calculator on the web assumes you already know which solid you are holding. Most of the time you do not, and picking the wrong one is a larger error than any rounding. This table maps ordinary objects onto the solid that actually describes them.
| Real object | Treat it as | Why, and what to measure |
|---|---|---|
| Builder's bucket, flower pot | Conical frustum | The sides taper, so calling it a cylinder overstates it by a tenth or more. Measure the inside radius at the rim, the inside radius at the base, and the vertical depth. |
| Gravel, sand or bark pile | Cone | A loose pile settles into a cone. Measure the base radius and the vertical height straight up the middle, never the slope down the side. |
| Propane bottle, air receiver | Capsule, lying down | A cylinder with a dome on each end. The straight section is h, and the two domes together make exactly one sphere of the same radius. |
| Fish tank, moving box, raised bed | Rectangular box | Measure the inside faces. Glass thickness and timber thickness are not capacity, and on a small tank that gap is worth several litres. |
| Dome tent, cloche, half a ball | Hemisphere | Half of a sphere, so half the volume. Measure the radius across the floor and halve it. |
| Ridge tent, roof void, wedge of cheese | Triangular prism | One triangle dragged along a length. Measure the triangle base, the triangle height, then the run. |
| Marble, ball bearing, football | Sphere | One measurement only, and it is the radius. Anything you measured across the whole ball is the diameter, so halve it first. |
| Post hole, storage drum, water butt | Cylinder | Radius and vertical height. A drum on its side holds the same amount, so orientation only matters once it is part full. |
| Drainpipe, tube, cored slab | Hollow cylinder | Work out the outer cylinder, work out the inner one, subtract. The formula is in the reference list, and the same subtract trick handles any drilled hole. |
| Rugby ball, egg, lozenge | Ellipsoid | Three different half widths rather than one radius. Reference formula only, because in practice displacement is faster. |
Rows in grey are reference formulas rather than picker options. If nothing in the table fits, the object is probably two solids stuck together, or a candidate for the displacement method described in the guide below.
Cubic units convert with a cube, not a straight line
This is the conversion people get wrong, and it is wrong by a factor of a million rather than a rounding. One metre is one hundred centimetres, so it feels as though one cubic metre should be a hundred cubic centimetres. It is not. The ratio between the lengths gets cubed, so one cubic metre is 100 × 100 × 100, which is 1,000,000 cubic centimetres. The same trap turns 3 feet per yard into 27 cubic feet per cubic yard.
| One of these | In cubic metres | In litres |
|---|---|---|
| Cubic centimetre (cm³) | 0.000001 | 0.001, which is 1 millilitre |
| Cubic inch (in³) | 0.000016387064 | 0.016387064 |
| Cubic foot (ft³) | 0.028316846592 | 28.316846592 |
| Cubic yard (yd³) | 0.764554857984 | 764.554857984 |
| Cubic metre (m³) | 1 | 1,000 |
Across the systems, one cubic metre is 35.3147 cubic feet, 1.30795 cubic yards and 61,023.744 cubic inches. One cubic foot is 1,728 cubic inches, and one cubic yard is 27 cubic feet.
Capacity, and the two gallons that are not the same size
A cubic figure answers how much space. Capacity answers how much liquid, and that is usually the real question behind a volume search. Litres are simple, because a cubic metre is exactly 1,000 litres and a cubic centimetre is exactly 1 millilitre. Gallons are not simple, because there are two of them still in daily use. A US liquid gallon is 3.785411784 litres and holds 128 fluid ounces. An Imperial gallon, used in the UK, is 4.54609 litres and holds 160 fluid ounces. That makes the Imperial gallon about 20 percent larger, which is why an unlabelled gallon figure is not a rounding error, it is a wrong answer for half the people reading it. Every gallon number on this page says which gallon it is.
| Volume | Litres | US gallons | Imperial gallons |
|---|---|---|---|
| 1 cubic metre | 1,000 | 264.172 | 219.969 |
| 1 cubic yard | 764.555 | 201.974 | 168.178 |
| 1 cubic foot | 28.3168 | 7.48052 | 6.22884 |
| 1 cubic inch | 0.0163871 | 0.0043290 | 0.0036047 |
| 10 litres | 10 | 2.64172 | 2.19969 |
How accurate does the tape measure need to be?
More accurate than you think, because volume multiplies three lengths together and a small error in one of them is multiplied three times over. Here is the whole derivation, and it takes two lines.
