Perimeter, area and volume. Attnom Academy

M3.6.1 Chapter 1

Changing metric units

How many metres are there in 3.5 kilometres?

Your guess

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

Your guess

A 35 m
You picked A. Read on. At the end, the chapter comes back to your guess.
B 350 m
You picked B. Read on. At the end, the chapter comes back to your guess.
C 3.5 m
You picked C. Read on. At the end, the chapter comes back to your guess.
D 3 500 m
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Change 3.5 km into metres and 450 mm into centimetres
Change between grams, kilograms and tonnes

A sign on a path says the river is 3.5 km away. How many metres is that? To find out, you convert the length. To convert is to change an amount from one unit to another.

We use metric units for length and mass: mm, cm, m and km, and g, kg and t. Metric units go up and down in tens, hundreds and thousands. That makes them quick to convert.

Draw it first

Climb down the ladder

1 Write the units in a ladder with the biggest at the top: km, m, cm, mm.
2 Start with 3.5 km. To reach metres, go down one rung.
3 Going down, you multiply by the number on the arrow: 3.5 × 1 000 = 3 500 m.

Going down the ladder, the unit gets smaller. You need more of them, so you multiply. Going up, the unit gets bigger. You need fewer of them, so you divide.

The length ladder has three steps. 1 km is 1 000 m. 1 m is 100 cm. 1 cm is 10 mm.

Worked example

Change 3.5 km into metres.

1 km is 1 000 m.
Going from km to m is going down the ladder. The unit gets smaller, so multiply.
3.5 × 1 000 = 3 500 m.

Answer: 3 500 m

From km to m you go down one rung and multiply by 1 000.

Multiplying by 1 000 moves every digit three places to the left, and zeros fill the gaps. So 3.5 becomes 3 500.

Worked example

Change 450 mm into centimetres.

10 mm is 1 cm.
Going from mm to cm is going up the ladder. The unit gets bigger, so divide.
450 ÷ 10 = 45 cm.
Check by going back down: 45 × 10 = 450 mm.

Answer: 45 cm

From mm to cm you go up one rung and divide by 10.

Mass has its own ladder. 1 kg is 1 000 g. The unit for a heavy load is the tonne (t). One tonne is 1 000 kg, about the mass of a small car.

1 t is 1 000 kg. 1 kg is 1 000 g. Each step is 1 000.

Worked example

Change 3 600 g into kilograms.

1 kg is 1 000 g.
Going from g to kg is going up the ladder. The unit gets bigger, so divide.
3 600 ÷ 1 000 = 3.6 kg.
Check by going back down: 3.6 × 1 000 = 3 600 g.

Answer: 3.6 kg

From g to kg you go up one rung and divide by 1 000.

Tonnes work the same way. To change 2.5 t into kilograms, go down one rung: 2.5 × 1 000 = 2 500 kg.

See it three ways

With things

Lay a ruler beside a pencil. Read its length in cm. Each cm has 10 small marks, so you can also say it in mm.

As a picture

In symbols

4.5 cm × 10 = 45 mm. And 45 mm ÷ 10 = 4.5 cm.

Try it

Measure in two units

You need: a ruler with cm and mm marks, a pencil, a sheet of paper.

Measure the length of your pencil in cm. Write it as a decimal, such as 15.2 cm.
Multiply by 10 to change it into mm. Write the answer.
Count the small marks on the ruler to read the length in mm. Compare the two answers.

What you will see: The two answers are the same length in two units. A pencil that is 7.5 cm long is also 75 mm long.

If you cannot try it, this is what happens.

A wrong idea many people have

When you change to a smaller unit, the number gets smaller.

Why it feels right: The word "smaller" makes people think of a smaller number.

The test that breaks it: Take a piece of string 1 m long. Measure it in cm. You count 100 cm. The unit is smaller, but you count more of them.

What is true: A smaller unit fits more times into the same length. So the number gets bigger. 1 m is 100 cm.

Key words

convert: to change an amount from one unit to another.
metric units: units that go up and down in tens, hundreds and thousands, such as m, cm and kg.
tonne: a unit of mass. 1 tonne is 1 000 kg.

Back to your guess

The answer is D: 3 500 m

One kilometre is 1 000 m. So 3.5 km is 3.5 × 1 000 = 3 500 m. A metre is a much smaller unit than a kilometre, so you need many more of them.

