Chapter 1 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

M4.5.1 Chapter 1

Prisms and cylinders

How much does a tin hold, and how much metal is in 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 of a prism as end area × length
  • Find the volume and surface area of a cylinder

A loaf tin, a roof beam and a tin of beans have one thing in common. Cut any of them across and you get the same flat shape, wherever you cut.

That flat shape is the cross section. A prism is a solid whose cross section stays the same all the way along. A cylinder works the same way, with a circle as its cross section.

You already know one prism. A box with a rectangle at the end has volume length × width × height. The length × width part is the area of the end face. So for a box, volume = end area × length.

Draw it first

Slice a prism

  1. A triangular prism whose end face is a right angled triangle, labelled 6 cm and 4 cm, and whose length is labelled 10 cm. Step 1 shows the prism. Step 2 cuts it into 10 equal slices, each 1 cm thick. Step 3 adds the label each 1 cm slice: 12 cm³.6 cm4 cm10 cm

    1 Here is a prism. Its end face is a triangle, and it is 10 cm long.

  2. A triangular prism whose end face is a right angled triangle, labelled 6 cm and 4 cm, and whose length is labelled 10 cm. Step 1 shows the prism. Step 2 cuts it into 10 equal slices, each 1 cm thick. Step 3 adds the label each 1 cm slice: 12 cm³.6 cm4 cm10 cm

    2 Cut it into 10 slices, each 1 cm thick. Every slice has the same end face.

  3. A triangular prism whose end face is a right angled triangle, labelled 6 cm and 4 cm, and whose length is labelled 10 cm. Step 1 shows the prism. Step 2 cuts it into 10 equal slices, each 1 cm thick. Step 3 adds the label each 1 cm slice: 12 cm³.6 cm4 cm10 cmeach 1 cm slice: 12 cm³

    3 Each slice holds 12 cm³, the end area of 12 cm² times 1 cm. Ten of them make 120 cm³.

In these pictures, the edges that go back into the page are drawn shorter. This makes the solid look solid. Read lengths from the labels, not from the drawing.

The slices show why the rule works. Make the slices thinner and thinner and the rule does not change. The volume is the end area, counted once for each unit of length.

For a prism whose cross section has area A and whose length is l, the volume is V = A × l. The rule needs a cross section that stays the same. If the shape changes along the solid, it is not a prism.

Watch the units. If A is in cm² and l is in cm, then V is in cm³. A volume is an area in cm², stretched along a length in cm.

A triangular prism. The end face is shaded. Its two straight sides are labelled 6 cm and 4 cm. The length of the prism is labelled 10 cm.6 cm4 cm10 cm
A triangular prism. The shaded end face is the cross section.

Worked example

Find the volume of the prism in the picture. The end face is a right angled triangle with base 6 cm and height 4 cm. The prism is 10 cm long.

  1. The end face is a triangle. Its area is ½ × 6 × 4 = 12 cm².
  2. The prism is 10 cm long, and the end face stays the same all the way along.
  3. Volume = end area × length = 12 × 10 = 120 cm³.

Answer: 120 cm³

The end face can be any shape you can find the area of. Here it is a trapezium. Find the area of the end face first, then multiply by the length.

A prism with a trapezium for its end face. The long parallel side is labelled 8 cm and the short one 6 cm. A dashed line, labelled 4 cm, joins the two parallel sides at a right angle. The length of the prism is labelled 6 cm.4 cm8 cm6 cm6 cm
A trapezium prism. The dashed line is the height of the trapezium.

Worked example

Find the volume of the trapezium prism in the picture. Its parallel sides are 8 cm and 6 cm, and they are 4 cm apart. The prism is 6 cm long.

  1. The end face is a trapezium. Its area is ½ × (8 + 6) × 4 = 28 cm².
  2. The prism is 6 cm long, and the end face stays the same all the way along.
  3. Volume = end area × length = 28 × 6 = 168 cm³.

Answer: 168 cm³

Now the cylinder. Its cross section is a circle of radius r, with area πr². So the volume of a cylinder of height h is V = πr² × h.

A cylinder. The radius of the top circle is drawn as a line labelled 3 cm. The height, on the right, is labelled 8 cm.3 cm8 cm
A cylinder with radius 3 cm and height 8 cm.

