Chapter 4 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

M4.5.4 Chapter 4

Joined and cut solids

How much does a flower pot hold, when it is a cone with its tip cut off?

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 of a cone with its tip cut off
  • Find the volume of a solid made of a cylinder and a cone

A sharpened pencil, a rocket and a flower pot are not one of the basic solids. But each one is made of basic solids that you know. The method is always the same.

Split the solid into parts you know, such as a cylinder and a cone. Find the volume of each part. Then add the volumes if the parts are joined. Subtract if a part has been cut away.

First, joined solids. Here a cylinder with radius 3 cm and height 6 cm has a cone on top. The cone has the same radius, 3 cm, and height 4 cm.

A cylinder with a cone on top of it. The radius of the base is labelled 3 cm. The height of the cylinder, on the left, is labelled 6 cm. The height of the cone, on the right, is labelled 4 cm.3 cm6 cm4 cm
A cylinder and a cone with the same radius.

Worked example

Find the volume of the solid in the picture. Give the answer to 1 decimal place.

  1. The cylinder has volume π × 3² × 6 = 54π cm³.
  2. The cone has volume ⅓ × π × 3² × 4 = 12π cm³.
  3. Add the two parts: 54π + 12π = 66π cm³.
  4. To 1 decimal place, 66π is 207.3 cm³.

Answer: 207.3 cm³

Take care with the heights. The cone has only its own height, 4 cm. Do not use the height of the whole solid, 10 cm, for the cone or for the cylinder.

Now a cut solid. Cut the tip off a cone with a straight cut parallel to the base. The part that is left is a frustum. The tip you cut off is a small cone, and the whole cone is a big cone.

The frustum is the big cone with the small cone taken away. So volume of frustum = volume of big cone − volume of small cone.

Draw it first

A cone with its tip cut off

  1. A cone with base radius 6 cm and height 8 cm. The radius is labelled 6 cm and the height, on the left, is labelled 8 cm. Step 2 shows the cone with its tip cut off. The frustum has base radius 6 cm labelled 6 cm and top radius labelled 3 cm. Its height is labelled 4 cm and the height of the tip is labelled 4 cm. Step 3 adds the label frustum = big cone − small cone.6 cm8 cm

    1 A cone with base radius 6 cm and height 8 cm.

  2. A cone with base radius 6 cm and height 8 cm. The radius is labelled 6 cm and the height, on the left, is labelled 8 cm. Step 2 shows the cone with its tip cut off. The frustum has base radius 6 cm labelled 6 cm and top radius labelled 3 cm. Its height is labelled 4 cm and the height of the tip is labelled 4 cm. Step 3 adds the label frustum = big cone − small cone.6 cm3 cm4 cm4 cm

    2 Cut the tip off halfway up. The tip is a small cone with radius 3 cm and height 4 cm.

  3. A cone with base radius 6 cm and height 8 cm. The radius is labelled 6 cm and the height, on the left, is labelled 8 cm. Step 2 shows the cone with its tip cut off. The frustum has base radius 6 cm labelled 6 cm and top radius labelled 3 cm. Its height is labelled 4 cm and the height of the tip is labelled 4 cm. Step 3 adds the label frustum = big cone − small cone.6 cm3 cm4 cm4 cmfrustum = big cone − small cone

    3 What is left is the frustum: the big cone minus the small cone.

Worked example

A pot is a frustum. Its base radius is 6 cm, its top radius is 3 cm and its height is 4 cm. The tip that was cut off is a cone with height 4 cm. Find its volume, to 1 decimal place.

  1. The big cone has radius 6 cm and height 4 + 4 = 8 cm.
  2. Its volume is ⅓ × π × 6² × 8 = 96π cm³.
  3. The tip is a cone with radius 3 cm and height 4 cm. Its volume is ⅓ × π × 3² × 4 = 12π cm³.
  4. The frustum is the big cone minus the tip: 96π − 12π = 84π cm³.
  5. To 1 decimal place, 84π is 263.9 cm³.

