Chapter 3 of 8

All 8 chapters
  1. 1Prisms and cylinders, readunderstood
  2. 2Pyramids and cones, readunderstood
  3. 3Spheres, readunderstood
  4. 4Joined and cut solids, readunderstood
  5. 5Changing units of area and volume, readunderstood
  6. 6Speed, distance and time, readunderstood
  7. 7Density and pressure, readunderstood
  8. 8Rates of flow and rate graphs, readunderstood

M4.5.3 Chapter 3

Spheres

How much air is in a football, and how much leather covers it?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the volume and surface area of a sphere
  • Find the volume of a half sphere

A football, a marble and the Earth are all close to spheres. Every point of a sphere is the same distance from its centre. That distance is the radius r.

A sphere. A line from the centre to the surface is labelled 3 cm. The equator is drawn as an ellipse, dashed at the back.3 cm
A sphere with radius 3 cm.

A sphere has no flat faces, so we cannot find its volume by slicing it into equal layers. We use a famous fact instead. Put the sphere inside the cylinder that just holds it.

That cylinder has the same radius r. Its height is the width of the sphere, 2r. So its volume is πr² × 2r = 2πr³.

Draw it first

A sphere in a cylinder

  1. A sphere with its radius labelled 3 cm. Step 2 draws the cylinder round it with dashed lines. The cylinder height is labelled 6 cm. Step 3 adds the label two thirds of the cylinder.3 cm

    1 Draw a sphere of radius 3 cm.

  2. A sphere with its radius labelled 3 cm. Step 2 draws the cylinder round it with dashed lines. The cylinder height is labelled 6 cm. Step 3 adds the label two thirds of the cylinder.3 cm6 cm

    2 Draw the cylinder that just holds it. It is 3 cm in radius and 6 cm high.

  3. A sphere with its radius labelled 3 cm. Step 2 draws the cylinder round it with dashed lines. The cylinder height is labelled 6 cm. Step 3 adds the label two thirds of the cylinder.3 cm6 cmtwo thirds of the cylinder

    3 Archimedes showed that the sphere fills exactly two thirds of this cylinder.

You can test it with water. Fill the cylinder, drop in a ball that just fits, and collect the water that spills. The spilled water is two thirds of the cylinder. One third is left in the cylinder.

Two thirds of 2πr³ is ⅔ × 2πr³ = 4/3 × πr³. The volume of a sphere is V = 4/3 πr³.

Worked example

Find the volume of a sphere of radius 3 cm. Give the answer to 1 decimal place.

  1. Work out r³ = 3 × 3 × 3 = 27.
  2. V = 4/3 × π × 27. Now 4/3 of 27 is 36, so V = 36π cm³.
  3. Put in the π key of a calculator. 36π is 113.1 cm³, to 1 decimal place.

Answer: 113.1 cm³

Archimedes also found the surface of the sphere. It has exactly the area of the curved surface of the cylinder that holds it, without the ends. That area is 2πr × 2r = 4πr².

Another way to remember it: the surface of a sphere is 4 times the area of a circle with the same radius. So the surface area of a sphere is A = 4πr².

A sphere. A line from the centre to the surface is labelled 5 cm. The equator is drawn as an ellipse, dashed at the back.5 cm
Amina's ball, with radius 5 cm.

Worked example

Amina has a ball of radius 5 cm. Find its surface area, to 1 decimal place.

  1. The surface area of a sphere is A = 4πr².
  2. Work out r² = 5 × 5 = 25.
  3. A = 4 × π × 25 = 100π cm².
  4. Put in the π key of a calculator. 100π is 314.2 cm², to 1 decimal place.

Answer: 314.2 cm²

Cut a sphere through its centre and you get two half spheres. Each has half the volume, so V = ½ × 4/3 πr³ = 2/3 πr³.

The surface of a half sphere needs care. The curved part is half of 4πr², which is 2πr². A solid half sphere also has a flat face, a circle of area πr². With the flat face, the total is 2πr² + πr² = 3πr².

A half sphere with its flat face up, like a bowl. A line from the centre of the flat face to its edge is labelled 6 cm.6 cm
A half sphere with radius 6 cm.

Worked example

A bowl is a half sphere of radius 6 cm. Find the volume it can hold, to 1 decimal place.

  1. The whole sphere has V = 4/3 × π × 6³, and 6³ = 216.
  2. 4/3 of 216 is 288, so the whole sphere has volume 288π cm³.
  3. The bowl is half of it, so its volume is 288π ÷ 2 = 144π cm³.
  4. Put in the π key of a calculator. To 1 decimal place, 144π is 452.4 cm³.

