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CalcMax

Cuboid Volume Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

80.0000 cm³

Volume

Total surface area
112.0000 cm²

A cuboid volume calculator takes the length, width and height of a rectangular box and returns two figures: how much space the box encloses, in cubic centimetres, and how much surface its six faces add up to, in square centimetres. Both are asked for because a box is usually wanted for both at once — how much it holds, and how much material it takes to make or cover. The volume is the length multiplied by the width multiplied by the height, which is the one formula here that everyone already knows without being told: it is how many unit cubes fit inside. The surface area is two times the sum of the three pairs of faces, because opposite faces of a box are identical, so there are only three distinct rectangles to work out and each of them appears twice. The two quantities behave differently when the box changes size, and that difference is the single most useful thing on the page. Scale a box up by a factor of two in every direction and its surface area goes up fourfold while its volume goes up eightfold. That is why a large container holds disproportionately more than a small one while needing disproportionately less material per unit of contents, and it is why an animal twice the length of another is not twice its weight. The table shows a pair of rows for exactly this reason: 3, 4 and 5 scales to 6, 8 and 10, the surface area going from 94 to 376 and the volume from 60 to 480. The units need a word because they are the commonest place to go wrong. Each of the three boxes accepts centimetres, metres, millimetres, inches or feet, and both answers come back in centimetres — cubed for the volume, squared for the surface area. A box measured in inches therefore gives a volume that has to be divided by 16.387064 to be read in cubic inches, not by 2.54 and not by 6.4516. Volumes convert by the cube of the length factor, and reaching for the plain length factor is the mistake to watch for. Every one of the three dimensions must be filled in. This page has no arrangement where you supply two of them and it works out the third, and it has no dropdown for choosing a shape — it is a box and only a box. If the figure is a cylinder, a cone or a sphere, that is another page; if it is a round tank, the aquarium page takes the same three measurements and answers in litres and gallons. One thing about the output order. The volume is the first line and the surface area is the second, whichever one you came for. That is because the page is named after the volume, and because the volume is the figure with a single, universally known formula while the surface area is the one people have to look up.

Common boxes: the volume and the surface area from the three edge lengths

Length (cm)Width (cm)Height (cm)Volume (cm³)Surface area (cm²)
11116
3456094
54480112
6810480376
1010101000600
1286576432
20151030001300
0.50.50.50.1251.5

Eight boxes and five columns: three dimensions going in, two figures coming out. The second row and the fourth are the pair that carries the point of the page — 3, 4 and 5 doubled to 6, 8 and 10, with the volume going from 60 to 480 while the surface area only goes from 94 to 376. The third row is the set the page loads with, 5, 4 and 4, and it is also the case the surface area page works through for its prism option, 112 on both. The first row is the unit cube, volume one and surface six, which is the smallest box there is and a useful thing to have in mind when reading a surface figure. The fifth row is the cube, and the sixth and seventh are flatter and longer boxes that hold a good deal more; comparing them with the cube is what shows that a box pays for its shape, not just its size. The sixth holds 576 cubic centimetres behind 432 square centimetres of surface, three quarters of a square centimetre per cubic centimetre, where the cube manages six tenths — so the cube is the most economical box for what it holds, and every step away from it costs cardboard. The last row is a half-centimetre cube, where the volume on a calculator is written out with its three decimals, 0.125, because the arithmetic is exact rather than rounded to the four places the display allows.

Formula

V = lwh S = 2(lw + lh + wh)

l
The length of the box in centimetres — any of the three edges, as long as the other two are the width and the height. A box does not care which name goes on which edge
w
The width, in centimetres. Together with the length it fixes the rectangle of the base, which is the face the box stands on
h
The height, in centimetres: how far the box rises above its base. This is the third direction, and the one that turns a rectangle of area into a solid of volume
lwh
The volume — three lengths multiplied. It is how many cubes one centimetre on a side would fit inside, which is what a cubic centimetre means
lw, lh, wh
The three distinct rectangles among the six faces: the base, the front and the side. The other three faces are the same three rectangles again, which is why the surface formula does not have to add up six separate areas
2(lw + lh + wh)
The surface area, in square centimetres. The factor of two stands for the fact that opposite faces of a box are identical, and it is the step people leave out when they add up six faces and count each pair twice
Four decimal places
How wide the readings are printed. Both figures are exact whenever the three lengths are, so the four decimals are a display width rather than a sign of approximation — 0.125 is written out in full