Answer: 263.9 cm³

Often you are not told the height of the tip. The small cone has the same shape as the big cone, so they are similar. Their lengths are in the same ratio. Use this to find the height of the tip.

A cone with its tip cut off, with the tip drawn dotted above it. The radius of the base is labelled 5 cm. The radius of the cut face is labelled 2 cm. The height of the cut cone, on the left, is labelled 6 cm. The height of the tip, on the right, is labelled x cm.5 cm2 cm6 cmx cm
A flower pot, with the tip drawn dotted. The height of the tip, x cm, is not given. You find it.

Worked example

Diego has a flower pot that is a frustum. Its base radius is 5 cm, its top radius is 2 cm and its height is 6 cm. Find its volume, to 1 decimal place.

  1. Let the tip have height x cm. The tip and the whole cone are similar, so x ÷ (x + 6) = 2 ÷ 5.
  2. Cross multiply: 5x = 2(x + 6) = 2x + 12. So 3x = 12, and x = 4.
  3. The whole cone has height 4 + 6 = 10 cm. Radius squared times height is 5² × 10 = 250 for it, and 2² × 4 = 16 for the tip.
  4. The volume is ⅓ × π × (250 − 16) = ⅓ × π × 234 = 78π cm³.
  5. To 1 decimal place, 78π is 245.0 cm³.

Answer: 245.0 cm³

Try it

Cut a cone in half

You need: stiff paper, scissors, tape and a ruler.

  1. Roll the paper into a cone and tape it. Trim the base flat.
  2. Mark the middle of the sloping side, halfway from the base to the apex.
  3. Cut straight across at that mark, level with the base.
  4. Stand the tip beside the frustum and compare their widths.

What you will see: The tip is a small copy of the whole cone. It is half as wide and half as tall. That is why the tip is similar to the whole cone, and why we can use ratios.

If you cannot try it, this is what happens.
A cone with its tip cut off halfway up, with the tip drawn dotted. The radius of the base is labelled 4 cm. The radius of the cut face is labelled 2 cm. The height of the cut cone, on the left, is labelled 3 cm. The height of the tip, on the right, is labelled 3 cm.4 cm2 cm3 cm3 cm

A wrong idea many people have

A frustum is a cylinder with the average of the two radii.

Why it feels right: The radius goes from big to small, so the average radius looks like a fair single radius.

The test that breaks it: Take the frustum with radius 6 cm and 3 cm and height 4 cm. The average radius is 4.5 cm. π × 4.5² × 4 = 81π, which is 254.5 cm³ to 1 decimal place. The true volume is 84π, which is 263.9 cm³. The answers differ.

What is true: A frustum is a big cone minus a small cone. Find both volumes and subtract.

A wrong idea many people have

A cylinder with a cone on top has volume πr² × the whole height.

Why it feels right: The cone sits on the cylinder and has the same radius, so the whole thing looks like one tall tube.

The test that breaks it: Take the joined solid with radius 3 cm, a cylinder of height 6 cm and a cone of height 4 cm. A cylinder of height 10 cm has volume π × 9 × 10 = 90π, which is 282.7 cm³ to 1 decimal place. The true volume is 66π, which is 207.3 cm³ to 1 decimal place.

What is true: The cone is only one third as big as a cylinder of its height. Find the two parts separately and add.

Key words

frustum
what is left of a cone when its tip is cut off by a cut parallel to the base.
similar
the same shape but a different size, with all the lengths in the same ratio.

Back to your guess

The answer is B: Take the volume of the tip away from the volume of the whole cone.

The pot is what is left when you cut the tip off a whole cone. So its volume is the whole cone minus the tip. Adding would count the tip twice. Taking away radii makes a different cone altogether.

Check yourself

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

  1. A solid is a cylinder with a cone on top. Both have radius 4 cm. The cylinder is 10 cm high and the cone is 6 cm high. What is the volume of the whole solid, to 1 decimal place?