Answer: 452.4 cm³

Try it

Measure a ball with water

You need: a ball of clay about 4 cm across, a full bowl on a tray, a measuring jug marked in ml.

  1. Stand the bowl on the tray. Fill it to the brim.
  2. Lower the ball of clay in gently. Water spills onto the tray.
  3. Pour the spilled water into the jug and read it.
  4. Work out 4/3 × π × 2³ for a ball with radius 2 cm, and compare.

What you will see: Each ml of water takes up 1 cm³. A ball of radius 2 cm has volume about 33.5 cm³, so about 33 or 34 ml spills.

If you cannot try it, this is what happens.
A sphere. A line from the centre to the surface is labelled 2 cm. The equator is drawn as an ellipse, dashed at the back.2 cm

A wrong idea many people have

If the radius of a sphere doubles, its volume doubles.

Why it feels right: Twice the radius looks like twice the size, and a ball twice as wide looks twice as big.

The test that breaks it: The sphere of radius 3 cm has volume 36π, which is about 113.1 cm³. The sphere of radius 6 cm has volume 288π, which is about 904.8 cm³. That is 8 times as big, because r is cubed and 2³ = 8.

What is true: Volume goes with r³. Double the radius and the volume is multiplied by 8. The surface area goes with r², so it is multiplied by 4.

A wrong idea many people have

The total surface of a solid half sphere is 2πr².

Why it feels right: A sphere has surface 4πr², and half of that is 2πr².

The test that breaks it: Press a solid half sphere on a table. The curved part is 2πr², but you also see the flat face on the table. That flat circle has area πr² and it belongs to the surface of the solid.

What is true: The curved surface is 2πr². A solid half sphere has 3πr² in all. Always say whether the flat face counts.

Key words

sphere
a solid ball shape: every point of its surface is the same distance from the centre.
half sphere
half of a sphere, cut through its centre, with a flat circle for a face.

Back to your guess

The answer is A: Two thirds of the cylinder that just holds it.

A ball that just fits inside a cylinder fills exactly two thirds of it. The ball touches the sides, the top and the bottom, but it leaves gaps in the corners. This fact was found by Archimedes, over two thousand years ago.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is the volume of a sphere of radius 5 cm, to 1 decimal place?

    A sphere. A line from the centre to the surface is labelled 5 cm. The equator is drawn as an ellipse, dashed at the back.5 cm
    Tap your answer.

    1. Not this one. That is 4 × π × 5², the formula for the surface area. The volume has r cubed and a 4/3 in front. Go to M4.5.3. Try another.

    2. Yes, that is right. V = 4/3 × π × 5³ = 4/3 × π × 125 = 500π/3, which is 523.6 cm³ to 1 decimal place.

    3. Not this one. You left out the 4/3. The volume of a sphere is 4/3 × π × 5³, not π × 5³. Go to M4.5.3. Try another.

    See the answer

    523.6 cm³

    V = 4/3 × π × 5³ = 4/3 × π × 125 = 500π/3, which is 523.6 cm³ to 1 decimal place.

    • If you chose A: That is 4 × π × 5², the formula for the surface area. The volume has r cubed and a 4/3 in front. Go to M4.5.3.
    • If you chose C: You left out the 4/3. The volume of a sphere is 4/3 × π × 5³, not π × 5³. Go to M4.5.3.

  2. What is the surface area of a sphere of radius 4 cm, to 1 decimal place?

    A sphere. A line from the centre to the surface is labelled 4 cm. The equator is drawn as an ellipse, dashed at the back.4 cm
    Tap your answer.

    1. Not this one. That number comes from the volume formula, 4/3 × π × 4³. The surface area is 4 × π × 4², and it is measured in cm². Go to M4.5.3. Try another.

    2. Not this one. You used the diameter, 8 cm, in place of the radius, 4 cm. Go to M4.5.3. Try another.

    3. Yes, that is right. A = 4 × π × 4² = 4 × π × 16 = 64π, which is 201.1 cm² to 1 decimal place.

    See the answer

    201.1 cm²

    A = 4 × π × 4² = 4 × π × 16 = 64π, which is 201.1 cm² to 1 decimal place.

    • If you chose A: That number comes from the volume formula, 4/3 × π × 4³. The surface area is 4 × π × 4², and it is measured in cm². Go to M4.5.3.
    • If you chose B: You used the diameter, 8 cm, in place of the radius, 4 cm. Go to M4.5.3.

