Area, volume and compound units. Attnom Academy

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

A Multiply the three measurements you can see.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Multiply the area of the end face by the length.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Add the area of the end face to the length.
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 Here is a prism. Its end face is a triangle, and it is 10 cm long.
2 Cut it into 10 slices, each 1 cm thick. Every slice has the same end face.
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 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.

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

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

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

Cut the cylinder open. The curved surface is a rectangle as long as the circumference of a circle.
The circumference is 2 × π × 3 = 6π cm. The curved surface is 6π × 8 = 48π cm².
Each end is a circle of area π × 3² = 9π cm². The two ends make 18π cm².
The total surface area is 48π + 18π = 66π cm².
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.

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

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.

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.

M4.5.2 Chapter 2

Pyramids and cones

Does a cone hold as much as the cylinder it fits inside?

Your guess

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

Your guess

A Half as much as the cylinder.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The same amount as the cylinder.
You picked B. Read on. At the end, the chapter comes back to your guess.
C One third as much as the cylinder.
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 pyramid as 1/3 × base area × height
Find the volume and curved surface area of a cone

A pyramid and a cone both rise from a flat base to a single point, the apex. A tent, a funnel and an ice cream cone all have this shape.

Take a pyramid and a prism with the same base and the same height. The pyramid fits inside the prism. How much of the prism does it fill?

Draw it first

A pyramid inside a prism

1 Draw a pyramid with a square base. Its height is the straight line from the apex down to the base.
2 Draw the box with the same base and the same height round the pyramid.
3 Three pyramids like this one, poured as sand, fill the box exactly once.

This is a fact you can test with sand, and it can be proved in more advanced mathematics. The pyramid fills one third of the box. The box has volume base area × height. So for a pyramid, V = ⅓ × base area × height.

The height must be the perpendicular height. It is measured at a right angle to the base, straight down from the apex. The slant height runs down a sloping face. It is longer, and it is not used in this volume.

The perpendicular height is 4 cm. The slant height of the front face is 5 cm.

Worked example

Mei builds a model pyramid with a square base of side 6 cm. Its perpendicular height is 4 cm. Find its volume.

The base is a square. Its area is 6 × 6 = 36 cm².
Volume = ⅓ × base area × perpendicular height = ⅓ × 36 × 4.
⅓ × 36 × 4 = 12 × 4 = 48 cm³.

Answer: 48 cm³

A cone is a pyramid with a circle for its base. The base area is πr², so the volume of a cone is V = ⅓πr²h, with h the perpendicular height.

The cone fills one third of the cylinder round it.

Worked example

A cone has base radius 3 cm and perpendicular height 4 cm. Find its volume, to 1 decimal place.

The base is a circle of radius 3 cm. Its area is π × 3² = 9π cm².
Volume = ⅓ × base area × height = ⅓ × 9π × 4.
⅓ × 9π × 4 = 3π × 4 = 12π cm³.
Put in the π key of a calculator. 12π is 37.7 cm³, to 1 decimal place.

Answer: 37.7 cm³

Now the curved surface of a cone. Cut it open and lay it flat. You get a sector of a circle, with a circle for the base beside it.

The sector has radius l, the slant height. Its curved edge is the circumference of the base.

The straight sides of the sector are the slant height l. The curved edge is the circumference of the base circle, 2πr. A whole circle of radius l has circumference 2πl and area πl².

Our sector is the fraction 2πr ÷ 2πl = r ÷ l of that whole circle. So its area is (r ÷ l) × πl² = πrl. The curved surface area of a cone is πrl.

To find l, use Pythagoras on the right angled triangle inside the cone. Its sides are r, the perpendicular height h, and l: l² = r² + h². So l = √(r² + h²).

Worked example

A cone has base radius 3 cm and perpendicular height 4 cm. Find the area of its curved surface, to 1 decimal place.