Keep it

Copy this into your exercise book.

Down the ladder: multiply. Up the ladder: divide. 1 km = 1 000 m. 1 m = 100 cm. 1 cm = 10 mm. 1 t = 1 000 kg. 1 kg = 1 000 g.

Draw: The length ladder: km, m, cm, mm. Write × 1 000, × 100 and × 10 on the arrows going down. Write ÷ on the arrows going up.

M3.6.2 Chapter 2

Perimeter

How much fence do I need to go all the way round my garden?

Your guess

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

Your guess

A Add the lengths of all the sides.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Add the length and the width.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Multiply the length by the width.
You picked C. Read on. At the end, the chapter comes back to your guess.
D Count the squares inside the garden.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the perimeter of a shape with straight sides
Find a missing side when you know the perimeter

A farmer wants a fence round a field. She needs to know how long the fence must be. The distance all the way round a shape is its perimeter.

To find a perimeter, add the lengths of all the sides. Every length needs its unit, such as cm or m.

Draw it first

Add every side

1 Draw the shape. This rectangle is 6 cm long and 4 cm wide.
2 Write the length of every side beside it, with its unit.
3 Add all four sides: 6 + 4 + 6 + 4 = 20 cm.

A rectangle has two long sides and two short sides. The opposite sides are equal. So the perimeter is 2 × length + 2 × width. You can also add the length and width, then double it: 2 × (length + width).

Worked example

A garden is a rectangle 8 m long and 5 m wide. How much fence goes round it?

The fence goes round all four sides. We need the perimeter.
The garden has two sides of 8 m and two sides of 5 m.
8 + 5 + 8 + 5 = 26 m.

Answer: 26 m

The garden. The fence is 8 + 5 + 8 + 5 = 26 m long.

A square has four sides that are all equal. So its perimeter is 4 × the side. A square with sides of 7 m has a perimeter of 4 × 7 = 28 m.

Sometimes a side is missing, and you know the perimeter instead. Add the sides you know. Then take that total away from the perimeter.

Worked example

A triangle has sides of 5 cm and 6 cm, and a third side. Its perimeter is 18 cm. How long is the third side?

The three sides add to the perimeter: 5 + 6 + the third side = 18.
The two known sides add to 5 + 6 = 11 cm.
The third side is 18 − 11 = 7 cm.
Check: 5 + 6 + 7 = 18 cm.

Answer: 7 cm

The side marked ? cm is 18 − 5 − 6 = 7 cm.

Some shapes hide a side. Look at an L shape. Two of its sides have no label. You can find each one from the sides on the opposite edge.

Worked example

Find the perimeter of this L shape. Two sides are marked ? m.

Find the two missing sides first.
The flat missing side is 11 − 4 = 7 m.
The upright missing side is 8 − 3 = 5 m.
Add all six sides: 11 + 3 + 7 + 5 + 4 + 8 = 38 m.
A rectangle 11 m by 8 m has 2 × (11 + 8) = 38 m. The same.

Answer: 38 m

The flat ? side is 11 − 4 = 7 m. The upright ? side is 8 − 3 = 5 m.

Look again at the L shape. Its perimeter is the same as the rectangle that would hold it. That works for any L shape with one corner cut out.

See it three ways

With things

Wrap a string round a book cover. Pull the string straight and measure it with a ruler.

As a picture

In symbols

Perimeter = 6 + 4 + 6 + 4 = 20 cm. Or 2 × (6 + 4) = 20 cm.

Try it

The string round a book

You need: a book or an exercise book, a piece of string, a ruler, a pencil.

Measure the long side and the short side of the cover in cm.
Add the four sides to find the perimeter.
Lay the string round the edge of the cover. Cut it where it meets.
Straighten the string and measure it with the ruler.

What you will see: The string and the sum give the same length. A cover is 21 cm by 15 cm. Its perimeter is 21 + 15 + 21 + 15 = 72 cm.

If you cannot try it, this is what happens.

A wrong idea many people have

The perimeter of a rectangle is its length plus its width.

Why it feels right: Those are the two numbers you see on the picture.

The test that breaks it: Walk round a rectangle that is 5 m by 3 m. You walk 5 m, then 3 m, then 5 m, then 3 m. That is 16 m, not 8 m.

What is true: A rectangle has four sides. Add all four, or add the length and width and then double the total.

