Volume of a cube and a cuboid · Lập trình với Martin

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. Bắt đầu 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.