The slant height l joins the apex to the rim. It is the longest side of a right angled triangle with sides 3 and 4.
l² = 3² + 4² = 9 + 16 = 25.
l = √25 = 5 cm.
Curved surface area = π × r × l = π × 3 × 5 = 15π cm².
Put in the π key of a calculator. 15π is 47.1 cm², to 1 decimal place.

Answer: 47.1 cm²

Try it

Three cones fill a cylinder

You need: a sheet of stiff paper, a piece of tape, dry rice, sand or salt.

Roll the paper into a cone and tape it. Cut the wide end flat, so the cone can stand on it.
Roll a second sheet into a cylinder with the same wide end and the same height.
Fill the cone with rice and tip it into the cylinder.
Count how many cones of rice fill the cylinder.

What you will see: You need three cones. The cone holds one third as much as the cylinder, which is why there is a ⅓ in the volume of a cone.

If you cannot try it, this is what happens.

A wrong idea many people have

In the volume of a cone, h is the slant height.

Why it feels right: The slant height is the length you can see along the side, and you can measure it with a ruler.

The test that breaks it: Take the cone with radius 3 cm, perpendicular height 4 cm and slant height 5 cm. The true volume is ⅓ × π × 9 × 4 = 12π, which is 37.7 cm³ to 1 decimal place. With 5 you get ⅓ × π × 9 × 5 = 15π, which is 47.1 cm³, and that is too big.

What is true: Use the perpendicular height, at a right angle to the base. The slant height belongs in the curved surface, πrl.

A wrong idea many people have

The volume of a pyramid is base area × height, like a prism.

Why it feels right: A pyramid has a base and a height, and for prisms we multiply them.

The test that breaks it: Take the pyramid with a 6 cm square base and a height of 4 cm. Base area × height is 36 × 4 = 144 cm³. That is the whole box. Three pyramids fill the box, so one pyramid is 144 ÷ 3 = 48 cm³.

What is true: A pyramid fills only one third of the prism with the same base and height. Multiply by ⅓.

Key words

apex: the point at the top of a pyramid or a cone.
perpendicular height: the height from the apex straight down to the base, at a right angle to it.
slant height: the length from the apex down the sloping side to the base edge.

Back to your guess

The answer is C: One third as much as the cylinder.

A cone with the same base and height as a cylinder holds exactly one third as much. Fill the cone with rice, tip it into the cylinder, and you need three cones to fill it.

Keep it

Copy this into your exercise book.

Pyramid and cone: V = ⅓ × base area × perpendicular height. Cone V = ⅓πr²h. Curved surface πrl, with l² = r² + h². Not the slant height in V.

Draw: A cone with r, h and l marked, the height as a dashed upright line with a right angle at the base.

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

A Two thirds of the cylinder that just holds it.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Half of the cylinder that just holds it.
You picked B. Read on. At the end, the chapter comes back to your guess.
C All of the cylinder that just holds it.
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 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 Draw a sphere of radius 3 cm.
2 Draw the cylinder that just holds it. It is 3 cm in radius and 6 cm high.
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.

Work out r³ = 3 × 3 × 3 = 27.
V = 4/3 × π × 27. Now 4/3 of 27 is 36, so V = 36π cm³.
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.

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.

The surface area of a sphere is A = 4πr².
Work out r² = 5 × 5 = 25.
A = 4 × π × 25 = 100π cm².
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 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.

The whole sphere has V = 4/3 × π × 6³, and 6³ = 216.
4/3 of 216 is 288, so the whole sphere has volume 288π cm³.
The bowl is half of it, so its volume is 288π ÷ 2 = 144π cm³.
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.

Stand the bowl on the tray. Fill it to the brim.
Lower the ball of clay in gently. Water spills onto the tray.
Pour the spilled water into the jug and read it.
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 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.

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.

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

A Add the volume of the whole cone and the volume of the tip.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Take the volume of the tip away from the volume of the whole cone.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Take the radius of the top away from the radius of the base and use the cone formula.
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 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.

