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GCSE Maths · EdexcelChapter 19 · Volume, surface area and 3D shapes
GCSE Maths · Volume, Surface Area and 3D Shapes · Foundation · Lesson 19.6

GCSE Maths: Volume of cylinders, and density (Foundation)

Part of the GCSE Maths volume, surface area and 3d shapes chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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19.6

Volume of cylinders, and density

Edexcel 1MA1: G16, G17, R11
Grade 3-6

A cylinder is a prism with a circle as its cross-section. So the volume rule for prisms still works:

Volume of a cylinder = area of the circular end × height = πr2h

Here r is the radius of the circle (half the diameter) and h is the height (the distance between the two circular ends). You met the area of a circle, πr2, in Chapter 18. For a cylinder with radius 4 cm and height 10 cm the area of the end is π × 42 = 16π cm2, and so the volume is 16π × 10 = 160π cm3.

10 cm4 cm
8 cm6 cm
Left: a cylinder given by its radius. Right: a cylinder given by its diameter, so halve it first.

Exact answers and decimal answers. When a question says “give your answer in terms of π”, leave the π symbol in the answer: 160π cm3. This is the exact answer. When it says “give your answer to 3 significant figures” (or 1 decimal place), use the π button on a calculator: 160π = 502.65… = 503 cm3 to 3 s.f. Never use 3.14 unless the question tells you to.

Density. Lead is heavy and cork is light, even for the same size piece. Density measures how much mass is in each unit of volume: for example, steel has a density of about 7.8 g/cm3, so every cm3 of steel has a mass of 7.8 g. Density, mass and volume are linked by

density = mass ÷ volume,  mass = density × volume,  volume = mass ÷ density

Always keep the units together: a mass in grams and a volume in cm3 give a density in g/cm3; a mass in kg and a volume in m3 give a density in kg/m3. Compound measures are covered in more depth in Chapter 16.

Key ideas

  • Volume of a cylinder = πr2h, where r is the radius. If you are given the diameter, halve it.
  • “In terms of π” means leave π in the answer: for example 160π.
  • For a decimal answer use the π button, and only round at the end.
  • Density = mass ÷ volume. Mass = density × volume. Volume = mass ÷ density.
  • To find h from the volume, divide the volume by πr2.
Worked example 19.6.1
Work out the volume of this cylinder. Give your answer in terms of π.10 cm4 cm

Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.

Method. Volume of a cylinder = area of the circular end × height = πr2h.

  1. Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.
    V = πr2h
  2. Substitute the radius and the height.
    V = π × 42 × 10
  3. Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.
    42 = 4 × 4 = 16
  4. Multiply by the height.
    16 × 10 = 160, so V = 160π
  5. Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.
    V = 160π cm3

Check. Estimate with π ≈ 3: 3 × 16 × 10 = 480, which is close to 503. ✓

Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.

Worked example 19.6.2
A cylinder has diameter 10 cm and height 6 cm. Work out its volume. Give your answer correct to 3 significant figures.6 cm10 cm

Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.

Method. Volume of a cylinder = area of the circular end × height = πr2h.

  1. Find the radius. The formulas use the radius, which is half the diameter.
    r = 10 ÷ 2 = 5
  2. Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.
    V = πr2h
  3. Substitute the radius and the height.
    V = π × 52 × 6
  4. Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.
    52 = 5 × 5 = 25
  5. Multiply by the height.
    25 × 6 = 150, so V = 150π
  6. Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.
    π × 52 × 6 = 471.2389…
  7. Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.
    471.2389… = 471 cm3

Check. Estimate with π ≈ 3: 3 × 25 × 6 = 450, which is close to 471. ✓

Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.

Worked example 19.6.3
A cylinder has radius 6 cm and volume 180π cm3. Work out its height.

Method. Put the known numbers into V = πr2h, then cancel the π and divide to find h.

  1. Write the volume formula with the numbers. We know V and r. The height h is unknown.
    180π = π × 62 × h
  2. Work out r2.
    62 = 36, so 180π = 36π × h
  3. Divide both sides by π. This cancels the π on each side.
    180 = 36 × h
  4. Divide by the number in front of h. Undo the multiplication.
    h = 180 ÷ 36 = 5 cm

Check. π × 62 × 5 = 180π. ✓

Watch out. Divide by π as well as by r2. If the volume is given in terms of π, the π simply cancels.