Key words

perimeter: the distance all the way round a shape.
rectangle: a four sided shape with four right angles. Its opposite sides are equal.
missing side: a side with no length written. You work it out from the sides you know.

Back to your guess

The answer is A: Add the lengths of all the sides.

A fence goes round every side. So you add the lengths of all the sides. This total is the perimeter. Multiplying or counting squares gives the space inside, which is a different job.

Keep it

Copy this into your exercise book.

Perimeter is the distance all the way round. Add every side and write the unit. Rectangle: 2 × (length + width). Square: 4 × side. Missing side: perimeter minus the sides you know.

Draw: A rectangle 6 cm by 4 cm with each side labelled. Write 6 + 4 + 6 + 4 = 20 cm under it.

M3.6.3 Chapter 3

Area of rectangles

How many tiles will cover my floor?

Your guess

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

Your guess

A Add the length and the width of the floor.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Add all four sides of the floor.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Multiply the length by the width of the floor.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the area of a rectangle as length × width
Find the area of an L shape by splitting it
Explain why 1 m² is 10 000 cm²

You want to cover a floor with square tiles. How many tiles do you need? The space inside a shape is its area. We measure area by counting squares.

A square centimetre is a square that is 1 cm long and 1 cm wide. We write it as 1 cm². An area of 24 cm² means 24 of these squares fit inside the shape.

Draw it first

Count the squares in rows

1 Draw the rectangle on squared paper. Each square is 1 cm by 1 cm.
2 One row has 6 squares. There are 4 rows.
3 Count all the squares with a sum: 6 × 4 = 24 cm².

Rows of equal size make a short cut. The number of squares in a row is the length. The number of rows is the width. So the area of a rectangle is its length × its width.

Worked example

A floor is a rectangle 7 m long and 3 m wide. What is its area?

Imagine squares that are 1 m by 1 m. Each is a square metre, written 1 m².
There are 3 rows with 7 squares in each row.
7 × 3 = 21 m².

Answer: 21 m²

Each square is 1 m by 1 m. There are 3 rows of 7 squares.

A square is a rectangle with equal sides. A square with sides of 6 m has an area of 6 × 6 = 36 m². Always write the unit with a small 2, such as cm² or m², because you are counting squares.

Some shapes are not rectangles. An L shape is made of two rectangles joined together. To find its area, split it into rectangles, find each area, and add them.

Worked example

Find the area of this L shape by splitting it with a flat line.

Draw a dashed line to split the L into two rectangles.
The bottom rectangle is 8 m by 3 m: 8 × 3 = 24 m².
The top rectangle is 4 m by 3 m: 4 × 3 = 12 m².
Add the two areas: 24 + 12 = 36 m².

Answer: 36 m²

Split with a flat line: 24 + 12 = 36 m².

You can split the L another way, with an upright line. The two parts change, but the total does not.

Split with an upright line: 4 × 6 = 24 and 4 × 3 = 12. The total is 36 m² again.

There is a third way. Fill in the missing corner to make one big rectangle. Find its area, then take the corner away.

Worked example

Find the area of the same L shape by filling in the corner.

Draw the rectangle that would hold the L. It is 8 m by 6 m.
Its area is 8 × 6 = 48 m².
The corner that is missing is 4 m by 3 m: 4 × 3 = 12 m².
Take the corner away: 48 − 12 = 36 m².

Answer: 36 m²

The whole rectangle is 8 × 6 = 48 m². The dotted corner is 4 m by 3 m, so 4 × 3 = 12 m². The L is 48 − 12 = 36 m².

Now think small. A square metre is a square that is 1 m long and 1 m wide. But 1 m is also 100 cm. So a square metre is a square 100 cm long and 100 cm wide.

The square metre. Each small square is 10 cm by 10 cm.

One small square from the picture above, made bigger. It holds 10 × 10 = 100 squares of 1 cm².

Worked example

Why is 1 m² the same as 10 000 cm²?

1 m is 100 cm.
A square metre is 100 cm long and 100 cm wide.
Cut it into squares of 10 cm by 10 cm. You get 10 across and 10 down, which is 100 squares.
Each small square is 10 × 10 = 100 cm².
100 squares of 100 cm² each make 100 × 100 = 10 000 cm².

Answer: 10 000 cm²

So 1 m² is not 100 cm². It is 100 times 100, which is 10 000 cm². Area counts squares. A square has a length and a width, so the 100 is used twice.