Packing and shipping are the plainest case. The volume of a parcel is what a courier charges by, and the surface area is the cardboard it takes to make the box — the two numbers a business needs before it commits to a carton size, and the volume is also what decides whether a given item fits. Storage is the second: how much a crate, a drawer, a storage box, a filing box or a shelf unit holds, and how much of a wall its outside will cover. Materials are the third: the surface area is what a box has to be painted, wrapped, lined, powder-coated or covered in, so a job quoted per square metre starts with this formula, and the volume is what a box has to be filled with when the filling is sold by volume. In construction and civil engineering it is the concrete in a rectangular footing, the spoil from a rectangular excavation, the water in a rectangular tank or pond, and the fill in a rectangular bed — all of them are this one multiplication, and getting the three dimensions in the same unit is the part that actually goes wrong. In manufacturing it is the material in a machined block, the capacity of a hopper, and the volume of a billet before it is worked. In the sciences it is the density of anything that can be cut into a rectangular solid, since density is mass divided by volume, and the volume of a sample cell or a rectangular specimen. In the classroom it is the demonstration that volume and surface area are different things that both come from the same three numbers: build a box of centimetre cubes, count the cubes for the volume, count the squares on the outside for the surface, and the fact that the two counts scale differently when the box is doubled is the whole lesson. And in everyday arithmetic it is whether a piece of furniture will fit through a doorway, how much soil a raised bed needs, and whether the box in the cupboard is big enough for what is going into it.

Worked examples

  1. A box 5 by 4 by 4

    1. Base rectangle: 5 × 4 = 20
    2. The other two rectangles: 5 × 4 = 20 and 4 × 4 = 16
    3. Sum the three and double it: 2 × (20 + 20 + 16) = 112 square centimetres
    4. Multiply all three lengths: 5 × 4 × 4 = 80 cubic centimetres

    The dimensions the page loads with, and the case the surface area page's worked example uses for its prism option. Both pages print 112 for it, which is worth checking: the two pages arrive at the figure by different routes and a disagreement between them would be a bug in one of the two. Note that two of the three dimensions are equal here, so two of the three distinct faces are the same rectangle.

  2. A box 3 by 4 by 5

    1. The three rectangles: 3 × 4 = 12, 3 × 5 = 15 and 4 × 5 = 20
    2. Sum and double: 2 × (12 + 15 + 20) = 94 square centimetres
    3. Multiply all three: 3 × 4 × 5 = 60 cubic centimetres

    The classic set of dimensions, and the pair to watch next to the row below it: doubling every edge is not just a bigger box, it is a different shape of answer. This row is the smaller half of the demonstration the page is really about.

  3. The same box doubled: 6 by 8 by 10

    1. The three rectangles: 6 × 8 = 48, 6 × 10 = 60 and 8 × 10 = 80
    2. Sum and double: 2 × (48 + 60 + 80) = 376 square centimetres
    3. Multiply all three: 6 × 8 × 10 = 480 cubic centimetres

    Every edge doubled, so the surface area is four times the previous row's 94 and the volume is eight times its 60. This is the most useful thing on the page: doubling a box does not double what it holds or what it is made of, and the two do not even grow at the same rate as each other. It is the same relationship that makes a large container cheaper per litre than a small one.

  4. A twelve by eight by six box

    1. The three rectangles: 12 × 8 = 96, 12 × 6 = 72 and 8 × 6 = 48
    2. Sum and double: 2 × (96 + 72 + 48) = 432 square centimetres
    3. Multiply all three: 12 × 8 × 6 = 576 cubic centimetres

    A flatter and longer box, and the row to compare with the cube in the table. It holds 576 cubic centimetres and has 432 square centimetres of surface, which is three quarters of a square centimetre of cardboard for every cubic centimetre it holds; the cube holds 1000 and has 600, which is six tenths. Among all boxes of a given volume the cube is the one with the least surface, and every departure from it costs material — which is why a cube-shaped carton is the efficient one and a flat box of the same capacity is not.