The cylinder has volume π × 3² × 6 = 54π cm³.
The cone has volume ⅓ × π × 3² × 4 = 12π cm³.
Add the two parts: 54π + 12π = 66π cm³.
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.
2 Cut the tip off halfway up. The tip is a small cone with radius 3 cm and height 4 cm.
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.

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

Let the tip have height x cm. The tip and the whole cone are similar, so x ÷ (x + 6) = 2 ÷ 5.
Cross multiply: 5x = 2(x + 6) = 2x + 12. So 3x = 12, and x = 4.
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.
The volume is ⅓ × π × (250 − 16) = ⅓ × π × 234 = 78π cm³.
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.

Roll the paper into a cone and tape it. Trim the base flat.
Mark the middle of the sloping side, halfway from the base to the apex.
Cut straight across at that mark, level with the base.
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 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.

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.

M4.5.5 Chapter 5

Changing units of area and volume

How many square centimetres are in one square metre?

Your guess

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

Your guess

A 100 square centimetres.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 1 000 square centimetres.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 10 000 square centimetres.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Change between cm² and m², and between cm³ and m³
Change between cm³, litres and m³

You know that 1 m = 100 cm. It is tempting to think that 1 m² = 100 cm² as well. It is not. This chapter shows why, and gives one rule that works for every unit.

A square metre is the area of a square 1 m long and 1 m wide. We can cut that square into smaller squares, 10 cm by 10 cm. The picture shows them.

One square metre, cut into 100 squares of 10 cm by 10 cm.

Each small square is 10 cm by 10 cm, so its area is 10 × 10 = 100 cm². There are 10 rows of 10, which is 100 small squares. So the whole area is 100 × 100 = 10 000 cm².

The same idea works for any unit. The square has 100 cm along each edge, and the area is 100 × 100. So 1 m² = 10 000 cm². Likewise 1 cm = 10 mm, so 1 cm² = 10 × 10 = 100 mm².

The area 3.5 m² is three whole square metres and half of another.

Worked example

Change 3.5 m² into cm².

One square metre is 10 000 cm².
There are 3.5 square metres, so multiply: 3.5 × 10 000.
3.5 × 10 000 = 35 000 cm².

Answer: 35 000 cm²

To change cm² into m², go the other way and divide by 10 000. A small unit gives a big number, and a big unit gives a small number. So 20 000 cm² is 2 m².

Now volume. A cube 1 m on each edge has volume 1 m³. It is also 100 cm by 100 cm by 100 cm. The picture cuts it into small cubes, 10 cm on each edge.

One cubic metre. It holds 10 × 10 × 10 = 1 000 small cubes.

A cube with 100 cm along each edge has volume 100 × 100 × 100 = 1 000 000 cm³. So 1 m³ = 1 000 000 cm³. A million small cubes, each 1 cm on an edge, fill a cubic metre.

When lengths change by a factor, areas change by that factor squared and volumes by that factor cubed.

The table gives the rule. Change the unit of length by a factor k. The unit of area changes by k × k. The unit of volume changes by k × k × k. For m to cm, k = 100.

A litre is a unit for liquids. One litre is the volume of a cube 10 cm on each edge. That is 10 × 10 × 10 = 1 000 cm³. One millilitre (ml) is one thousandth of a litre, so 1 ml = 1 cm³.

One litre is a cube 10 cm on each edge.

The cubic metre picture held 1 000 small cubes of 10 cm. Each is one litre. So 1 m³ = 1 000 litres. It also agrees with 1 000 000 cm³ ÷ 1 000 cm³ = 1 000.

Worked example

Lucía has a fish tank that is 80 cm long, 40 cm wide and 50 cm high. How many litres does it hold when it is full?

The volume is length × width × height = 80 × 40 × 50.
80 × 40 × 50 = 160 000 cm³.
One litre is 1 000 cm³, so divide by 1 000.
160 000 ÷ 1 000 = 160 litres.

Answer: 160 litres

The fish tank, full to the top.

