Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9: a cylinder has radius 3 cm and height 5 cm. Find its volume in terms of π.
Answer
45π cm³, because π × 3² × 5 = 45π. - From Year 9: work out 4³.
Answer
64, because 4 × 4 × 4 = 64. - From lesson 1: a cone has radius 9 cm and vertical height 40 cm. How long is its slant height?
Answer
41 cm, because 81 + 1600 = 1681 and √1681 = 41.
Words used in this lesson
- volume
- the amount of space inside a solid. For example: cm³ or litres.
- sphere
- a perfectly round solid, like a ball. For example: a football.
- hemisphere
- half of a sphere. For example: a bowl.
- pyramid
- a solid with a flat base and triangular faces that meet at a point. For example: a roof.
- composite solid
- a solid made from two or more simpler solids. For example: a silo: a cylinder with a dome.
- exact form
- an answer left with π in it, not rounded. For example: a volume of 20π cm³.
Kulfi in a cone
A kulfi seller in Jaipur has a steel cup and a cone-shaped mould. Both are exactly the same width and the same height. He fills the cone with kulfi and tips it into the cup, again and again, until the cup is full.
Predict first: do you think it takes 2 cones, 3 cones or 4 cones to fill the cup? Write down a reason, then use the first figure.
Radius 2, height 6: the cylinder is πr²h = 24π ≈ 75.40; the cone is 8π ≈ 25.13, which is 1/3 of the cylinder.
Show me the steps
- Radius 2, vertical height 6: cylinder = π × 2² × 6 = 24π ≈ 75.40; cone = 8π ≈ 25.13. The shaded band is one third of the way up the cylinder.
- Radius 1, vertical height 1: cylinder = π ≈ 3.14; cone = π/3 ≈ 1.05. The fraction does not depend on the size.
- Radius 4, vertical height 3: cylinder = 48π ≈ 150.80; cone = 16π ≈ 50.27. Make the height 6 and the cone doubles to 32π.
- Radius 5, vertical height 7: cylinder = 175π ≈ 549.78; cone = 175π/3 ≈ 183.26. In every state the cone is one third of its cylinder.
A pyramid or cone holds one third of the prism or cylinder with the same base and vertical height: V = ⅓ × base area × h. For a cone, V = ⅓πr²h.
A sphere has volume 4πr³ ÷ 3. A hemisphere is half of that, 2πr³ ÷ 3. Use the vertical height h, the one measured straight up, never the slant height.
Radius 2: the tube is πr² × 2r = 16π ≈ 50.27; the sphere is 4πr³ ÷ 3 = 32π/3 ≈ 33.51, which is 0.667 of the tube.
Show me the steps
- Radius 2: the tube holds π × 2² × 4 = 16π ≈ 50.27 and the sphere holds 4π × 8 ÷ 3 = 32π/3 ≈ 33.51. That is 0.667 of the tube.
- Radius 1: tube = 2π ≈ 6.28, sphere = 4π/3 ≈ 4.19. The fraction is again 0.667.
- Radius 3: tube = 54π ≈ 169.65, sphere = 36π ≈ 113.10. The fraction is 0.667.
- Radius 5: tube = 250π ≈ 785.40, sphere = 500π/3 ≈ 523.60. Whatever the radius, the sphere fills the same fraction of its tube.
Two slips to avoid. The one third belongs to pyramids and cones, not to cylinders or spheres. And r is the radius: if a question gives the diameter, halve it before cubing.
Worked examples
1. A square-based pyramid has base 12 cm by 12 cm and vertical height 8 cm. Find its volume.
- Base area = 12 × 12 = 144 cm².
- V = ⅓ × 144 × 8 = 384 cm³.
2. A cone has radius 10 cm and vertical height 6 cm. Find its volume to 1 decimal place.
- V = ⅓ × π × 10² × 6 = 200π.
- 200π = 628.3 cm³.
3. A sphere has radius 12 cm. Find its volume in terms of π, then to 1 decimal place.
- V = 4π × 12³ ÷ 3 = 4π × 1728 ÷ 3 = 2304π.
- 2304π ≈ 7238.2 cm³.
Your go. The first step is given. A pyramid has a rectangular base 8 cm by 5 cm and vertical height 9 cm. Find its volume.
- Base area = 8 × 5 = 40 cm².
- Now multiply by the height and by one third.
Check the answer
V = ⅓ × 40 × 9 = 120 cm³.
With a GDC (optional)
Enter a sphere volume in one go, with brackets: (4 ÷ 3) × π × 7³. Many GDCs will show 1436.755 and more digits; round only at the end, to the accuracy the question asks for.
Check it against the exact form 4πr³ ÷ 3 first. Everything here can be done by hand with π left in the answer.
Check before you go on
1. A cone and a cylinder have the same radius. The cone is 3 times as tall as the cylinder. The cylinder holds 90 cm³. How much does the cone hold?
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
2. A square-based pyramid has base edge 9 cm and vertical height 10 cm. What is its volume, in cm³?
3. A cone has base radius 6 cm and vertical height 7 cm. Its volume is ?π cm³. What number goes in front of π?
Level 2 · Building
4. A sphere has radius 6 cm. Its volume is ?π cm³. What number goes in front of π?
5. A spherical water tank has a diameter of 14 m. What is its volume, in m³, to the nearest whole number?
6. A cone has base radius 3 cm and volume 36π cm³. What is its vertical height, in cm?
Level 3 · Stretch
7. A bowl is a hemisphere of radius 9 cm. When it is full, how much does it hold, in cm³, to the nearest whole number?
8. An ice-cream cone has base radius 3 cm and vertical height 10 cm. A hemisphere of ice cream of radius 3 cm sits on top. The total volume is ?π cm³. What number goes in front of π?
Something to think about
A hemisphere has radius r. A cone has the same radius and a vertical height of r. Work out the volume of each in terms of r, and compare them.
One way to think about it
Hemisphere: 2πr³ ÷ 3. Cone with height r: πr² × r ÷ 3 = πr³ ÷ 3. So the hemisphere holds exactly twice the cone. Cone, hemisphere and cylinder (radius r, height r) hold in the ratio 1 : 2 : 3, which is the pattern Archimedes admired.
Exit ticket
9. A cone has base radius 5 cm and slant height 13 cm. Which is its volume?