  5. A box with no height: 5 by 5 by 0

    1. The three rectangles: 5 × 5 = 25, 5 × 0 = 0 and 5 × 0 = 0
    2. Sum and double: 2 × (25 + 0 + 0) = 50 square centimetres
    3. Multiply all three: 5 × 5 × 0 = 0 cubic centimetres

    A box pressed flat into a sheet. The volume is zero, which is correct and is not the page failing to answer, and the surface area is not zero — it is 50, the two faces that are still there. Zero is a legal value in any of the three boxes: measuring zero is a measurement, and only leaving a box empty is the case with nothing to work from.

Limitations

This page computes a rectangular box and nothing else: not a cylinder, a cone, a sphere, a pyramid, nor an irregular solid, and it does not accept a volume and work backwards to a missing edge. All three dimensions are required, and none of them may be negative. Both answers come back in centimetres — cubed for the volume, squared for the surface — whatever unit each box is set to, because the conversion happens on the way in. A box measured in inches therefore gives a volume that has to be divided by 16.387064 to be read in cubic inches: volumes convert by the cube of the length factor, and dividing by 2.54 or by 6.4516 by analogy with a length or an area is the mistake to avoid. There is no output for the diagonal through the middle of the box, and the diagonal a reader may have in mind from a screen specification is the diagonal of a face, which is a different length. Nothing here handles a box with rounded corners, a box with an open top, or the thickness of the material it is made from — an open-top box takes less material than the figure printed here, and the difference is the face you left off. Four decimal places is a display width; the arithmetic itself is exact whenever the three inputs are, so a figure like 0.125 is written out in full rather than rounded. And nothing here converts between cubic centimetres and litres: the aquarium page takes the same three measurements and answers in litres and gallons if that is what is wanted.

Frequently asked questions

Why does doubling every side multiply the volume by eight?
Because each of the three lengths is doubled and they are multiplied together, so the factor of two enters three times: two times two times two is eight. The surface area only doubles twice, because area is a product of two lengths, so it goes up fourfold. That gap is why a big container holds far more than a small one while using proportionally less material, and why a small box of the same shape is never a scaled-down bargain.
I entered inches. Why is the volume in cubic centimetres?
Because the conversion happens on the way in, not on the way out. Your inches are turned into centimetres and both answers come back in centimetres. To read the volume in cubic inches, divide by 16.387064 — that is 2.54 cubed. Dividing by 2.54 instead, the way you would for a length, gives an answer that is wrong by a factor of about forty.
Which of the three boxes is the height?
Whichever one you are treating as vertical — a box does not care which name goes on which edge, because the volume and the surface area are symmetric in all three. The page uses the word height in the sense of a figure's height rather than a tank's or a person's, but the arithmetic will not change if you swap two of them round.
Is this the same as the aquarium volume page?
The geometry is identical and the three measurements are the same three, but the answers are not. That page treats the figure as a container and answers in litres and gallons, and it is capped at dimensions a tank could sensibly have. This page treats it as a geometric solid and answers in cubic centimetres with the surface area alongside, which is what you want when the box is a material object rather than a vessel.
Can I get the surface area on its own, or a different shape?
The surface area page takes a sphere, a cone, a cylinder or a rectangular prism and returns only the outside area, so it covers this box among three other figures. Its prism option is the same formula this page uses, and the two agree: a box of 5 by 4 by 4 reads 112 on both. If you want one figure at a time with no choice to make, this page is the quicker route.
Why is there no diagonal for the box?
Because the diagonal most people have in mind, and the one this site already reports elsewhere, is the diagonal across a flat face — the figure quoted for screens. A box also has a diagonal through the middle from one corner to the opposite corner, and it is a different length with a different formula. Printing one under the plain name diagonal would make the label wrong on one of the two pages, so this page leaves it out.

References

Related calculators

Surface Area CalculatorSurface area of spheres, cones, cylinders and prisms.Aquarium Volume CalculatorTank volume in litres and gallons from its dimensions.Cylinder Volume CalculatorVolume of a cylinder, plus the areas of its ends and its curved side.Rectangle CalculatorRectangle area, perimeter and diagonal from length and width.