Why is the volume of a cylinder the area of its circle times its height?

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

1. From Year 8 (circles): a circle has radius 5 cm. What is its area, in terms of π?
   Answer

   25π cm². The area is π × 5 × 5.
2. From Year 7 (volume of cuboids): a cuboid is 4 cm by 5 cm by 6 cm. What is its volume?
   Answer

   120 cm³. The base is 4 × 5 = 20 cm², and 20 × 6 = 120.
3. Work out 4² × 10.
   Answer

   160. Square first: 4² = 16, then 16 × 10 = 160.

Words used in this lesson

cylinder

a solid with two equal circles as its ends and a curved side. For example: a tin of beans.

radius

the distance from the centre of a circle to its edge. For example: 5 cm.

diameter

the distance across a circle through its centre, twice the radius. For example: 10 cm.

base area

the area of the face the solid stands on. For example: π × 5² = 25π cm².

height

how tall the solid is, at right angles to its base. For example: 8 cm.

volume

how much space a solid takes up. For example: 200π cm³.

in terms of π

written with the symbol π instead of a decimal. For example: 160π cm³.

Two cooking pots

Two cylindrical pots sit on a stove in a kitchen in Chennai. Pot A has radius 6 cm and height 10 cm. Pot B is half as wide and twice as tall: radius 3 cm and height 20 cm.

Predict first: does Pot B hold more than Pot A, less, or exactly the same? Halving one measurement and doubling another feels like it should cancel. Then use the figures to find out how a cylinder's volume depends on each measurement.

Radius 3 cm, height 5 cm: each layer has area π × 3² = 9π cm², and 5 layers make 45π ≈ 141.4 cm³.

Show me the steps

1. Radius 3 cm, height 5 cm: one layer is π × 3² = 9π cm² (about 28.3 cm²). Five layers: 9π × 5 = 45π ≈ 141.4 cm³.
2. Make the height 10 cm and keep the radius 3 cm: 10 layers of 9π cm² give 90π ≈ 282.7 cm³. Twice the layers, twice the volume.
3. Return the height to 5 cm and make the radius 6 cm: one layer is π × 6² = 36π cm², so the volume is 36π × 5 = 180π ≈ 565.5 cm³. Twice the radius gives four times the area of each layer.
4. Radius 1 cm, height 1 cm: a single layer, π × 1² × 1 = π ≈ 3.1 cm³. The volume of the smallest cylinder here is exactly π.

Nothing changes: radius 2 cm and height 5 cm give π × 2² × 5 = 20π ≈ 62.8 cm³, which is × 1 the original.

Show me the steps

1. Nothing changes: radius 2 cm, height 5 cm, volume π × 2² × 5 = 20π cm³. This is the original.
2. Double the height to 10 cm: π × 2² × 10 = 40π cm³. The volume is × 2, because the height is a factor on its own.
3. Double the radius to 4 cm and keep the height 5 cm: π × 4² × 5 = 80π cm³. The volume is × 4, because the radius is squared and 2² = 4.
4. Double both: π × 4² × 10 = 160π cm³. The volume is × 8, which is 4 × 2.

Worked example: a tin of chickpeas

Your go. The first step is given. A section of bamboo water pipe is 10 cm across and 40 cm long. Find the volume it holds, in terms of π and to one decimal place.

With a calculator

Keep π as a symbol for as long as you can: write 160π cm³ and only turn it into a number at the end. To do that, press 160 × the π key. The calculator shows 502.654… and the answer to one decimal place is 502.7.

Every answer on this page that is rounded uses the π key, not 3.14 (3.14 would give 502.4). Estimate first, with 3 for π: 3 × 160 = 480, so an answer near 500 is sensible and an answer near 50 or 5000 is a slip.

Check before you go on

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

Level 2 · Building

Level 3 · Stretch

Something to think about

Exit ticket