Measure a radius 1 percent too large. The measurement you use is 1.01 times the true radius.
A sphere scales with the radius cubed. So the volume comes out 1.01 × 1.01 × 1.01 = 1.030301 times the truth, which is 3.03 percent too large. One percent in, three percent out.
A cylinder squares the radius and uses the height once. So 1 percent on the radius alone gives 1.01 × 1.01 = 1.0201, about 2 percent, and 1 percent on every side of a box gives 3 percent again.
The rule of thumb falls straight out of that: multiply your percentage measuring error by the power the dimension is raised to. Cubed means triple it, squared means double it. A 2 percent error on every dimension of a garden pond becomes about 6 percent of its volume, and on a 10,000 litre pond that is 600 litres of water you did not plan for. Measure twice, measure the inside faces, and keep every dimension in the same unit before you start.
Ordering materials: because the error runs one way as easily as the other, trades usually take the calculated volume and add a margin rather than trusting it exactly. The arithmetic here is exact, but your tape measure is not.
Every volume formula on one page
Ten of these are in the picker above. The rest are reference formulas for shapes that come up less often.
- CubeV = a³One sideEvery face is identical, so one measurement is enough.
- Rectangular boxV = l × w × hLength, width, heightMeasure the inside if you want capacity.
- Triangular prismV = ½ b h lTriangle base, triangle height, runWith three side lengths instead, use ¼l√((a+b+c)(b+c-a)(c+a-b)(a+b-c)).
- SphereV = (4/3) π r³RadiusDouble the radius and the volume goes up eight times.
- HemisphereV = (2/3) π r³RadiusExactly half a sphere, but not half the surface area.
- CylinderV = π r² hRadius, heightTanks, pipes and part filled depths live on the cylinder volume calculator.
- CapsuleV = π r² h + (4/3) π r³Radius, straight sectionh is the straight part only, not the overall length.
- ConeV = (1/3) π r² hRadius, perpendicular heightUse the vertical height, not the slant. Full derivation on the cone volume calculator.
- Conical frustumV = (1/3) π h (r₁² + r₁r₂ + r₂²)Both radii, vertical depthSet the two radii equal and it becomes a cylinder, which is a quick way to check your entry.
- Square pyramidV = (1/3) a² hBase side, perpendicular heightAny pyramid is (1/3) × base area × height, whatever the base shape.
- Hollow cylinderV = π h (R² − r²)Outer radius, inner radius, lengthReference. Outer solid minus the hole.
- EllipsoidV = (4/3) π a b cThree semi axesReference. Each one is a half width, not a full width.
- Spherical capV = (1/3) π h² (3R − h)Sphere radius, cap heightReference. R is the radius of the whole ball it was sliced from.
- Truncated pyramidV = (h/3)(A₁ + A₂ + √(A₁A₂))Both base areas, heightReference. Works for any base shape, not just squares.
- TetrahedronV = a³ / (6√2)Edge lengthReference. All four faces are the same triangle.
- TorusV = 2 π² R r²Ring radius, tube radiusReference. R is centre to tube centre.
- Hexagonal prismV = (3√3 / 2) a² hSide, heightReference. a is one edge of the hexagon.
- Pentagonal prismV = (5/4) a² √(25 + 10√5) hSide, heightReference. Same base area times height idea.
Prisms and cylinders all share one idea: base area multiplied by height. Anything that narrows to a point takes a third of that.
Volume, from a tape measure to litres and gallons
Ten solids, the cubic units they produce, and the two steps either side of the formula that people actually get wrong.
Volume is the amount of three dimensional space a solid fills, and it is always measured in cubic units. This volume calculator covers ten shapes in one place, gives its result in whatever unit your tape used, then turns that into capacity so the litres, the US gallons and the Imperial gallons sit side by side instead of leaving you to guess which gallon somebody meant.
People rarely ask for a volume for its own sake. They want to know how much soil fills a raised bed, or whether one drum will take the whole batch. The multiplication in between is routine. The errors creep in at either end: naming the solid before you start and turning cubic units into litres or gallons once you finish. Those two steps get most of the space below.
Cubic units, and the one unit the world agrees on
Area multiplies two lengths and produces square units. Volume multiplies three and produces cubic units, written with a small three: cm³, m³, in³, ft³ and yd³. The SI unit of volume is the cubic metre, and every other unit here is defined against it. Because three lengths are multiplied together, the factor between two cubic units is the length factor cubed. One metre is 100 centimetres, so one cubic metre is 1,000,000 cubic centimetres rather than 100. One yard is 3 feet, so one cubic yard is 27 cubic feet rather than 3. That is the most expensive mistake in the whole subject, because it is wrong by a factor of thousands rather than by a few percent.