    A cylinder with a cone on top of it. The radius of the base is labelled 4 cm. The height of the cylinder, on the left, is labelled 10 cm. The height of the cone, on the right, is labelled 6 cm.4 cm10 cm6 cm
    Tap your answer.

    1. Yes, that is right. The cylinder is π × 4² × 10 = 160π. The cone is ⅓ × π × 4² × 6 = 32π. The total is 192π, which is 603.2 cm³ to 1 decimal place.

    2. Not this one. You treated the whole solid as one cylinder of height 16 cm. The cone is only a third of a cylinder of its own height. Go to M4.5.4. Try another.

    3. Not this one. That is the cylinder alone, π × 4² × 10. You also need the cone on top. Go to M4.5.4. Try another.

    See the answer

    603.2 cm³

    The cylinder is π × 4² × 10 = 160π. The cone is ⅓ × π × 4² × 6 = 32π. The total is 192π, which is 603.2 cm³ to 1 decimal place.

    • If you chose B: You treated the whole solid as one cylinder of height 16 cm. The cone is only a third of a cylinder of its own height. Go to M4.5.4.
    • If you chose C: That is the cylinder alone, π × 4² × 10. You also need the cone on top. Go to M4.5.4.

  2. A pot is a cone with its tip cut off. Its base radius is 8 cm and its top radius is 4 cm. The pot is 5 cm high and the tip is also 5 cm high. What is the volume of the pot, to 1 decimal place?

    A cone with its tip cut off, with the tip drawn dotted. The radius of the base is labelled 8 cm. The radius of the cut face is labelled 4 cm. The height of the cut cone, on the left, is labelled 5 cm. The height of the tip, on the right, is labelled 5 cm.8 cm4 cm5 cm5 cm
    Tap your answer.

    1. Not this one. You used a cylinder with the average radius, 6 cm. A frustum is not a cylinder, so this is only close. Go to M4.5.4. Try another.

    2. Not this one. That is the whole big cone, with the tip still in it. The tip was cut away, so subtract its volume. Go to M4.5.4. Try another.

    3. Yes, that is right. The big cone has height 10, so ⅓ × π × 8² × 10 = 670.2. The tip is ⅓ × π × 4² × 5 = 83.8. The pot is 670.2 − 83.8 = 586.4 cm³, to 1 decimal place.

    See the answer

    586.4 cm³

    The big cone has height 10, so ⅓ × π × 8² × 10 = 670.2. The tip is ⅓ × π × 4² × 5 = 83.8. The pot is 670.2 − 83.8 = 586.4 cm³, to 1 decimal place.

    • If you chose A: You used a cylinder with the average radius, 6 cm. A frustum is not a cylinder, so this is only close. Go to M4.5.4.
    • If you chose B: That is the whole big cone, with the tip still in it. The tip was cut away, so subtract its volume. Go to M4.5.4.

  3. A cone has its tip cut off. The base radius is 6 cm, the top radius is 2 cm and the height of the part left is 8 cm. How tall is the tip that was cut off?

    Tap your answer.

    1. Not this one. You multiplied the height of the pot, 8 cm, by 2 ÷ 6. The ratio 2 ÷ 6 compares the tip with the whole cone, not with the pot. Go to M4.5.4. Try another.

    2. Yes, that is right. Let the tip have height x. Then x ÷ (x + 8) = 2 ÷ 6. So 6x = 2x + 16, 4x = 16, and x = 4 cm.

    3. Not this one. That is the height of the whole cone. The tip is only the top part, which is shorter. Go to M4.5.4. Try another.

    See the answer

    4 cm

    Let the tip have height x. Then x ÷ (x + 8) = 2 ÷ 6. So 6x = 2x + 16, 4x = 16, and x = 4 cm.

    • If you chose A: You multiplied the height of the pot, 8 cm, by 2 ÷ 6. The ratio 2 ÷ 6 compares the tip with the whole cone, not with the pot. Go to M4.5.4.
    • If you chose C: That is the height of the whole cone. The tip is only the top part, which is shorter. Go to M4.5.4.