A water store 1 m by 1 m by 1 m, filled to 75 cm. It holds 0.75 m³.

Worked example

A water store holds 0.75 m³. How many cm³ is that, and how many litres?

One cubic metre is 1 000 000 cm³.
So 0.75 m³ is 0.75 × 1 000 000 = 750 000 cm³.
One litre is 1 000 cm³. So 750 000 cm³ is 750 000 ÷ 1 000 = 750 litres.
Check with 1 m³ = 1 000 litres: 0.75 × 1 000 = 750 litres. The two ways agree.

Answer: 750 000 cm³ and 750 litres

Try it

A hundred small squares

You need: squared paper with 1 cm squares, a pencil and a ruler.

Draw a square 10 cm by 10 cm on the paper.
Count the squares inside it. Count the rows, then the columns.
Work out 10 × 10 and compare with your count.
Imagine 100 rows of 100 of these small squares. That is one square metre.

What you will see: The square holds 100 squares of 1 cm², so its area is 100 cm². A square 10 times as long on each side, 100 cm, would hold 10 × 10 times as many, which is 10 000.

If you cannot try it, this is what happens.

A wrong idea many people have

Since 1 m = 100 cm, it follows that 1 m² = 100 cm².

Why it feels right: We change metres to centimetres by multiplying by 100, and the little 2 only seems to say that it is an area.

The test that breaks it: Draw a square 1 m by 1 m. In centimetres it is 100 cm by 100 cm, and its area is 100 × 100 = 10 000 cm². It is not 100 cm². A 10 cm by 10 cm square has area 100 cm², which is a tiny part of a square metre.

What is true: Areas change by the square of the length factor, and volumes by the cube. So 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³.

A wrong idea many people have

A litre is 100 cm³, because a litre box is 10 cm long.

Why it feels right: The edge of the litre cube is 10 cm, and 10 × 10 = 100 is a number that is quick to remember.

The test that breaks it: A litre cube has three edges of 10 cm. Volume is length × width × height, so 10 × 10 × 10 = 1 000 cm³. A square of 10 × 10 is only a flat face of the cube.

What is true: One litre is 1 000 cm³, so 1 ml is 1 cm³. And 1 000 litres make 1 m³.

Key words

litre: a unit for liquids: the volume of a cube 10 cm on each edge, which is 1 000 cm³.
millilitre: one thousandth of a litre. 1 ml is the same as 1 cm³.

Back to your guess

The answer is C: 10 000 square centimetres.

A square metre is 100 cm long and 100 cm wide, so it holds 100 × 100 = 10 000 square centimetres. The number is not 100, because 100 is the number of centimetres along one edge.

Keep it

Copy this into your exercise book.

1 m² = 10 000 cm². 1 m³ = 1 000 000 cm³. 1 litre = 1 000 cm³. 1 ml = 1 cm³. 1 m³ = 1 000 litres. Small unit to big unit: divide.

Draw: A square metre cut into 100 squares of 10 cm, with one square shaded and labelled 100 cm².

M4.5.6 Chapter 6

Speed, distance and time

A car drives 60 km at 60 km per hour, then 40 km at 20 km per hour. What is its average speed?

Your guess

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

Your guess

A About 33 km per hour.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 40 km per hour, halfway between 60 and 20.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 80 km per hour, the two speeds added.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use speed = distance ÷ time with the right units
Change km per hour into metres per second
Find the average speed for a journey in two parts

Speed tells you how far something goes in one unit of time. A cyclist who goes 15 km in each hour has a speed of 15 km per hour, written 15 km/h.

Km per hour is a compound unit. It is made of two units, a unit of distance (km) and a unit of time (hours), joined by "per". Metres per second (m/s) is another.

The rule is speed = distance ÷ time. It can be turned round: distance = speed × time, and time = distance ÷ speed.

One rule, three ways to use it.

The units must fit. If the distance is in km and the time is in hours, the speed is in km/h. If the time is given in minutes, change it to hours first. There are 60 minutes in an hour.

