Volume of a cube and a cuboid · 跟马丁学编程

How many little cubes fit inside

Surface area was paper for wrapping. Volume is what fits inside: how much water into a tank, how much sand into a crate, how much Lego into a box.

Volume is measured in cubic units. A cubic centimetre (cm³) is a little cube 1 × 1 × 1 cm — about like a sugar cube. How many such little cubes fit into a box, that is the box’s volume.

And again there is a trick with multiplying instead of counting one by one. A cuboid 4 × 3 × 2 cm you fill with little cubes like this: on the bottom a layer of 4 × 3 = 12 little cubes fits. And there are two layers on top of each other (height 2 cm). In total 12 × 2 = 24 little cubes.

From that the formula: V = a × b × c — length times width times height. With a cube all the edges are the same: V = a × a × a.

From a surface (two multiplications of a side, cm²) you have moved into space (three multiplications, cm³). A third dimension, a third letter in the formula, a three in the unit.

Adam is building a storage box from Lego: “Will my whole collection fit?” The box has an inside 40 × 30 × 20 cm. Ema counts: “The bottom holds a layer 40 times 30 — that is 1 200 little cubes a centimetre by a centimetre. And there are twenty layers.” She multiplies: 1 200 × 20 = 24 000 cm³. “Twenty-four litres,” she adds. Adam does not get where the litres came from. “A litre is a cube ten by ten by ten — a thousand cm³. We will leave that for the next lesson.” The collection fitted with room to spare. Layers times height — volume is a times table in 3D.

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Picture volume in layers: the bottom (a × b little cubes) times the number of storeys (c). The formula V = a × b × c is just a shortcut for that picture.

The formula and counting in layers

Cuboid: V = a × b × c.

Bottom layer: a × b little cubes (that is actually the area of the bottom!).
Number of layers: c (height in centimetres = number of storeys of little cubes).
Volume: layer × storeys = a × b × c.

For a cuboid 5 × 4 × 3 cm: bottom 5 × 4 = 20, three storeys → V = 60 cm³.

Cube: V = a × a × a. Edge 4 cm: V = 4 × 4 × 4 = 64 cm³. Watch, a × a × a is not a × 3! Four times three is 12, but 4³ is 64.

The order of multiplying does not matter: 5 × 4 × 3 = 3 × 4 × 5 = 60. A box on its side has the same volume as a box standing up — nothing is lost by tipping.

Units: edges in the same unit → volume in that unit to the third. Centimetres → cm³. Metres → m³. And as always: a metre with a centimetre must not be multiplied, convert first.

The other way: I know the volume and two edges, I look for the third. A tank has 24 000 cm³, bottom 40 × 30 cm. Height = 24 000 : (40 × 30) = 24 000 : 1 200 = 20 cm. Volume divided by the area of the bottom = height.

Volume of a cuboid 6 × 4 × 5 cm

Layer on the bottom: 6 × 4 = 24 little cubes.

Storeys: 5.

Volume: V = 24 × 5 = 120 cm³.

An estimate check: a soap box, about a hundred sugar cubes fit in — it sits.

And for comparison with the last lesson: the surface area of the same cuboid is 2 × (24 + 30 + 20) = 148 cm². A different number, a different unit, a different question.

Volume of a cube with edge 5 cm

V = a × a × a = 5 × 5 × 5.

In steps: 5 × 5 = 25, then 25 × 5 = 125 cm³.

A common mistake would be 5 × 3 = 15 — but the three in “to the third power” says HOW MANY TIMES you multiply, not BY WHAT.

A staircase of cubes for a picture: edge 1 cm → 1 cm³, edge 2 → 8, edge 3 → 27, edge 4 → 64, edge 5 → 125. Twice the edge = eight times the volume!

How much water is in a tank?

The tank has a bottom 50 × 30 cm. The water reaches a height of 24 cm (not to the rim!).

Volume of water: V = 50 × 30 × 24.

In steps: 50 × 30 = 1 500 (area of the bottom). Then 1 500 × 24 = 36 000 cm³.

Notice: you count the height of the WATER, not the height of the tank. The volume of a solid is a formula — but what you put into it, the word problem decides. Next time we will show that 36 000 cm³ is 36 litres.

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Do not mix surface area and volume — and their units. Volume comes out in cm³ (three multiplied lengths), surface area in cm² (two). If for “how much water fits” you get cm², you counted the wrapping instead of the inside. A three in the unit = three dimensions in play.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Work out the volume of a cuboid: V = a × b × c, happily in layers. The answer is a number (in cm³).

Practise the questions

On paper

If you have cubes at home (Lego, sugar), build. Who once built a cuboid from little cubes does not forget the formula.

Build (or draw in layers) a cuboid 4 × 3 × 2 from little cubes. Count the little cubes in layers and check with the formula.
Work out the volumes: a cuboid 10 × 6 × 4 cm, a cube with edge 3 cm, a cube with edge 6 cm. How many times is the volume of the second cube bigger than the first?
A box has volume 480 cm³, bottom 12 × 8 cm. How high is it? Do a check by multiplying.
A sandpit is 2 m long, 2 m wide and the sand in it is 0.25 m high. How many m³ of sand is in it?

Now you 💪

I can work out the volume of a cuboid with the formula V = a × b × c and explain it through layers.
I know that a cube with edge 5 has volume 125, not 15.
I can work out the third edge from the volume and the area of the bottom.

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Done when: You work out the volume of a cube and a cuboid, you understand it as a count of little cubes in layers, and you do not mix it up with surface area.

What to take from this lesson

Volume = how many unit little cubes fit inside. Cubic units: cm³, m³.
Cuboid: V = a × b × c (a layer on the bottom times the number of storeys). Cube: V = a × a × a.
a × a × a is not a × 3: a cube with edge 5 has volume 125 cm³.
The order of multiplying does not matter, the units of the edges must be the same.
Height = volume : area of the bottom. The formula works backwards too.