  4. A model rocket is a cylinder with radius 3 cm and height 5 cm. A cone 6 cm high sits on top. What is its volume, to 1 decimal place?

    A cylinder with a cone on top of it. The radius of the base is labelled 3 cm. The height of the cylinder, on the left, is labelled 5 cm. The height of the cone, on the right, is labelled 6 cm.3 cm5 cm6 cm
    Tap your answer.

    1. Yes, that is right. The cylinder is π × 3² × 5 = 45π. The cone is ⅓ × π × 3² × 6 = 18π. The total is 63π, which is 197.9 cm³ to 1 decimal place.

    2. Not this one. You treated the cone as a cylinder, with no ⅓. π × 3² × 11 = 311.0 is for one tube 11 cm high. Go to M4.5.2. Try another.

    3. Not this one. That is the cylinder alone, π × 3² × 5. The cone on top has volume too. Go to M4.5.4. Try another.

    See the answer

    197.9 cm³

    The cylinder is π × 3² × 5 = 45π. The cone is ⅓ × π × 3² × 6 = 18π. The total is 63π, which is 197.9 cm³ to 1 decimal place.

    • If you chose B: You treated the cone as a cylinder, with no ⅓. π × 3² × 11 = 311.0 is for one tube 11 cm high. Go to M4.5.2.
    • If you chose C: That is the cylinder alone, π × 3² × 5. The cone on top has volume too. Go to M4.5.4.

  5. A cone is cut halfway up its height, parallel to the base. The tip is a cone with every length 2 times smaller than the whole cone. What fraction of the whole cone's volume is the tip?

    Tap your answer.

    1. Not this one. The tip is half as tall, but it is also half as wide. A volume has three lengths in it, so the tip is much less than half. Go to M4.4.6. Try another.

    2. Yes, that is right. Every length is halved, so the volume is multiplied by ½ × ½ × ½ = 1/8. The frustum is the other 7/8.

    3. Not this one. A factor of 2 × 2 = 4 is for areas. A volume has three lengths in it, so you multiply three times. Go to M4.4.6. Try another.

    See the answer

    1/8

    Every length is halved, so the volume is multiplied by ½ × ½ × ½ = 1/8. The frustum is the other 7/8.

    • If you chose A: The tip is half as tall, but it is also half as wide. A volume has three lengths in it, so the tip is much less than half. Go to M4.4.6.
    • If you chose C: A factor of 2 × 2 = 4 is for areas. A volume has three lengths in it, so you multiply three times. Go to M4.4.6.

  6. From an earlier chapter

    A sphere has radius 6 cm. What is its volume, to 1 decimal place?

    See the answer

    904.8 cm³

    V = 4/3 × π × 6³ = 4/3 × π × 216 = 288π, which is 904.8 cm³ to 1 decimal place.

  7. A solid is a cylinder of radius 4 cm and height 7 cm, with a cone of height 9 cm on top. What is its volume, to 1 decimal place?

    See the answer

    502.7 cm³

    The cylinder is π × 4² × 7 = 112π. The cone is ⅓ × π × 4² × 9 = 48π. The total is 160π, which is 502.7 cm³ to 1 decimal place.

  8. A pot has base radius 6 cm, top radius 3 cm and height 5 cm. It is a cone with its tip cut off. What is its volume, to 1 decimal place?

    See the answer

    329.9 cm³

    The tip x has x ÷ (x + 5) = 3 ÷ 6, so x = 5. The whole cone is ⅓ × π × 6² × 10 = 120π. The tip is ⅓ × π × 3² × 5 = 15π. The pot is 105π, which is 329.9 cm³ to 1 decimal place.

Keep it

Copy this into your exercise book.

Joined solid: add the parts. Cut solid: subtract the part that is gone. Frustum = big cone − small cone. Similar triangles give the missing height.

Draw: A frustum with its tip dotted above it, and the radii and the two heights marked.

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

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