See it three ways

With things

On squared paper, make 3 rows with 5 squares in each row. Count all the squares.

As a picture

In symbols

Area = length × width = 5 × 3 = 15 cm².

Try it

Count the squares, then multiply

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

Draw a rectangle that is 7 squares long and 5 squares wide.
Count the squares one by one. Write the number.
Now multiply 7 × 5. Compare the two answers.

What you will see: Counting and multiplying give the same area, 35 cm². Multiplying is faster, and it works for any size.

If you cannot try it, this is what happens.

A wrong idea many people have

A shape with a longer perimeter must have a bigger area.

Why it feels right: A longer edge seems to hold more.

The test that breaks it: Draw a strip 8 cm by 1 cm and a square 3 cm by 3 cm. The strip has the longer perimeter, 18 cm against 12 cm. The square has the bigger area, 9 cm² against 8 cm².

What is true: Perimeter is the distance round. Area is the space inside. They are two different measures, and one does not tell you the other.

Key words

area: the space inside a shape, counted in squares.
square centimetre: a square 1 cm long and 1 cm wide. Written cm².
square metre: a square 1 m long and 1 m wide. Written m².

Back to your guess

The answer is C: Multiply the length by the width of the floor.

Tiles cover the space inside the floor. A floor 7 tiles long and 3 tiles wide has 3 rows of 7 tiles. That is 7 × 3 = 21 tiles. So you multiply the length by the width.

Keep it

Copy this into your exercise book.

Area is the space inside, counted in squares. Rectangle: length × width. L shape: split it, or fill and take away. 1 m² = 100 cm × 100 cm = 10 000 cm².

Draw: A rectangle on squared paper, 6 by 4, with its squares drawn. Write 6 × 4 = 24 cm² beside it.

M3.6.4 Chapter 4

Area of triangles and parallelograms

A leaning shape has the same base and height as a rectangle. Does it hold less space?

Your guess

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

Your guess

A No. It covers the same space, because the base and the height are the same.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Yes. It is leaning, so it covers less.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Yes. The slanted sides make it smaller.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the area of a parallelogram as base × height
Find the area of a triangle as half of base × height

A parallelogram is a four sided shape with two pairs of parallel sides. It looks like a rectangle that leans. Does it hold less space than a rectangle with the same base and height?

One side of the shape is its base. The height is the distance straight up from the base to the top side. It is measured at a right angle to the base. A dashed line shows it.

Draw it first

Turn a parallelogram into a rectangle

1 Draw a parallelogram. Its base is 6 cm.
2 Draw the height from the top corner straight down to the base. It is 4 cm. The slanted side is 5 cm, which is not the height.
3 Cut off the triangle at one end and slide it to the other end. You make a rectangle 6 cm by 4 cm.

The rectangle has the same area as the parallelogram, because nothing was added or lost. The rectangle is 6 cm long and 4 cm wide. So its area is 6 × 4 = 24 cm².

Pushing a frame over is different. Its sides keep their length, so the height falls and the space inside shrinks.

The area of a parallelogram is base × height. Use the height, not the slanted side. The slanted side is longer than the height, so it gives an answer that is too big.

Worked example

A parallelogram has a base of 8 cm and a height of 5 cm. What is its area?

The base is 8 cm. The height is the dashed line, 5 cm.
The area is base × height.
8 × 5 = 40 cm².

Answer: 40 cm²

The height is the dashed line, at a right angle to the base.

Now a triangle. Make a copy of any triangle and turn the copy half way round. Fit the two together along a side. They make a parallelogram.

Two copies of the triangle make a parallelogram with base 8 cm and height 5 cm.

The parallelogram is 8 × 5 = 40 cm². The triangle is only half of it. So the area of a triangle is half of base × height.

Worked example

A triangle has a base of 8 cm and a height of 5 cm. What is its area?

Two copies make a parallelogram with base 8 cm and height 5 cm.
That parallelogram has an area of 8 × 5 = 40 cm².
The triangle is half of it: 40 ÷ 2 = 20 cm².

Answer: 20 cm²

A right angled triangle has two sides at the right angle. These can be the base and the height. The third side is slanted, so you do not use it.

Worked example

A right angled triangle has sides of 6 cm and 8 cm at the right angle. The slanted side is 10 cm. What is its area?

