Rectangular Prism Volume Calculator
Calculate the volume of a rectangular prism or box from length, width and height.
About
Rectangular Prism Volume Calculator
The volume of a rectangular prism is V = l × w × h. Enter the three dimensions above and the calculator returns the volume with the substitution shown, converted into litres and US gallons.
What the formula means
A rectangular prism is a box: six rectangular faces meeting at right angles. Its volume counts how many unit cubes fit inside. A box 4 units long, 3 wide and 2 high holds 4 × 3 = 12 cubes in the bottom layer, and 2 layers, so 24 cubes in total.
The order of multiplication makes no difference, which is worth knowing when you are working from a drawing and cannot tell which dimension the draughtsman called length. The names matter for describing the box; the arithmetic does not care.
The shape is also called a cuboid, and a rectangular tank, a shipping carton and a room are all examples. When all three dimensions are equal it is a cube.
A worked example
Take dimensions of 1, 2 and 3 units.
- Multiply the first two: 1 × 2 = 2
- Multiply by the third: 2 × 3 = 6 cubic units
A practical version: a room 4.2 m long, 3.6 m wide and 2.4 m high contains 4.2 × 3.6 × 2.4 = 36.288 m³, which is 36,288 litres of air, or 1281.5 cubic feet.
Room volume and ventilation
Room volume is the figure behind air changes per hour, the measure used to size extract fans and ventilation systems. A bathroom requiring 8 air changes an hour with a volume of 20 m³ needs a fan moving 160 m³ per hour.
It also drives heating. Older heat load estimates worked from volume rather than floor area, because the air in the space has to be warmed as well as the surfaces around it. A room with a high ceiling needs more heat than its floor area suggests, which is why converted warehouses are expensive to keep warm.
Capacity against outside dimensions
A container's internal volume is smaller than its external one by the wall thickness on both sides of each dimension. For a box with 20 mm walls, each dimension loses 40 mm.
A crate measuring 1.2 × 0.8 × 0.6 m externally with 20 mm walls has an internal volume of 1.16 × 0.76 × 0.56 = 0.494 m³, against 0.576 m³ externally. That is a 14% difference, which matters when you are quoting capacity or working out how much a tank holds.
Volumetric weight
Carriers charge on whichever is greater, actual weight or volumetric weight, because a van fills up on space before it runs out of payload. Volumetric weight is the volume in cubic centimetres divided by a divisor, commonly 5000 for air freight.
A box 40 × 30 × 25 cm has a volume of 30,000 cm³, so its volumetric weight is 30,000 / 5000 = 6 kg. A parcel of pillows weighing 2 kg in that box is charged as 6 kg. Reducing box size cuts cost even when nothing is removed, which is why packaging design is a shipping cost decision.
Units
| Dimensions in | Volume in | Equivalent |
|---|---|---|
| centimetres | cm³ | millilitres |
| metres | m³ | 1000 litres |
| feet | ft³ | 7.481 US gallons |
| inches | in³ | 231 in³ per US gallon |
All three dimensions must share a unit before multiplying. Length in metres with a height in centimetres gives an answer a hundred times out, and the error is easy to miss because the number still looks plausible.
Filling a space: soil, water and concrete
Most rectangular volume problems end in an order for material, and the conversion from volume to quantity is where the money is.
| Job | Dimensions | Volume | Order |
|---|---|---|---|
| Raised bed | 2.4 × 1.2 × 0.3 m | 0.864 m³ | about 1.1 tonnes of topsoil |
| Aquarium | 1.2 × 0.5 × 0.6 m | 0.36 m³ | 360 litres, 360 kg of water |
| Patio base | 5 × 4 × 0.1 m | 2.0 m³ | about 3.4 tonnes of hardcore |
Note the aquarium. Water weighs a kilogram per litre, so that tank puts 360 kg on the floor before the glass, gravel and stand are counted. Floor loading is the reason large tanks go against a load-bearing wall.
Diagonals inside a box
The longest straight line inside a rectangular prism runs corner to opposite corner and measures √(l² + w² + h²). For a box 4 by 3 by 2 that is √(16 + 9 + 4) = 5.39.
This decides whether a long item fits. A 5.3 m beam will pass diagonally through a room 4 by 3 by 2 m even though no single dimension comes close, which is worth checking before deciding something will not go in.
Surface area alongside volume
A closed box has surface area 2(lw + lh + wh). For 4 by 3 by 2 that is 2(12 + 8 + 6) = 52 square units against a volume of 24.
For a fixed volume, the cube has the smallest surface area of any rectangular box, which is why compact packaging uses less cardboard and why a cubic tank loses the least heat. Long thin boxes waste material on both counts.
Stacking, packing and what actually fits
Geometric volume tells you what a space could hold if the contents poured in like liquid. Real objects do not, and the shortfall is often large.
A pallet 1.2 by 1.0 m loaded to 1.8 m has a geometric volume of 2.16 m³. Filled with cartons 0.4 by 0.3 by 0.25 m, each 0.03 m³, it could in theory take 72 of them. In practice the layer pattern rarely divides evenly, and 60 to 66 is a realistic count once the arrangement is fixed.
The gap is why warehouse planning uses a cube utilisation figure, typically 70 to 85%, rather than the raw volume. Applying that factor before ordering avoids the discovery that the last three cartons have nowhere to go.
Working backwards to a dimension
With the volume and two dimensions known, the third is V divided by their product. A tank holding 500 litres, which is 0.5 m³, measuring 1.0 by 0.5 m in plan must be 0.5 / 0.5 = 1.0 m deep.
This is the everyday form of the calculation. Space is usually constrained in two directions by the room, and the third is what you solve for. It is also how a required depth of material is found: spreading 2 m³ of gravel over a 20 m² drive gives a layer 0.1 m deep before compaction.
Checking a result
Volume answers are easy to get wrong by a power of ten, so a rough check is worth the few seconds it takes. Round each dimension to one significant figure and multiply those instead.
For 4.2 by 3.6 by 2.4, that gives 4 × 4 × 2 = 32, against the true 36.288. Close enough to confirm the scale is right, and far enough from 3.6 or 363 to catch a misplaced decimal point immediately.
Common mistakes
Mixing units. Convert all three dimensions first.
Using external dimensions for capacity. Subtract the wall thickness twice from each dimension.
Assuming a fill is complete. Liquids need headspace and solids leave voids, so usable capacity is below the geometric figure.
Converting after multiplying. Volume conversions cube the linear factor. A cubic metre is a million cubic centimetres, not a hundred.
Common questions
Frequently asked questions
Length times width times height. A box of 1, 2 and 3 units holds 6 cubic units. The order of multiplication makes no difference.
Multiply length, width and ceiling height. A room 4.2 by 3.6 with a 2.4 m ceiling contains 36.288 cubic metres, which is 36,288 litres of air.
A cuboid, or simply a box. When all three dimensions are equal it becomes a cube.
Subtract twice the wall thickness from each dimension. A 1.2 by 0.8 by 0.6 m crate with 20 mm walls holds 0.494 cubic metres against 0.576 externally.
A shipping charge based on space rather than mass. Divide the volume in cubic centimetres by 5000 for air freight. A 40 by 30 by 25 cm box gives 6 kg regardless of what is inside.
1000. A cubic metre of water weighs one tonne, which is the easiest way to picture the unit.
Multiply the room volume by the required air changes per hour. A 20 cubic metre bathroom at 8 changes an hour needs 160 cubic metres per hour.
Volume conversions cube the linear factor. One metre is 100 centimetres, so one cubic metre is 100 cubed, a million cubic centimetres.