A cyclist rides 24 km in 90 minutes.

Worked example

Ravi rides his bicycle 24 km in 90 minutes. What is his speed in km/h?

The time is in minutes. Change it to hours: 90 ÷ 60 = 1.5 hours.
Speed = distance ÷ time = 24 ÷ 1.5.
24 ÷ 1.5 = 16 km/h.

Answer: 16 km/h

Now change units. In science, speeds are often in metres per second. One km/h is 1 000 m in 3 600 seconds, which is 1 ÷ 3.6 m/s. So to change km/h into m/s, divide by 3.6. To change m/s into km/h, multiply by 3.6.

Divide the number in km/h by 3.6 to get m/s.

Worked example

A train goes at 72 km/h. What is its speed in m/s?

One hour is 3 600 seconds, and 1 km is 1 000 m.
So 1 km/h = 1 000 ÷ 3 600 m/s, which is 1 ÷ 3.6 m/s.
To change km/h to m/s, divide by 3.6.
72 ÷ 3.6 = 20 m/s.

Answer: 20 m/s

Now a journey in two parts. A car drives 60 km in 1 hour, then 40 km in the next 2 hours. The bar below shows the two parts. Each part is as long as its distance.

A journey in two parts. The speeds are 60 km/h and 20 km/h.

It is tempting to say the average speed is halfway between 60 and 20, which is 40 km/h. That is wrong, because the car spends 2 hours at 20 km/h and only 1 hour at 60 km/h.

The average speed is the one steady speed that covers the same total distance in the same total time. So average speed = total distance ÷ total time.

Worked example

A car drives 60 km in 1 hour, then 40 km in 2 hours. What is its average speed, to 1 decimal place?

Total distance = 60 + 40 = 100 km.
Total time = 1 + 2 = 3 hours.
Average speed = total distance ÷ total time = 100 ÷ 3.
To 1 decimal place, 100 ÷ 3 = 33.3 km/h.
This is not the mean of the two speeds, 60 and 20, which would be 40.

Answer: 33.3 km/h

A distance time graph shows the same journey. Time goes across and distance goes up. The steeper the line, the faster the speed. The dashed line is the average, a steady journey from the start to the end.

The dashed line is steady. It reaches the same end point in the same time.

Try it

Measure your walking speed

You need: a long tape or string, marked at 20 m, a clock or phone that shows seconds, a friend to time you.

Lay the string flat along a straight path, 20 m long.
Walk along it at your usual pace. Your friend times you in seconds.
Work out speed = distance ÷ time, in m/s.
Multiply by 3.6 to change it to km/h.

What you will see: Most people cover 1 to 1.5 m every second when they walk. That is about 3.6 to 5.4 km/h.

If you cannot try it, this is what happens.

A wrong idea many people have

The average speed of a journey is the mean of the two speeds.

Why it feels right: We find an average of numbers by adding them and dividing, so it looks right to do it with speeds.

The test that breaks it: Drive 60 km at 60 km/h, then 40 km at 20 km/h. The mean of 60 and 20 is 40 km/h. But the car goes 100 km in 3 hours, which is 33.3 km/h. The car spent longer at the slow speed.

What is true: Average speed is the total distance divided by the total time. Use the mean of two speeds only if the times are equal.

A wrong idea many people have

A cyclist who rides 24 km in 90 minutes has a speed of 24 ÷ 90 km/h.

Why it feels right: The rule says divide the distance by the time, and the time is the number 90.

The test that breaks it: 24 ÷ 90 = 0.27, which is far slower than anyone walks. But km/h needs the time in hours. 90 minutes is 1.5 hours, and 24 ÷ 1.5 = 16 km/h.

What is true: Make the units fit first. For km/h the time must be in hours.

Key words

compound unit: a unit made of two units, such as km per hour or metres per second.
average speed: the total distance of a journey divided by the total time.

Back to your guess

The answer is A: About 33 km per hour.