The two sides at the right angle are the base and the height: 8 cm and 6 cm.
The slanted side, 10 cm, is not used.
Base × height = 8 × 6 = 48 cm².
The triangle is half of that: 48 ÷ 2 = 24 cm².
Check: a rectangle 8 cm by 6 cm has an area of 48 cm². The triangle is half of it, 24 cm².

Answer: 24 cm²

The base and the height are the two sides at the right angle.

See it three ways

With things

Cut a parallelogram from paper. Cut a triangle off one end and slide it to the other end.

As a picture

In symbols

Parallelogram: area = base × height. Triangle: area = half of base × height.

Try it

Two triangles make a parallelogram

You need: a sheet of paper, scissors, a pencil.

Draw a triangle of any shape on the paper. Cut it out.
Trace round it on a second sheet. Cut out the copy.
Turn the copy half way round, flat on the table. Do not flip it over. Fit one of its sides to the same side of the first triangle.
Look at the new shape.

What you will see: The two triangles fit together without a gap and make a parallelogram. Each triangle is half of it. This works for every triangle you try.

If you cannot try it, this is what happens.

A wrong idea many people have

To find the area of a parallelogram, multiply the base by the slanted side.

Why it feels right: The slanted side is a real side of the shape. It often has a length written beside it.

The test that breaks it: Cut the triangle off and slide it across. The rectangle is 6 cm by 4 cm, so the area is 24 cm². Using the slanted side gives 6 × 5 = 30, which is more squares than the shape holds.

What is true: Use the height, the dashed line that meets the base at a right angle.

Key words

parallelogram: a four sided shape with two pairs of parallel sides.
base: the side of a shape that you measure the height from.
height: the distance from the base straight up to the top, at a right angle to the base.

Back to your guess

The answer is A: No. It covers the same space, because the base and the height are the same.

Cut a triangle off one end of the leaning shape and slide it to the other end. You get the rectangle. Nothing was added or lost, so the area is the same.

Keep it

Copy this into your exercise book.

Height is at a right angle to the base, never the slanted side. Parallelogram: base × height. Triangle: half of base × height.

Draw: A parallelogram with a dashed height and a small square at the right angle. Beside it, a triangle with its height.

M3.6.5 Chapter 5

Trapeziums and joined shapes

How do I find the area of a flower bed with two parallel sides of different lengths?

Your guess

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

Your guess

A Multiply the longer parallel side by the height.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Add the two parallel sides, multiply by the height, then halve.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Add all four sides.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the area of a trapezium
Find the area of a shape made of rectangles and triangles

A trapezium is a four sided shape with one pair of parallel sides. Parallel sides run side by side and never meet. A flower bed with a long side and a short side can be a trapezium.

You know the area of a parallelogram. Use it. Make a second copy of the trapezium, turn it half way round, and fit it beside the first.

Draw it first

Two trapeziums make a parallelogram

1 Draw the trapezium. Its parallel sides are 8 cm and 4 cm long.
2 Add a copy, turned half way round. The height is 5 cm.
3 The two make a parallelogram. Its base is 8 + 4 = 12 cm.

The parallelogram has a base of 8 + 4 = 12 cm and a height of 5 cm. Its area is 12 × 5 = 60 cm². The trapezium is one of the two copies, so it is half of that.

To find the area of a trapezium, add the two parallel sides. Multiply by the height. Then halve the answer.

Worked example

A trapezium has parallel sides of 8 cm and 4 cm. Its height is 5 cm. What is its area?

Add the parallel sides: 8 + 4 = 12 cm.
Multiply by the height: 12 × 5 = 60 cm².
Halve it: 60 ÷ 2 = 30 cm².

Answer: 30 cm²

The height is the dashed line, at a right angle to both parallel sides.

There is another way to say the same thing. Half way between the two parallel sides is (8 + 4) ÷ 2 = 6 cm. A rectangle 6 cm by 5 cm has the same area: 30 cm².

Worked example

A trapezium has parallel sides of 10 cm and 6 cm. Its height is 4 cm. What is its area?

Add the parallel sides: 10 + 6 = 16 cm.
Multiply by the height: 16 × 4 = 64 cm².
Halve it: 64 ÷ 2 = 32 cm².

Answer: 32 cm²

The parallel sides are 10 cm and 6 cm. The height is 4 cm.