Worked example

Kofi has a tin shaped like a cylinder, with radius 3 cm and height 8 cm. Find its volume, to 1 decimal place.

  1. The end face is a circle of radius 3 cm. Its area is π × 3² = 9π cm².
  2. Volume = end area × height = 9π × 8 = 72π cm³.
  3. Put in the π key of a calculator. To 2 decimal places, 72π is 226.19 cm³.
  4. To 1 decimal place, the volume is 226.2 cm³.

Answer: 226.2 cm³

Surface area is the area of all the faces added up. To see a cylinder's faces, cut it open and lay it flat. This flat picture is its net.

The net has two circles and one rectangle. The rectangle wraps once round the circle, so one side is the height h and the other side is the circumference 2πr. That rectangle is the curved surface.

The net of a cylinder: a rectangle with a circle above it and a circle below it. A line across the top of the rectangle is labelled 18.85 cm (2πr). The right side is labelled 8 cm. The radius of the top circle is labelled 3 cm.3 cm18.85 cm (2πr)8 cm
The net of the cylinder. The rectangle is as long as the circumference of the circle.

The curved surface has area 2πr × h = 2πrh. Each circle has area πr², and there are two. The total surface area of a closed cylinder is 2πrh + 2πr².

Worked example

Find the total surface area of the cylinder with radius 3 cm and height 8 cm. Give the answer to 1 decimal place.

  1. Cut the cylinder open. The curved surface is a rectangle as long as the circumference of a circle.
  2. The circumference is 2 × π × 3 = 6π cm. The curved surface is 6π × 8 = 48π cm².
  3. Each end is a circle of area π × 3² = 9π cm². The two ends make 18π cm².
  4. The total surface area is 48π + 18π = 66π cm².
  5. Put in the π key of a calculator. To 1 decimal place, 66π is 207.3 cm².

Answer: 207.3 cm²

Try it

Wrap a tin

You need: a round tin or a cup, a strip of paper, a pencil and a ruler.

  1. Wrap the paper once round the tin. Mark where the paper meets itself.
  2. Unwrap it and measure from the end to the mark. This is the circumference.
  3. Measure straight across the top of the tin. This is the diameter.
  4. Divide the circumference by the diameter.

What you will see: The answer is a little over 3 for every round tin, big or small. That number is π. It is why the paper rectangle is 2πr long.

If you cannot try it, this is what happens.
The net of a tin: a rectangle with a circle above it and a circle below it. A line across the top is labelled 25.13 cm (2πr). The right side is labelled 10 cm. The radius of the top circle is labelled 4 cm.4 cm25.13 cm (2πr)10 cm

A wrong idea many people have

The volume of a cylinder is π × diameter² × height.

Why it feels right: The diameter is the width you measure across a tin, so it feels like the number to use.

The test that breaks it: Take a cylinder with radius 3 cm and height 8 cm. The true volume is π × 9 × 8 = 72π, which is 226.2 cm³ to 1 decimal place. With the diameter, 6 cm, you get π × 36 × 8 = 288π, which is 904.8 cm³. That is four times too big.

What is true: The area of a circle is πr², with r the radius. Halve the diameter first.

A wrong idea many people have

The volume of a triangular prism is base × height × length.

Why it feels right: For a box this is exactly right, and a triangle looks like a box with a corner cut off.

The test that breaks it: Take the prism with an end face of base 6 cm and height 4 cm, 10 cm long. 6 × 4 × 10 = 240 cm³. But the end face is a triangle, with half the area of the 6 by 4 rectangle. Half of 240 is 120 cm³, the answer from the slices.

What is true: Find the area of the end face first. A triangle is ½ × base × height.

Key words

cross section
the flat shape you see when you cut a solid straight across.
prism
a solid whose cross section stays the same all the way along.
net
the faces of a solid laid out flat.
curved surface
the side of a cylinder, without its two ends.

Back to your guess

The answer is B: Multiply the area of the end face by the length.

A prism is built from the same flat slice, again and again, all the way along. The volume is the area of one slice times the length. Multiplying three measurements only works for a box, and adding an area to a length mixes two kinds of measure.

