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Capacity and volume together

Measurement · level 4, lesson 6 of 7

A question to start

A fish tank is 60 cm long, 30 cm wide and 40 cm tall. How many litres of water does it hold when full? And could you lift it?

The tank holds 72 litres. That much water has a mass of about 72 kilograms, more than many grown-ups. That is why you should never try to move a full fish tank, and why big tanks sit on strong stands. In this lesson you join two ideas you already know, volume (cubic centimetres) and capacity (litres), with one exact link.

The idea

Two names for the same space

In level 3 you measured capacity, how much a container holds, in millilitres and litres. In the last lesson you measured volume, how much space something takes up, in cubic centimetres and cubic metres.

These are really the same thing seen in two ways. The capacity of a jug is just the volume of the space inside it. We tend to say "capacity" and use litres for liquids and containers, and say "volume" and use cubic units for solid things and spaces. But they measure the same stuff, so there must be a link between the units. Here it is.

When the metric system was planned, the litre was defined as the space inside a cube 10 cm long, 10 cm wide and 10 cm tall. (Today the litre is defined as exactly 1000 cm³, which says the same thing.)

10 cm 1 small cube = 1 cm³ = 1 mL one layer: 10 × 10 = 100 cm³ 10 layers: 1000 cm³ the whole cube = 1 litre

How many centimetre cubes fill that litre cube? Use the layer method: one layer is \(10 \times 10 = 100\) cubes, and there are 10 layers, so \(100 \times 10 = 1000\) cubes. A litre is 1000 cm³.

But a litre is also 1000 mL. So 1000 mL and 1000 cm³ are the same amount of space, and sharing each into 1000 equal parts gives

\[1 \text{ mL} = 1 \text{ cm}^3, \qquad 1 \text{ L} = 1000 \text{ cm}^3\]

A millilitre is a cubic centimetre, with a different name. Doctors and nurses often write "cc" (cubic centimetre) on a syringe for the same amount as 1 mL.

Cubic metres and litres

A cubic metre is a cube 100 cm on each edge. How many litre cubes (10 cm each) fit inside? Along each edge, \(100 \div 10 = 10\) litre cubes fit. So one layer holds \(10 \times 10 = 100\) litre cubes, and there are 10 layers:

\[1 \text{ m}^3 = 10 \times 10 \times 10 = 1000 \text{ L}\]

A cubic metre is a thousand litres. In cubic centimetres it is \(100 \times 100 \times 100 = 1\,000\,000\) cm³: a million. (Changing between square and cubic units is the main idea of level 5.)

And mass, for water

In level 3 you saw that 1 mL of water has a mass of about 1 g, and 1 L of water about 1 kg. Now we can add:

Space Water in it has a mass of about
1 cm³ = 1 mL 1 g
1000 cm³ = 1 L 1 kg
1 m³ = 1000 L 1000 kg = 1 tonne

A cubic metre of water weighs about a tonne, as much as a small car. A full bathtub, at about 100 to 150 L, holds only a small part of that.

Inside and outside

The capacity of a container is the space inside it. If a tank has glass walls 1 cm thick, the inside is a little smaller than the outside measurements. For a rough answer we often ignore this, but a careful answer uses the inside measurements.

Measuring odd shapes with water

How do you find the volume of a stone, a toy or a key? They are not boxes, so length × width × height does not work. But water can measure them for you.

100200300400500 100200300400500 300 mL of waterwith the stone: 380 mL

Pour some water into a measuring jug and read the level: 300 mL. Lower the stone in gently, so that it sinks and is completely under water. The water rises to 380 mL. The stone has pushed aside its own volume of water, so the stone's volume is the rise: \(380 - 300 = 80\) mL, which is 80 cm³. This is called measuring by displacement.

The story goes that the Greek thinker Archimedes, about 2200 years ago, realised this while getting into a full bath and watching it overflow, and that he used it to test whether a king's crown was pure gold. Whether or not the bath story is true, the method works.

Worked examples

1. The fish tank. 60 cm by 30 cm by 40 cm.

Volume: \(60 \times 30 \times 40 = 72\,000\) cm³. Litres: \(72\,000 \div 1000 = 72\) L. Mass of the water: about 72 kg.