The average speed is the total distance divided by the total time. The car goes 100 km in 3 hours, which is 33.3 km per hour. It spends longer at the slow speed, so the average is pulled towards 20, not halfway.

Keep it

Copy this into your exercise book.

Speed = distance ÷ time. Distance = speed × time. Time = distance ÷ speed. km/h to m/s: ÷ 3.6. Average speed = total distance ÷ total time.

Draw: A bar in two parts with its distances and times, and the total under a bracket.

M4.5.7 Chapter 7

Density and pressure

Why does a big block of wood float, while a small iron nail sinks?

Your guess

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

Your guess

A Heavy things sink and light things float.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Big things always float and small things always sink.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A thing floats when it has less mass in each cm³ than water has.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use density = mass ÷ volume to find any one of the three
Use pressure = force ÷ area
Find the mass of a block from its size and density

Two blocks of the same size can have very different masses. A block of iron is much heavier than a block of wood of the same size. The iron has more mass packed into each cm³.

Density says how much mass there is in each unit of volume. Density = mass ÷ volume. Its unit is a compound unit: grams per cm³, written g/cm³. Another is kg per m³.

Water has a density of about 1 g/cm³. So 1 cm³ of water has a mass of about 1 g. A litre of water, which is 1 000 cm³, has a mass of about 1 kg.

Some densities. A solid object made of one material floats in water if its density is less than 1 g/cm³.

The rule density = mass ÷ volume can be turned round, in the same way as speed. Use the table to choose.

One rule, three ways to use it.

Yusuf's block of metal: 10 cm by 5 cm by 4 cm, with a mass of 540 g.

Worked example

Yusuf has a block of metal with a mass of 540 g and a volume of 200 cm³. What is its density?

Density = mass ÷ volume.
Density = 540 ÷ 200 g/cm³.
540 ÷ 200 = 2.7 g/cm³. That matches aluminium in the table.

Answer: 2.7 g/cm³

To find a mass from a size, first find the volume of the block. Then multiply the volume by the density. The mass is in grams if the volume is in cm³ and the density is in g/cm³.

A block of iron, 10 cm by 5 cm by 4 cm.

Worked example

Iron has a density of about 7.9 g/cm³. Find the mass of an iron block 10 cm by 5 cm by 4 cm.

The volume of the block is 10 × 5 × 4 = 200 cm³.
Mass = density × volume = 7.9 × 200.
7.9 × 200 = 1 580 g.
That is 1.58 kg, because 1 000 g is 1 kg.

Answer: About 1 580 g

Now pressure. A force is a push or a pull, measured in newtons (N). Pressure is how hard a force pushes on each unit of area. Pressure = force ÷ area.

Pressure is measured in newtons per cm² (N/cm²), or in newtons per m² (N/m²). One newton per m² is one pascal (Pa).

The same force can give a small or a big pressure. A block lying on its large face spreads its weight over more area than the same block standing on its small face.

Weight is a force, the pull of the Earth on something. Every 1 kg of mass weighs about 10 N. The iron block has a mass of 1.58 kg, so it weighs about 16 N.

The block lies on its bottom face, 10 cm by 5 cm, which is 50 cm².

The block stands on its bottom face, 5 cm by 4 cm, which is 20 cm².

Worked example

A block weighing about 16 N stands on a face of 5 cm by 4 cm. Find the pressure on the table, first in N/cm² and then in pascals.

The area of the face is 5 × 4 = 20 cm².
Pressure = force ÷ area = 16 ÷ 20 = 0.8 N/cm².
One m² is 10 000 cm², so 1 N/cm² is the same as 10 000 N/m².
So 0.8 N/cm² is 0.8 × 10 000 = 8 000 N/m², which is about 8 000 pascals.

Answer: About 0.8 N/cm², or about 8 000 pascals

Try it

Sink or float

You need: a bowl of water, a stone, a coin, a piece of wood and a candle.