Worked example 19.6.4
A can is a cylinder with diameter 8 cm and height 12 cm. How many litres of drink does it hold when full? Give your answer to 2 decimal places.

Method. Find the volume in cm3 using πr2h, then change cm3 to litres by dividing by 1000.

  1. Find the radius. The formula uses the radius, which is half of the diameter.
    r = 8 ÷ 2 = 4 cm
  2. Write the volume formula and substitute.
    V = π × 42 × 12 = π × 16 × 12 = 192π
  3. Use your calculator. Use the π button and keep the whole display.
    192π = 603.1858… cm3
  4. Change to litres. 1 litre = 1000 cm3, so divide by 1000.
    603.1858… ÷ 1000 = 0.60319… litres
  5. Round to 2 decimal places. The third decimal place decides whether the second goes up.
    0.60 litres

Check. Estimate: 3 × 16 × 12 = 576 cm3, so about 0.6 litres. ✓

Watch out. Divide by 1000 to change cm3 to litres. Do not round before the last step.

Worked example 19.6.5
A prism has a semicircle of radius 4 cm as its cross-section. The prism is 10 cm long. Work out its volume in terms of π.

Method. A semicircle is half a circle, so the prism is half of a cylinder with the same radius and length.

  1. Volume of the full cylinder. V = πr2h with r = 4 and h = 10 (the length).
    π × 42 × 10 = 160π
  2. Halve it. The prism is half of the cylinder.
    160π ÷ 2 = 80π
  3. Write the unit.
    80π cm3

Check. Area of a semicircle = ½πr2 = 8π, and 8π × 10 = 80π. ✓

Watch out. Halve the volume of the cylinder; do not halve the radius.

Worked example 19.6.6
A metal block has mass 450 g and volume 50 cm3. Work out its density.

Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.

Method. Density = mass ÷ volume.

  1. Write down the formula. Density is mass shared out over each unit of volume.
    Density = mass ÷ volume
  2. Substitute. Mass is in g and volume in cm3, so the answer will be in g/cm3.
    Density = 450 ÷ 50
  3. Divide.
    450 ÷ 50 = 9
  4. Write the unit.
    Density = 9 g/cm3

Check. Mass = density × volume = 9 × 50 = 450 g. ✓

Watch out. Divide the mass by the volume, not the other way round.

Worked example 19.6.7
Oak has a density of 750 kg/m3. Work out the mass of 2 m3 of oak.

Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.

Method. Mass = density × volume.

  1. Write down the formula. Rearrange Density = mass ÷ volume by multiplying both sides by the volume.
    Mass = density × volume
  2. Substitute. Density is 750 kg/m3: every m3 has a mass of 750 kg.
    Mass = 750 × 2
  3. Multiply.
    750 × 2 = 1500
  4. Write the unit.
    Mass = 1500 kg

Check. 1500 ÷ 2 = 750, the density we started with. ✓

Watch out. Check that the volume is in the same unit as in the density.

Worked example 19.6.8
A solid cylinder has radius 2 cm and height 5 cm. It is made of a metal with density 8 g/cm3. Work out the mass of the cylinder. Give your answer to the nearest gram.

Method. First find the volume of the cylinder, then use mass = density × volume.

  1. Volume of the cylinder. V = πr2h.
    V = π × 22 × 5 = 20π cm3
  2. Use the calculator for the volume. Keep the whole display.
    20π = 62.8319… cm3
  3. Use mass = density × volume. Every cm3 has a mass of 8 g.
    8 × 62.8319… = 502.655… g
  4. Round to the nearest gram.
    503 g

Check. Exact: mass = 8 × 20π = 160π g, and 160π is about 502.65. ✓

Watch out. Find the volume first. Do not multiply the density by the radius or the height.

Watch out

Use the radius, not the diameter, in πr2h. If the diameter is given, halve it first. Square only the radius. And for density, make sure the volume is in the same unit as in the density (cm3 with g/cm3, m3 with kg/m3).