  3. A solid half sphere has radius 3 cm. What is its volume, to 1 decimal place?

    A half sphere standing on its flat face. A line from the centre of the flat face to its edge is labelled 3 cm.3 cm
    Tap your answer.

    1. Yes, that is right. The whole sphere has V = 4/3 × π × 27 = 36π. The half sphere has 36π ÷ 2 = 18π, which is 56.5 cm³ to 1 decimal place.

    2. Not this one. That is the volume of the whole sphere. The half sphere has half of it. Go to M4.5.3. Try another.

    3. Not this one. You halved the radius, 1.5 cm, instead of halving the volume. A half sphere has the same radius as the whole sphere. Go to M4.5.3. Try another.

    See the answer

    56.5 cm³

    The whole sphere has V = 4/3 × π × 27 = 36π. The half sphere has 36π ÷ 2 = 18π, which is 56.5 cm³ to 1 decimal place.

    • If you chose B: That is the volume of the whole sphere. The half sphere has half of it. Go to M4.5.3.
    • If you chose C: You halved the radius, 1.5 cm, instead of halving the volume. A half sphere has the same radius as the whole sphere. Go to M4.5.3.

  4. A solid half sphere has radius 5 cm. Count its flat face. What is its total surface area, to 1 decimal place?

    Tap your answer.

    1. Not this one. That is the curved part alone, 2π × 5². The question says to count the flat face too. Go to M4.5.3. Try another.

    2. Not this one. That is 4π × 5², the surface of the whole sphere. The half sphere has half of that, plus the flat face. Go to M4.5.3. Try another.

    3. Yes, that is right. The curved part is 2π × 5² = 50π. The flat face is π × 5² = 25π. The total is 75π, which is 235.6 cm² to 1 decimal place.

    See the answer

    235.6 cm²

    The curved part is 2π × 5² = 50π. The flat face is π × 5² = 25π. The total is 75π, which is 235.6 cm² to 1 decimal place.

    • If you chose A: That is the curved part alone, 2π × 5². The question says to count the flat face too. Go to M4.5.3.
    • If you chose B: That is 4π × 5², the surface of the whole sphere. The half sphere has half of that, plus the flat face. Go to M4.5.3.

  5. One sphere has radius 5 cm. A second sphere has radius 10 cm. How many times as big is the volume of the second sphere?

    Tap your answer.

    1. Yes, that is right. The volume goes with r³. The radius is multiplied by 2, so the volume is multiplied by 2³ = 8.

    2. Not this one. The radius is twice as big, but the volume has r³ in it. A sphere that is twice as wide is much more than twice as big. Go to M4.5.3. Try another.

    3. Not this one. That is how many times the surface area grows. The volume goes with r³, not r². Go to M4.4.6. Try another.

    See the answer

    8 times as big

    The volume goes with r³. The radius is multiplied by 2, so the volume is multiplied by 2³ = 8.

    • If you chose B: The radius is twice as big, but the volume has r³ in it. A sphere that is twice as wide is much more than twice as big. Go to M4.5.3.
    • If you chose C: That is how many times the surface area grows. The volume goes with r³, not r². Go to M4.4.6.

  6. From an earlier chapter

    A cone has radius 5 cm and perpendicular height 9 cm. What is its volume, to 1 decimal place?

    See the answer

    235.6 cm³

    The base area is π × 5² = 25π. The volume is ⅓ × 25π × 9 = 75π, which is 235.6 cm³ to 1 decimal place.

  7. A sphere has radius 9 cm. What is its volume, to 1 decimal place?

    See the answer

    3 053.6 cm³

    V = 4/3 × π × 9³ = 4/3 × π × 729 = 972π, which is 3 053.6 cm³ to 1 decimal place.

  8. A sphere has radius 7 cm. What is its surface area, to 1 decimal place?

    See the answer

    615.8 cm²

    A = 4 × π × 7² = 4 × π × 49 = 196π, which is 615.8 cm² to 1 decimal place.

  9. A solid half sphere has radius 4 cm. What is its volume, to 1 decimal place?

    See the answer

    134.0 cm³

    The whole sphere has volume 4/3 × π × 64, which is 268.08 cm³ to 2 decimal places. Half of that is 134.04, which is 134.0 cm³ to 1 decimal place.

Keep it

Copy this into your exercise book.

Sphere: V = 4/3 πr³, A = 4πr². Half sphere: V = 2/3 πr³, curved surface 2πr², with the flat face 3πr². Radius doubles, volume × 8.

Draw: A sphere inside a cylinder that just holds it, with r and 2r marked.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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