Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8 (circles): a circle has radius 6 cm. What is its area, in terms of π?
Answer
36π cm². The area is π × 6 × 6. - From Year 9 (cylinders): a cylinder has radius 3 cm and height 6 cm. What is its volume, in terms of π?
Answer
54π cm³. The base area is 9π, and 9π × 6 = 54π. - Work out 4/3 of 27.
Answer
36. One third of 27 is 9, and 4 × 9 = 36.
Words used in this lesson
- sphere
- a perfectly round solid, like a ball. For example: a football.
- hemisphere
- half of a sphere. For example: a bowl.
- radius
- the distance from the centre to the surface. For example: 9 cm.
- diameter
- the distance across through the centre, twice the radius. For example: 18 cm.
- surface area
- the area of the outside of a solid. For example: 324π cm².
- volume
- the space inside a solid. For example: 972π cm³.
- ratio
- a comparison of amounts, such as 1 : 2 : 3. For example: cone : sphere : cylinder.
An orange, peeled
A fruit seller in Nairobi peels an orange in one go. She lays the peel in pieces on the counter and traces circles on a sheet of paper, each one the same size as the orange seen from the side: a circle with the radius of the orange.
Predict first: how many of these circles does the peel of one orange cover? One? Two? Three? Four? Write your guess and a reason. Then use the first figure to cover them one at a time, and the second to see how the volume of a sphere fits between a cone and a cylinder.
Radius 2 cm: one circle is π × 2² = 4π cm². The peel has covered no circles yet.
Show me the steps
- Radius 2 cm, nothing covered yet: one circle has area π × 2² = 4π cm². Move the peel on to 2 circles and 8π cm² is covered. The whole sphere is known only when the peel is used up: 4 circles, 16π cm².
- Move the peel to 4 circles: 4 × 4π = 16π cm². The peel exactly covers four circles and nothing is left over.
- Make the radius 5 cm: one circle is π × 5² = 25π cm², and the whole sphere is 4 × 25π = 100π cm².
- Make the radius 1 cm: one circle is π cm², and the whole sphere is 4π cm². The rule is the same for every radius: four circles.
The surface area of a sphere is four circles of the same radius: A = 4πr². The volume is V = 4/3 πr³.
Why four? The label of the tin that just holds the sphere, 2πr × 2r, has the same area: 4πr².
A hemisphere is half a sphere. Its curved surface is 2πr², and with its flat circle of πr² the whole solid hemisphere has surface 3πr². Its volume is 2/3 πr³.
Put a cone, a sphere and a cylinder side by side, with the same radius and a height equal to the sphere's diameter: their volumes are in the ratio 1 : 2 : 3.
Radius 5 cm, cone and cylinder 10 cm tall: the cone holds 250π/3 cm³, the sphere 500π/3 cm³ and the cylinder 250π cm³, which is 1 : 2 : 3.
Show me the steps
- Radius 5 cm, height 10 cm: cone ⅓ × π × 5² × 10 = 250π/3 cm³, sphere 4/3 × π × 5³ = 500π/3 cm³, cylinder π × 5² × 10 = 250π cm³. In thirds: 250 : 500 : 750, which is 1 : 2 : 3.
- Make the radius 6 cm and the height 12 cm: cone 144π cm³, sphere 288π cm³, cylinder 432π cm³. Still 1 : 2 : 3.
- Make the radius 2 cm and the height 4 cm: cone 16π/3 cm³, sphere 32π/3 cm³, cylinder 16π cm³. In thirds: 16 : 32 : 48, still 1 : 2 : 3.
- Look at the sphere and the cylinder with radius 5 cm: 500π/3 is 2/3 of 250π. The sphere fills two thirds of the cylinder that just holds it. The cone holds half of what the sphere holds, so cone + sphere = cylinder.
A common mistake: mixing up the two formulas, or using the diameter in them. Area has r squared and a 4; volume has r cubed and 4/3. The volume of a sphere is not 4/3 × π × r × r (two radii only) and not 4 × π × r³. Always halve the diameter first.
Worked example: a ball
A ball has a radius of 9 cm. Find its surface area and its volume, in terms of π.
- The surface area is 4 × π × 9² = 4 × 81 × π = 324π cm².
- The volume is 4/3 × π × 9³. Now 9³ = 729, and 729 ÷ 3 = 243, so 4 × 243 = 972: 972π cm³.
- Check the sphere against its cylinder: a cylinder with radius 9 and height 18 holds π × 81 × 18 = 1458π. The sphere should be two thirds of it: 2/3 × 1458 = 972. It is.
Your go. The first step is given. A solid metal sphere has a diameter of 30 cm. Find its surface area and its volume, in terms of π.
- The diameter is 30 cm, so the radius is 15 cm.
- Now find 4πr², then 4/3 πr³, and check.
Check the answer
The surface area is 4 × 15² = 4 × 225 = 900π cm². The volume is 4/3 × 15³ = 4/3 × 3375 = 4 × 1125 = 4500π cm³.
Check before you go on
1. A sphere has radius 5 cm. Which expression gives its surface area in cm²?
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 sphere has radius 6 cm. Its surface area is ?π cm². What number goes in front of π?
3. A sphere has radius 3 cm. Its volume is ?π cm³. What number goes in front of π?
Level 2 · Building
4. A hemisphere (half a sphere) has radius 6 cm. Its volume is ?π cm³. What number goes in front of π?
5. A ball is 14 cm across. Its surface area is ?π cm². What number goes in front of π?
6. A solid hemisphere has radius 4 cm. Its whole surface, curved part and flat circle together, is ?π cm². What number goes in front of π?
Level 3 · Stretch
7. A sphere has surface area 576π cm². Its volume is ?π cm³. What number goes in front of π?
8. Three balls, each with radius 4 cm, are stacked in a tube that is a cylinder with radius 4 cm and height 24 cm. The empty space in the tube is ?π cm³. What number goes in front of π?
Something to think about
The radius of a sphere is halved. By what fraction do its surface area and its volume change? Check with the radius 4 cm and the radius 2 cm sphere.
One way to think about it
The surface area has r² in it, so halving r gives (1/2)² = 1/4 of the area. The volume has r³ in it, so it gives (1/2)³ = 1/8 of the volume. Check: radius 4 gives an area of 64π and a volume of 256π/3; radius 2 gives 16π and 32π/3. And 64 ÷ 4 = 16, while 256 ÷ 8 = 32.
Exit ticket
9. A solid hemisphere has whole surface area 27π cm², curved part and flat circle together. What are its radius and its volume?