2. Not quite full. The same tank is filled only to 35 cm deep. How much water?

The water is a box 60 by 30 by 35: \(60 \times 30 \times 35 = 63\,000\) cm³ = 63 L.

3. How deep? You pour 2 L of water into an empty box-shaped container whose base is 20 cm by 10 cm. How deep is the water?

2 L = 2000 cm³. Each 1 cm layer of water holds \(20 \times 10 = 200\) cm³. \(2000 \div 200 = 10\) layers, so the water is 10 cm deep.

4. A hot tub. A hot tub is 2 m long, 2 m wide and filled 0.9 m deep. How many litres?

\(2 \times 2 \times 0.9 = 3.6\) m³, and each m³ is 1000 L, so \(3.6 \times 1000 = 3600\) L. The water alone has a mass of about 3600 kg: 3.6 tonnes.

5. Displacement. A jug has 450 mL of water. A toy car is dropped in and sinks; the level rises to 520 mL. The toy's volume is \(520 - 450 = 70\) mL = 70 cm³.

6. Rain. In a storm, 1 mm of rain falls on a flat roof of area 1 m². How many litres of water is that?

The rain makes a thin box 100 cm by 100 cm by 0.1 cm (1 mm = 0.1 cm): \(100 \times 100 \times 0.1 = 1000\) cm³ = 1 L. So "1 mm of rain" means 1 litre on every square metre. A roof of 80 m² collects 80 L from each millimetre of rain.

Try it

  1. Change: (a) 250 cm³ to mL; (b) 3.5 L to cm³; (c) 4000 cm³ to L; (d) 2 m³ to L.
  2. A box-shaped vase is 10 cm by 10 cm by 25 cm inside. How many litres does it hold?
  3. An aquarium is 80 cm by 35 cm by 45 cm. How many litres when full? About how heavy is the water?
  4. A jug holds 600 mL of water. A stone is added and the water rises to 735 mL. What is the volume of the stone?
  5. A rectangular water tank on a roof is 2 m by 1.5 m by 1 m. How many litres? How many 250 mL glasses could it fill?
  6. You pour 1.5 L of juice into a box-shaped jug with base 15 cm by 10 cm. How deep is the juice?
  7. A garden pond is a box shape 3 m by 2 m by 0.5 m deep. How many litres to fill it? If a hose delivers 20 L each minute, how long will it take?
  8. 25 mm of rain falls on a roof of 60 m². How many litres could be collected?
Answers
  1. (a) 250 mL. (b) \(3.5 \times 1000 = 3500\) cm³. (c) \(4000 \div 1000 = 4\) L. (d) \(2 \times 1000 = 2000\) L.
  2. \(10 \times 10 \times 25 = 2500\) cm³ = 2.5 L.
  3. \(80 \times 35 \times 45 = 126\,000\) cm³ = 126 L, so about 126 kg of water.
  4. \(735 - 600 = 135\) mL = 135 cm³.
  5. \(2 \times 1.5 \times 1 = 3\) m³ = 3000 L. Four 250 mL glasses make 1 L, so it fills \(3000 \times 4 = 12\,000\) glasses.
  6. 1.5 L = 1500 cm³. Each 1 cm layer holds \(15 \times 10 = 150\) cm³. \(1500 \div 150 = 10\) cm deep.
  7. \(3 \times 2 \times 0.5 = 3\) m³ = 3000 L. At 20 L per minute, \(3000 \div 20 = 150\) minutes, which is 2 hours 30 minutes.
  8. Each mm gives 1 L per m², so 25 mm gives 25 L per m². On 60 m²: \(25 \times 60 = 1500\) L.

Common mistake

Thinking 1 m³ is 100 L or 1000 cm³. A cubic metre is huge: 1000 litres, or a million cubic centimetres. Picture the 10 by 10 by 10 stack of litre cubes inside it. Dividing by 1000 turns cm³ into litres, not cm³ into m³.

Forgetting to sink the object. With displacement, the object must be completely under water. A floating cork only pushes aside part of its volume, so the rise is too small. Push it under with a thin wire.

Where it leads