Choose the solid before you choose the formula
Almost every calculator of this kind takes for granted that you know which solid is in front of you. Naming it correctly is where most jobs go wrong. A builder's bucket tapers, so it is a conical frustum, and treating it as a cylinder overstates the contents by more than a fifth. A pile of gravel settles into a cone. A propane bottle is a capsule lying on its side. Check the decision table above for these cases and several others before entering anything, since picking the wrong solid is a far bigger error than any rounding.
Two of the ten shapes have pages of their own, so the one third derivation, slant height and cone surface area live on the cone volume calculator, while tanks, pipes, hollow cylinders and part filled depths live on the cylinder volume calculator.
Turning a volume into a capacity
A cubic figure answers how much space. Capacity answers how much will fit inside, and that is usually the real question. Metric makes it painless, because one cubic centimetre is exactly one millilitre and one cubic metre is exactly 1,000 litres, so the conversion is a decimal point shift. Gallons are where it falls apart, because two different gallons are still in daily use. A US liquid gallon is 3.785411784 litres and splits into 128 fluid ounces. An Imperial gallon, the one used in the United Kingdom, is 4.54609 litres and splits into 160 fluid ounces. An Imperial gallon is therefore about 20 percent larger, so an unlabelled gallon figure is not a small rounding difference, it is the wrong answer for whichever reader was assumed away. This page prints both, labelled, every time.
Working figures: 3.785 litres and 4.546 litres are close enough for a shopping list, but the exact statutory values are 3.785411784 and 4.54609, and those are what the calculator uses.
When the object is not a shape at all
Some things refuse to be a solid. A rock, a casting, an ornament, a shoe. For those, use water displacement. Part fill a measuring jug or a graduated cylinder, write down the level, lower the object in until it is completely covered, and write down the new level. The difference is the volume, and because one millilitre is one cubic centimetre the reading is already a cubic measurement. Anything too big for a jug works the same way in a bucket standing in a tray, where the water that overflows is the answer.
The other route is mass and density. Volume equals mass divided by density, so a known weight of a material with a known density gives the volume without measuring anything. That is the usual method for gravel, sand and potting soil, where measuring the heap is hopeless but weighing the bag is easy.
When the object is two shapes at once
Composite volumes are nothing more than addition and subtraction. A grain silo is a cylinder with a hemisphere on top, so work out both and add them. A block with a hole drilled through it is a box minus a cylinder, so work out both and subtract. A tapered planter with a recessed drainage well is a truncated pyramid minus a small box. Nothing new is needed, only the discipline of working in one unit and tracking which pieces are added and which are taken away. This is also the escape route when an object sits between two shapes: approximate it with the nearest solid, then correct it with a second, smaller one rather than starting again.
Three jobs worked through
A rectangular swimming pool. 8 m long, 4 m wide and 1.5 m deep is 8 × 4 × 1.5, which is 48 cubic metres. That is 48,000 litres, 12,680 US gallons or 10,559 Imperial gallons. The two gallon figures are more than 2,000 apart on the same pool, which is exactly why the label matters when you are dosing chemicals.
A raised bed, ordered in cubic yards. 8 ft by 4 ft, filled 1 ft deep, is 32 cubic feet. Soil is sold by the cubic yard and one cubic yard is 27 cubic feet, so that is 32 divided by 27, which is 1.19 cubic yards. Order one yard and the bed finishes short.
A builder's bucket, treated properly. Inside radius 15 cm at the rim, 12 cm at the base, 30 cm deep. As a conical frustum it holds 17,247 cm³, which is 17.2 litres, 4.56 US gallons or 3.79 Imperial gallons. Treated as a straight cylinder of the rim radius it comes out at 21,206 cm³, overstating the bucket by 23 percent. One wrong shape choice, one badly wrong answer.
Measuring well enough for the answer to matter
- 1Cubing triples your error
A radius measured 1 percent too large makes a sphere come out about 3 percent too large, because 1.01 cubed is 1.030301. Multiply your measuring error by the power that dimension is raised to.
- 2Radius, never diameter
Every curved solid here wants the radius. A tape laid across the full width reads the diameter, so halve it first; skip that and a sphere reads eight times too large.