Check yourself

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

  1. Which of these solids is a prism?

    Tap your answer.

    1. Not this one. A cone gets narrower to a point, so its cross section changes along its height. A prism keeps the same cross section. Go to M4.5.1. Try another.

    2. Yes, that is right. The tent has the same triangle all the way along, so it is a triangular prism. The cone and the ball change size as you cut across them.

    3. Not this one. A cut across a ball gives circles of different sizes. A prism gives the same shape every time. Go to M4.5.1. Try another.

    See the answer

    A tent shaped like a triangle, with the same triangle at both ends

    The tent has the same triangle all the way along, so it is a triangular prism. The cone and the ball change size as you cut across them.

    • If you chose A: A cone gets narrower to a point, so its cross section changes along its height. A prism keeps the same cross section. Go to M4.5.1.
    • If you chose C: A cut across a ball gives circles of different sizes. A prism gives the same shape every time. Go to M4.5.1.

  2. The end face of this prism is a right angled triangle with base 15 cm and height 8 cm. The prism is 6 cm long. What is its volume?

    A triangular prism. The end face is shaded. Its sides are labelled 15 cm, 17 cm and 8 cm. The length of the prism is labelled 6 cm.15 cm17 cm8 cm6 cm
    Tap your answer.

    1. Not this one. You multiplied 15 × 8 × 6 and forgot that the end face is a triangle. A triangle is half of the 15 by 8 rectangle. Go to M4.5.1. Try another.

    2. Not this one. You may have added the three sides of the triangle, 15 + 17 + 8 = 40, and multiplied by 6. The volume needs the area of the end face, not its perimeter. Go to M3.6.4. Try another.

    3. Yes, that is right. The end area is ½ × 15 × 8 = 60 cm². The volume is 60 × 6 = 360 cm³. The 17 cm side is a slanted side and is not needed.

    See the answer

    360 cm³

    The end area is ½ × 15 × 8 = 60 cm². The volume is 60 × 6 = 360 cm³. The 17 cm side is a slanted side and is not needed.

    • If you chose A: You multiplied 15 × 8 × 6 and forgot that the end face is a triangle. A triangle is half of the 15 by 8 rectangle. Go to M4.5.1.
    • If you chose B: You may have added the three sides of the triangle, 15 + 17 + 8 = 40, and multiplied by 6. The volume needs the area of the end face, not its perimeter. Go to M3.6.4.

  3. A cylinder has radius 5 cm and height 10 cm. What is its volume, to 1 decimal place?

    A cylinder. The radius is a line labelled 5 cm. The height, on the right, is labelled 10 cm.5 cm10 cm
    Tap your answer.

    1. Yes, that is right. The end area is π × 5² = 25π. The volume is 25π × 10 = 250π, which is 785.4 cm³.

    2. Not this one. You used the diameter, 10 cm, in place of the radius. That makes the answer four times too big. Go to M4.5.1. Try another.

    3. Not this one. That is π × 5 × 10. The end area is π × 5², not π × 5. Go to M4.5.1. Try another.

    4. Not this one. That is 2π × 5 × 10, the area of the curved surface. The volume is the end area times the height. Go to M4.5.1. Try another.

    See the answer

    785.4 cm³

    The end area is π × 5² = 25π. The volume is 25π × 10 = 250π, which is 785.4 cm³.

    • If you chose B: You used the diameter, 10 cm, in place of the radius. That makes the answer four times too big. Go to M4.5.1.
    • If you chose C: That is π × 5 × 10. The end area is π × 5², not π × 5. Go to M4.5.1.
    • If you chose D: That is 2π × 5 × 10, the area of the curved surface. The volume is the end area times the height. Go to M4.5.1.

  4. The end face of this prism is a trapezium. Its parallel sides are 10 cm and 6 cm, and they are 5 cm apart. The prism is 12 cm long. What is its volume?

    A prism with a trapezium for its end face. The long parallel side is labelled 10 cm and the short one 6 cm. A dashed line, labelled 5 cm, joins them at a right angle. The length of the prism is labelled 12 cm.5 cm10 cm6 cm12 cm
    Tap your answer.