Guess for each thing: will it sink or float?
Put the things in the water one at a time.
Write down what happened.

What you will see: The stone and the coin sink, and the wood and the candle float. It does not depend on how big a thing is. It depends on its density compared with water.

If you cannot try it, this is what happens.

A wrong idea many people have

Heavy things sink and light things float.

Why it feels right: Most heavy things we pick up, like stones, do sink, and a feather floats.

The test that breaks it: A big log has a mass of many kilograms, and it floats. A small steel nail has a mass of a few grams, and it sinks. Mass alone does not decide it.

What is true: A solid object made of one material sinks in water if its density is more than 1 g/cm³. It floats if its density is less. Compare the density, not the mass.

A wrong idea many people have

Pressure is the same as force.

Why it feels right: A big force feels like a big pressure when you push hard with your hand.

The test that breaks it: The block weighs about 16 N in both pictures. Lying down, the pressure is 16 ÷ 50 = 0.32 N/cm². Standing, it is 16 ÷ 20 = 0.8 N/cm². The force is the same, but the pressure is 2.5 times as big.

What is true: Pressure is force divided by area. The same force on a smaller area gives a bigger pressure. That is why a sharp knife cuts.

Key words

density: the mass in each unit of volume. Density = mass ÷ volume.
pressure: the force on each unit of area. Pressure = force ÷ area.
pascal: the unit of pressure. One pascal is one newton per m².

Back to your guess

The answer is C: A thing floats when it has less mass in each cm³ than water has.

A big log is heavy, but each cm³ of wood has less mass than each cm³ of water. So it floats. A small nail is light, but each cm³ of iron has much more mass than water. So it sinks. What counts is the mass in each cm³, which is called density.

Keep it

Copy this into your exercise book.

Density = mass ÷ volume. Water is about 1 g/cm³. Pressure = force ÷ area. N/m² is the pascal. 1 N/cm² = 10 000 N/m². Smaller area, bigger pressure.

Draw: A block standing on its small face, and the same block lying flat. Put a 16 N arrow on each.

M4.5.8 Chapter 8

Rates of flow and rate graphs

A tap fills a tank at 12 litres a minute. The tank holds 120 litres. How long does it take?

Your guess

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

Your guess

A 10 minutes, because 120 ÷ 12 = 10.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 108 minutes, because 120 − 12 = 108.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 1 440 minutes, because 120 × 12 = 1 440.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find how long a tank takes to fill at 12 litres a minute
Read the gradient of a straight line graph as a rate

A rate of flow says how much liquid moves in each unit of time. A tap that gives 12 litres every minute has a rate of 12 litres per minute, written 12 litres/min. It is a compound unit, like km/h.

The rule is the same shape as the rule for speed. Rate = volume ÷ time. It can be turned round: volume = rate × time, and time = volume ÷ rate.

One rule, three ways to use it.

The picture shows a tank with a tap above it. The tank is partly full, and more water is flowing in. The tank is 60 cm long, 40 cm wide and 50 cm deep.

A tank that fills at 12 litres a minute.

Worked example

Hana has the tank in the picture. It is 60 cm long, 40 cm wide and 50 cm deep. It starts empty, and the picture shows it part way through filling. A tap fills it at 12 litres per minute. How long does it take to fill the whole tank?

The volume of the tank is 60 × 40 × 50 = 120 000 cm³.
Change to litres. One litre is 1 000 cm³, so 120 000 ÷ 1 000 = 120 litres.
Time = volume ÷ rate = 120 ÷ 12.
120 ÷ 12 = 10 minutes.

Answer: 10 minutes

Now draw a graph. Put time in minutes across, and volume in litres up. When the tap runs steadily, each minute adds the same volume. So the points lie on a straight line through (0, 0).

See it three ways

With things

A tap fills a jug with 2 litres every minute. Pour each jugful into a tank and write down how much water the tank holds after each minute.

As a picture

In symbols

Volume = rate × time. After 5 minutes the tank holds 2 × 5 = 10 litres.