Exercise 19F  Cylinders and density: exact answers (non-calculator)

Non-calculator
1
FWork out the volume of each cylinder. Give your answers in terms of π.
(a)radius 3 cm, height 10 cm
(b)radius 5 cm, height 4 cm
(c)diameter 8 cm, height 5 cm
(d)radius 1.5 cm, height 8 cm
2
FThe diagram shows a cylinder. Work out its volume. Leave your answer in terms of π.
7 cm6 cm
3
FWork out the height of each cylinder.
(a)Radius 5 cm, volume 150π cm3
(b)Radius 6 cm, volume 252π cm3
(c)Radius 4 cm, volume 96π cm3
4
PSA tin of tomatoes is a cylinder with radius 3.5 cm and height 10 cm. Use π = 22 ÷ 7 to work out the volume of the tin.
5
RCylinder A has radius 2 cm and height 9 cm. Cylinder B has radius 3 cm and height 4 cm. Show that the two cylinders have the same volume.
6
PSA prism has a semicircle as its cross-section. The semicircle has radius 6 cm. The prism is 10 cm long. Work out the volume, in terms of π.
7
FWork out each missing value.
(a)A block has mass 450 g and volume 50 cm3. Find the density.
(b)Aluminium has density 2.7 g/cm3. Find the mass of 200 cm3.
(c)A metal has density 8 g/cm3. A piece has mass 600 g. Find its volume.
8
PSA block of wood is a cuboid 10 cm by 5 cm by 4 cm. Its mass is 160 g. Work out the density of the wood.
9
PSThe density of a type of oak is 750 kg/m3. A plank has volume 0.04 m3. Work out the mass of the plank.
10
RA cylindrical water tank has radius 10 cm and height 20 cm. Show that its capacity is 2π litres.
11
PSA cylinder contains water of volume 900π cm3. The radius of the cylinder is 10 cm. Work out the depth of the water.
12
RAli works out the volume of a cylinder with diameter 10 cm and height 6 cm. He writes V = π × 102 × 6 = 600π cm3. What mistake has he made? Write the correct volume.
13
RA cylinder has radius r and height h. Write what happens to its volume when (a) the height is doubled, (b) the radius is doubled.
(a)The height is doubled
(b)The radius is doubled
14
FWhat is the area of a circle with radius 3 cm? Give your answer in terms of π.
15
FA cylinder has a circular end with area 25π cm2 and height 4 cm. Work out the volume in terms of π.
16
FWrite down the formula for density in terms of mass and volume.

Exercise 19G  Cylinders and density (calculator)

Calculator allowed
1
FWork out the volume of each cylinder. Give your answers to 3 significant figures.
(a)radius 4 cm, height 10 cm
(b)radius 2.5 cm, height 7.2 cm
(c)diameter 12 cm, height 15 cm
(d)radius 0.8 m, height 1.5 m
2
FWork out the volume of each cylinder. Give your answers correct to 1 decimal place.
(a)radius 6.4 cm, height 3 cm
(b)diameter 6.6 cm, height 11 cm
3
RA can of drink is a cylinder with diameter 6.6 cm and height 11 cm. A label says the can holds 330 ml. Is the label correct? Show your working and give a reason.
4
PSA water tank is a cylinder with radius 0.8 m and height 1.5 m. Work out the capacity of the tank in litres. Give your answer to the nearest litre.
5
PSA metal pipe is a hollow cylinder 20 cm long. The outer radius is 5 cm and the inner radius is 4 cm. Work out the volume of metal in the pipe, to 3 significant figures.
6
PSCan A has radius 3 cm and height 10 cm. Can B has radius 4 cm and height 6 cm. Which can holds more, and by how much? Give the difference to 3 significant figures.
7
PSAn iron rod is a cylinder with radius 1.5 cm and length 40 cm. The density of iron is 7.8 g/cm3. Work out the mass of the rod in kilograms, to 3 significant figures.
8
PSA cylindrical jug has radius 8 cm. It holds 2 litres when full. Work out the height of the jug, to 3 significant figures.
9
PSA trough is a prism whose cross-section is a semicircle of radius 15 cm. The trough is 2 m long. How many litres of water does it hold when full? Give your answer to 3 significant figures.
10
RA cylinder of radius 5 cm and height 10 cm fits exactly inside a cube of side 10 cm. What percentage of the volume of the cube is taken up by the cylinder? Give your answer to 3 significant figures.
11
PSA brass cylinder has radius 2 cm and height 5 cm. Its mass is 420 g. Work out the density of brass, to 1 decimal place.
12
PSWater flows through a cylindrical pipe of radius 2 cm. Work out the volume of water in a 5 m length of the pipe, in litres to 3 significant figures.

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