    1. Not this one. You found (10 + 6) × 5 × 12 and left out the ½ in the area of a trapezium. Go to M3.6.5. Try another.

    2. Yes, that is right. The end area is ½ × (10 + 6) × 5 = 40 cm². The volume is 40 × 12 = 480 cm³.

    3. Not this one. You used only the long parallel side, 10 × 5 × 12. A trapezium has two parallel sides, and the area uses both of them. Go to M3.6.5. Try another.

    See the answer

    480 cm³

    The end area is ½ × (10 + 6) × 5 = 40 cm². The volume is 40 × 12 = 480 cm³.

    • If you chose A: You found (10 + 6) × 5 × 12 and left out the ½ in the area of a trapezium. Go to M3.6.5.
    • If you chose C: You used only the long parallel side, 10 × 5 × 12. A trapezium has two parallel sides, and the area uses both of them. Go to M3.6.5.

  5. A closed cylinder has radius 2 cm and height 5 cm. What is its total surface area, to 1 decimal place?

    A cylinder. The radius is a line labelled 2 cm. The height, on the right, is labelled 5 cm.2 cm5 cm
    Tap your answer.

    1. Not this one. That is the curved surface alone, 2π × 2 × 5. A closed cylinder also has two circular ends. Go to M4.5.1. Try another.

    2. Not this one. That is the two circular ends alone, 2 × π × 2². The curved surface is missing. Go to M4.5.1. Try another.

    3. Not this one. That is right for a tin with no lid, with only one circular end. This cylinder is closed, so it has two. Go to M4.5.1. Try another.

    4. Yes, that is right. The curved surface is 2π × 2 × 5 = 20π. The two ends are 2 × π × 2² = 8π. The total is 28π, which is 88.0 cm², to 1 decimal place.

    See the answer

    88.0 cm²

    The curved surface is 2π × 2 × 5 = 20π. The two ends are 2 × π × 2² = 8π. The total is 28π, which is 88.0 cm², to 1 decimal place.

    • If you chose A: That is the curved surface alone, 2π × 2 × 5. A closed cylinder also has two circular ends. Go to M4.5.1.
    • If you chose B: That is the two circular ends alone, 2 × π × 2². The curved surface is missing. Go to M4.5.1.
    • If you chose C: That is right for a tin with no lid, with only one circular end. This cylinder is closed, so it has two. Go to M4.5.1.

  6. One cylinder has radius 3 cm. A second cylinder has the same height and radius 6 cm. How many times as big is the volume of the second cylinder?

    Tap your answer.

    1. Not this one. The radius doubles, but the radius is squared in πr². The area of the end face grows by more than 2 times. Go to M4.5.1. Try another.

    2. Not this one. A volume grows by 8 times when all three lengths double. Here the height stays the same, so only the end area changes. Go to M4.4.6. Try another.

    3. Yes, that is right. The end area is πr². Doubling r multiplies it by 2², which is 4. The height has not changed, so the volume is 4 times as big.

    See the answer

    4 times as big

    The end area is πr². Doubling r multiplies it by 2², which is 4. The height has not changed, so the volume is 4 times as big.

    • If you chose A: The radius doubles, but the radius is squared in πr². The area of the end face grows by more than 2 times. Go to M4.5.1.
    • If you chose B: A volume grows by 8 times when all three lengths double. Here the height stays the same, so only the end area changes. Go to M4.4.6.

  7. The end face of a prism is a right angled triangle with base 8 cm and height 5 cm. The prism is 9 cm long. What is its volume?

    See the answer

    180 cm³

    The end area is ½ × 8 × 5 = 20 cm². The volume is 20 × 9 = 180 cm³.

  8. A cylinder has radius 4 cm and height 7 cm. What is its volume, to 1 decimal place?

    See the answer

    351.9 cm³

    The end area is π × 4² = 16π. The volume is 16π × 7 = 112π, which is 351.9 cm³ to 1 decimal place.

Keep it

Copy this into your exercise book.

Prism: volume = end area × length. Cylinder: V = πr²h. Curved surface 2πrh. Two ends 2πr². Use the radius, not the diameter.

Draw: A cylinder, and beside it its net: a rectangle with two circles, the rectangle as long as 2πr.

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

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