The steepness of the line is its gradient. The gradient is how much the volume goes up for each 1 minute across. That is the rate of flow. Gradient = change in volume ÷ change in time.

Two points on the line. Across 4 minutes the volume goes up 32 litres.

Worked example

The graph shows water running steadily into a tank. The line goes through (2, 16) and (6, 48). Find the rate of flow.

Choose two points on the line that are far apart: (2, 16) and (6, 48).
The volume goes up by 48 − 16 = 32 litres.
The time goes up by 6 − 2 = 4 minutes.
Gradient = 32 ÷ 4 = 8 litres per minute.

Answer: 8 litres per minute

A graph can have more than one part. A flat part means that the volume is not changing, so the rate is 0. A line that goes down means water is leaving the tank.

Hana's tank holds 120 litres. The graph shows it filled, left, and then emptied.

A tank fills, rests, and empties.

Worked example

The graph shows Hana's tank, which holds 120 litres. It is filled, left, and then emptied. Find the rate of flow when it fills and when it empties.

Filling: the tank has 60 litres after 5 minutes. The rate is 60 ÷ 5 = 12 litres per minute.
The flat part, from 5 to 8 minutes, has a gradient of 0. No water flows.
Emptying: the water goes from 60 litres to 0 between 8 and 14 minutes. That takes 14 − 8 = 6 minutes.
Water leaves at 60 ÷ 6 = 10 litres per minute.

Answer: 12 litres per minute in, 10 litres per minute out

Try it

Time a tap

You need: a tap, a 1 litre bottle or jug, a clock that shows seconds.

Open the tap fully and hold the bottle under it.
Time how many seconds it takes to fill the bottle to the 1 litre mark.
Work out how many litres would flow in 60 seconds: 60 ÷ your time.

What you will see: The table shows the seconds taken to fill 1 litre, and the rate in litres per minute. If the bottle fills in 10 seconds, the tap gives 6 litres in a minute. If it fills in 5 seconds, it gives 12 litres a minute.

If you cannot try it, this is what happens.

A wrong idea many people have

A flat part of a volume time graph means that the tank is full.

Why it feels right: The line stops going up, and a full tank cannot take any more water.

The test that breaks it: Look at the graph of the tank. The line is flat at 60 litres from 5 to 8 minutes, but the tank holds 120 litres. It is half full, and the tap is turned off.

What is true: A flat part means the volume is not changing. The rate is 0. It does not tell you whether the tank is full or not.

A wrong idea many people have

The time to fill a tank is the volume times the rate.

Why it feels right: When we find a distance we multiply, so multiplying looks like the right thing to do.

The test that breaks it: Volume 120 litres times rate 12 litres per minute is 1 440. That would be a whole day for a small tank. Each minute adds 12 litres, so you count how many 12s fit in 120. That is a division, 10 minutes.

What is true: Time = volume ÷ rate. Multiply only when you want a volume: volume = rate × time.

Key words

rate of flow: how much liquid moves in each unit of time, such as litres per minute.
gradient: how steep a line is: the change up divided by the change across.

Back to your guess

The answer is A: 10 minutes, because 120 ÷ 12 = 10.

Each minute adds 12 litres, so you count how many lots of 12 litres make 120. That is 120 ÷ 12 = 10 minutes. Taking one number from the other has no meaning, and 1 440 minutes would be a whole day.

Keep it

Copy this into your exercise book.

Rate = volume ÷ time. Volume = rate × time. Time = volume ÷ rate. On a volume time graph the gradient is the rate. Flat means no flow. Down means water leaves.

Draw: A volume time graph with a rising line, a flat part and a falling line, each part labelled.

Slice a prism

Wrap a tin

A pyramid inside a prism

Three cones fill a cylinder

A sphere in a cylinder

Measure a ball with water

A cone with its tip cut off

Cut a cone in half

A hundred small squares

Measure your walking speed

Sink or float

Time a tap

Key words, A to Z