Cuboids and liquid volume

Count each layer, then the layers

A cube with every edge 1 cm fills one cubic centimetre, 1 cm³. Volume counts the space filled by these units, including units hidden behind the front layer.

A cuboid is 8 cm long, 5 cm wide and 3 cm high. The base fits 8 × 5 = 40 one-centimetre cubes in one layer. Three layers contain 40 × 3 = 120 cubes. Its volume is 120 cm³.

All three lengths must use the same unit. For a cube with edge 4 cm, the volume is 4 × 4 × 4 = 64 cm³. Do not confuse this with the area 4 × 4 = 16 cm² of one face.

A cube with every edge 1 m fills one cubic metre, 1 m³. A cuboid 2 m by 3 m by 4 m has volume 2 × 3 × 4 = 24 m³. Keep the three lengths in the same unit; these examples do not require converting between cm³ and m³.

Check your understanding 1 Use all three dimensions

A cuboid is 6 cm long, 4 cm wide and 2 cm high. A cube has edge 3 cm. Find each volume and explain why multiplying only two of the cuboid’s dimensions would give a different kind of measure.

Your cuboid volume, cube volume and explanation

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The cuboid has volume 6 × 4 × 2 = 48 cm³. Its 6-by-4 base has area 24 cm²; the height supplies the number of 1 cm layers. The cube has volume 3 × 3 × 3 = 27 cm³, not 3 × 3 = 9 cm².

A liquid fills to its own height

A rectangular tank has internal base dimensions 20 cm by 15 cm. Water is 12 cm deep. Its volume is 20 × 15 × 12 = 3600 cm³. Use the water depth rather than the tank’s full height. An empty tank has zero water volume even though its capacity is positive.

Connect cubic centimetres and liquid units

One millilitre occupies one cubic centimetre: 1 ml = 1 cm³. Therefore the water is 3600 ml, or 3.6 ℓ. This link compares volume measures; it does not say 1 cm is 1 ml.

The relationship also works backwards: 1.25 ℓ is 1250 ml, so it occupies 1250 cm³. This does not require converting between cm³ and m³.

If the tank is 20 cm high, its full capacity is 6000 cm³, or 6 ℓ. It has room for another 6-3.6 = 2.4 ℓ, assuming the dimensions are internal and no objects occupy space in it.

Check your understanding 2 Read the right height

A tank has internal base 30 cm by 20 cm and height 25 cm. It contains water to depth 10 cm. Find the water volume in litres and the extra amount needed to fill it.

Your current water volume and amount needed to fill the tank

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The water occupies 30 × 20 × 10 = 6000 cm³ = 6 ℓ. Capacity is 30 × 20 × 25 = 15 000 cm³ = 15 ℓ. Another 9 ℓ is needed. The 25 cm is the tank height, not the current water depth.

Check your understanding 3 Read the direction of the conversion

Write 2.4 ℓ in cm³. Write 850 cm³ in both ml and litres. Keep the quantity of liquid unchanged.

Your three equal-volume measurements

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2.4 ℓ = 2400 ml = 2400 cm³. Also, 850 cm³ = 850 ml = 0.85 ℓ. The number changes when the unit changes; the volume of liquid stays the same. A litre is a larger unit than a millilitre, so fewer litres name the same volume.

Transfer to a change in depth

The first tank’s 20 cm by 15 cm base receives another 600 ml. How far does the water level rise, if it does not overflow?

Your depth increase and new water depth