- 3One unit everywhere
Mixing metres with centimetres is the most common failure of all. Convert every dimension first, calculate second, convert the result last.
- 4Measure the inside
Capacity is set by the inside faces. Wall thickness, glass and liners all eat into the usable volume, and on a small container that is several litres gone.
Frequently Asked Questions
What units is volume measured in, and what is the SI unit?
Volume is measured in cubic units, because three lengths are multiplied together. The common ones are the cubic centimetre, cubic metre, cubic inch, cubic foot and cubic yard. The SI unit is the cubic metre. Capacity units such as the litre, the millilitre and the gallon measure the same quantity in a different form, and one cubic metre is exactly 1,000 litres.
How do I turn a cubic measurement into litres?
Convert the figure to cubic metres, then multiply by 1,000. One cubic centimetre is 0.001 litres, one cubic foot is 28.316846592 litres and one cubic yard is 764.554857984 litres. The calculator above does it for every shape and prints the litre figure next to the cubic one, so you can check both without a second tool.
Is a US gallon the same as an Imperial gallon?
No, and the gap is large. A US liquid gallon is 3.785411784 litres and contains 128 fluid ounces. An Imperial gallon, the one used in the United Kingdom, is 4.54609 litres and contains 160 fluid ounces, making it about 20 percent bigger. A container described as 10 gallons holds 37.9 litres in the United States and 45.5 litres in the United Kingdom, so an unlabelled gallon figure is worth nothing on its own.
How do I find the volume of an irregular object?
Use water displacement. Part fill a measuring jug or a graduated cylinder, note the level, submerge the object completely, note the new level, and the difference is the volume. Because one millilitre is exactly one cubic centimetre, the reading is already a cubic measurement. If the object floats, push it under with a thin rod. If it will not fit in a jug, stand a full bucket in a tray and measure what overflows.
Can I find a missing height or radius if I already know the volume?
For the simpler shapes, yes, by rearranging the formula. A box gives h = V / (l × w). A cylinder gives h = V / (π r²). A sphere gives r as the cube root of 3V / (4π). Shapes with two independent radii, such as the conical frustum, and shapes such as the spherical cap do not unwind cleanly, because more than one set of dimensions produces the same volume. For those, fix every dimension but one and adjust that one until the volume matches.
How do I handle an object made of two shapes stuck together?
Work out each piece separately in the same unit, then add or subtract. A grain silo is a cylinder plus a hemisphere. A block with a hole through it is a box minus a cylinder. A planter with a recessed well is a truncated pyramid minus a box. The same trick rescues an object that sits between two shapes: take the nearest solid, then correct it with a second, smaller one.
Why does volume use cubes when area uses squares?
It comes from how many lengths are multiplied. Area multiplies two, so the unit is a length squared, such as the square metre. Volume multiplies three, so the unit is a length cubed, such as the cubic metre. It also explains how scaling behaves: double every dimension and the area goes up four times while the volume goes up eight times.
Which shape should I use for a bucket or a tapered container?
A conical frustum, which is a cone with the point cut off. Measure the inside radius at the rim, the inside radius at the base, and the vertical depth. Treating a tapered bucket as a cylinder can overstate its contents by more than 20 percent, because a cylinder assumes the base is as wide as the rim. The decision table on this page maps other awkward objects the same way.
How accurate does my measurement need to be?
More accurate than feels necessary, because volume multiplies three lengths and any error is multiplied along with them. A radius measured 1 percent too large makes a sphere about 3 percent too large, since 1.01 cubed is 1.030301, and about 2 percent too large for a cylinder, where the radius is only squared. The rule is to multiply your percentage error by the power that dimension is raised to. On a 10,000 litre pond, a 2 percent error on every dimension is 600 litres.
Is a cubic centimetre the same as a millilitre?
Yes, exactly, by definition. That is why 1,000 cubic centimetres make one litre and 1,000 litres make one cubic metre. It is the most useful conversion in the subject, because it takes you from a measured size straight to a capacity with no arithmetic in between.
What is the difference between volume and surface area?
Volume is the space inside, measured in cubic units. Surface area is the skin on the outside, measured in square units. A cube with 3 cm sides has a volume of 27 cm³ and a surface area of 54 cm², and the two are never interchangeable. Volume tells you what fits inside or what it weighs, while surface area tells you how much paint, wrapping or heat loss to expect. This calculator reports both.