How to Find the Volume of a Rectangular Prism

Rectangular prism is the formal name for a box, and its volume formula is the shortest one in geometry. That brevity hides something worth seeing once, because the multiplication is not really a formula at all. It is a shortcut for counting cubes, and knowing that is what lets you handle the awkward versions: missing dimensions, litres, wall thickness, and the question of how many small boxes actually fit inside a big one.

You are counting cubes, not applying a formula

Fill the worked box with cubes one unit on each side. The bottom of the box is 8 by 5, so 40 cubes cover it with none left over and none overlapping. The box is 3 units tall, so three of those layers fill it completely. That is 40 plus 40 plus 40, which is 120, and multiplying is just the fast way to say it.

Nothing breaks when the numbers stop being whole. A box 8.4 by 5.1 by 2.7 still holds 115.67 cubic units even though no whole cube fits neatly inside it, because the cube picture is a way of understanding the rule rather than a condition for using it. But it is the picture that makes the rest of this lesson obvious rather than memorised.

Any face can be the base

Stand the same box on a different face and nothing about its contents changes. Take the 8 by 3 face as the floor and each layer holds 24 cubes, with 5 layers to stack: 120 again. Take the 5 by 3 face and it is 15 cubes in a layer, 8 layers high: 120 for the third time. The box does not know which way up it is.

That freedom is more than a curiosity. It means the real rule is base area times height, and once it is written that way it stops being about boxes. A triangular prism has a triangle for a base, a cylinder has a circle, and both of them multiply that base by the distance to the far end in exactly the same move. Length times width times height is that general rule applied to the one base shape simple enough to hide it.

A worked check you can repeat on paper

The last two rows are the interesting ones. The same box that holds 120 cubic units of space needs 158 square units of cardboard to build, and those two numbers answer completely different questions. A cube of side 4.93 holds that identical 120 but needs only 146 square units of material, about 8 percent less, which is why packaging designers push boxes towards cubes whenever the contents let them.

Litres, cubic metres and the factor of a million

Two conversions do almost all the work. A litre is 1000 cubic centimetres, and a cubic metre is 1000 litres. The trap sits between them: a metre is 100 centimetres, so a cubic metre is 100 cubed cubic centimetres, which is a million. Converting a length scales by 100, converting a volume scales by 1000000, and people lose whole orders of magnitude by forgetting to cube the factor.

The fix is to pick your unit before you multiply, not after. If the answer is going to be ordered in cubic metres, put every measurement into metres first, even the 150 mm slab that becomes an awkward looking 0.15.

Box	Dimensions	Volume	In useful units
Worked example	8 by 5 by 3	120	cubic units
Fish tank	60 by 30 by 36 cm	64800	64.8 litres, about 64.8 kg of water
Concrete slab	5 by 4 by 0.15 m	3.00	3 cubic metres, or 3000 litres
Air in a room	4.2 by 3.6 by 2.4 m	36.29	36288 litres
Shipping carton	40 by 30 by 20 cm	24000	24 litres

Volumetric weight, where couriers use this formula against you

Carriers do not want to fly air, so most of them bill on whichever is greater: the actual weight of a parcel or its volumetric weight. That second number is length times width times height in centimetres, divided by a fixed figure that is usually 5000 for road and express, or 6000 for standard air freight. Check the number your carrier uses, because it changes the bill.

Take the 40 by 30 by 20 cm carton from the table. Its volume is 24000 cubic centimetres, so at 5000 the volumetric weight is 4.8 kg and at 6000 it is 4 kg. If the contents actually weigh 2 kg you are still billed as though they weighed 4.8, and the only way to fix that is to make the box smaller. Cutting the 20 cm dimension to 15 takes the volumetric weight to 3.6 kg without touching what is inside.

Why volume does not tell you how many boxes fit

This is the honest limit of the formula, and it catches people who have done everything else right. A crate 120 by 100 by 100 cm holds 1200000 cubic centimetres. A carton 40 by 30 by 25 cm holds 30000. Divide one by the other and you get 40, so 40 cartons should fit.

They do not. Line them up the best way and you get 3 along the 120, 3 along one 100 and 4 up the other 100, which is 36. The 100 cm side takes three cartons of 30 with 10 cm left over, and 10 cm of empty air is still 10 cm. Volume division gives you an upper bound and nothing more. Whenever the answer matters, divide dimension by dimension, take the whole number part of each, and try the six ways of turning the carton before you settle on one.

Mistakes that catch people out

• Measuring the outside when you want the inside. Capacity is an internal question. That 40 by 30 by 20 crate with 2 cm walls holds 36 by 26 by 16, which is 14976 cubic centimetres rather than 24000. Wall thickness costs you 38 percent of the contents and none of it is visible from outside.
• Mixing units mid calculation. 4.2 m times 3.6 m times 240 cm is not a volume of anything. Convert everything into one unit before the first multiplication, never after the last.
• Converting without cubing the factor. A cubic metre is 1000000 cubic centimetres, not 100. The same trap turns a cubic foot into 1728 cubic inches rather than 12.
• Answering in square units. Three lengths were multiplied, so the unit carries a 3. Square units belong to the surface area, which for this box is 158 rather than 120.
• Assuming a container is a rectangular prism. Skips, buckets, hoppers and most plant pots taper, so their layers are not identical and this formula overstates them. A tapered container needs the frustum rule instead.

Try these

Common questions

What is the formula for the volume of a rectangular prism?

V = length × width × height. A box measuring 8 by 5 by 3 holds 120 cubic units. The same rule written as base area times height covers every prism, not just boxes.

Does the order of the three measurements matter?

No. Any face can act as the base. For an 8 by 5 by 3 box the three routes are 40 times 3, 24 times 5 and 15 times 8, and all of them give 120.

How do I find a missing side from the volume?

Divide the volume by the product of the two sides you know. A box holding 180 on a base of 6 by 5 has a height of 180 over 30, which is 6.

How do I turn the answer into litres?

Divide cubic centimetres by 1000, or multiply cubic metres by 1000. A tank 60 by 30 by 36 cm holds 64800 cubic centimetres, which is 64.8 litres, and water at that volume weighs about 64.8 kg.

What is the difference between volume and surface area?

Volume is the space inside, in cubic units. Surface area is the material needed to cover the outside, in square units. The 8 by 5 by 3 box gives 120 and 158, two right answers to two different questions.

Can I work out how many small boxes fit by dividing volumes?

Only as an upper bound. A crate 120 by 100 by 100 cm has 40 times the volume of a 40 by 30 by 25 cm carton but only holds 36 of them, because the leftover strips are too narrow to use.

Related lessons and calculators

• Surface area of a rectangular prism, the same box measured for what it takes to cover rather than for what it holds.
• Volume of a triangular prism, the same base times height rule with a triangle doing the work.
• Volume of a pyramid, where the layers shrink as you go up and the answer drops to a third.
• Volume of 3D shapes calculator, including the frustum rule for containers that taper.
• Area formulas for every common shape, the index this lesson sits inside.
• NIST on SI units, the reference for why volume is cubed and how the litre relates to the cubic metre.
• Volume of a cylinder, the same base area times height rule with a circle in the base slot.