Now think of a shape made of plain parts, such as the end wall of a house. It is a rectangle with a triangle on top. A shape like this is a joined shape. Split it into parts you know, find each area, then add.

Worked example

The end of a house is a rectangle 6 m wide and 4 m high. A triangle roof, 3 m high, sits on top. What is the area of the end?

Split the shape with a flat dashed line into a rectangle and a triangle.
The rectangle is 6 m by 4 m: 6 × 4 = 24 m².
The triangle has a base of 6 m and a height of 3 m. It is half of 6 × 3, which is 9 m².
Add the two areas: 24 + 9 = 33 m².

Answer: 33 m²

The rectangle is 24 m². The roof is 9 m². Together they make 33 m².

Sometimes it is quicker to take a part away. Find the whole rectangle. Then take away the corner that is missing.

Worked example

A rectangle is 8 m by 5 m. A triangle is cut off one corner. The top is now 5 m long and the left side is 3 m long. What is the area left?

Imagine the whole rectangle. It is 8 m by 5 m, so its area is 8 × 5 = 40 m².
The cut corner is a triangle. Across, it is 8 − 5 = 3 m.
Down, it is 5 − 3 = 2 m.
The triangle is half of 3 × 2, which is 3 m².
Take it away: 40 − 3 = 37 m².

Answer: 37 m²

The whole rectangle is 40 m². The dotted corner is 3 m across and 2 m down, so it is 3 m². The shape is 40 − 3 = 37 m².

See it three ways

With things

Cut a trapezium from paper. Cut a second one the same. Turn the second half way round, flat on the table. Do not flip it over. Fit it to the first.

As a picture

In symbols

Area of the trapezium = half of (8 + 4) × 5 = half of 60 = 30 cm².

Try it

Make a parallelogram from two trapeziums

You need: two sheets of paper, scissors, a ruler, a pencil.

Draw a trapezium on the paper. Make the top side shorter than the bottom side. Cut it out.
Trace round it and cut out a copy.
Turn the copy half way round, flat on the table. Do not flip it over. Fit its sloping side against the same side of the first.
Measure the base of the new shape. Compare it with the two parallel sides added.

What you will see: The base of the new shape is the two parallel sides added together. For a trapezium with sides 8 cm and 4 cm, the base is 12 cm. Each trapezium is half of the parallelogram.

If you cannot try it, this is what happens.

A wrong idea many people have

The area of a trapezium is its longer parallel side × its height.

Why it feels right: The longer side looks like the base, and base × height worked for a parallelogram.

The test that breaks it: A rectangle 8 cm by 5 cm has an area of 40 cm². The trapezium fits inside it with room left over, because its top side is only 4 cm. So its area must be less than 40 cm².

What is true: Add the two parallel sides, multiply by the height, and halve. For this trapezium that is 30 cm².

Key words

trapezium: a four sided shape with one pair of parallel sides.
parallel sides: sides that run side by side and never meet.
joined shape: a shape made by putting rectangles and triangles together.

Back to your guess

The answer is B: Add the two parallel sides, multiply by the height, then halve.

Two copies of the bed fit together to make a parallelogram. Its base is the two parallel sides added. Its area is that base × the height. One bed is half of it.

Keep it

Copy this into your exercise book.

Trapezium: add the two parallel sides, multiply by the height, then halve. Joined shape: split it into rectangles and triangles and add. Or take a missing corner away.

Draw: A trapezium with its parallel sides labelled and a dashed height. Beside it, a house end split by a flat dashed line.

M3.6.6 Chapter 6

Circles

How far is it all the way round a round table?

Your guess

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

Your guess

A About twice the distance across.
You picked A. Read on. At the end, the chapter comes back to your guess.
B About four times the distance across.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A little more than three times the distance across.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Name the radius, diameter and circumference of a circle
Measure round tins to see that circumference ÷ diameter is about 3.14
Find the circumference and area of a circle

A wheel, a coin and a plate are all circles. A circle has three parts that you will use again and again.

The radius is a straight line from the centre to the edge. The diameter is a straight line across, through the centre. The circumference is the distance all the way round the edge.

Draw it first

The parts of a circle

1 Draw a circle. Mark its centre with a dot.
2 The radius goes from the centre to the edge. Here it is 3 cm.
3 The diameter goes across, through the centre. It is two radii, 6 cm.
4 The circumference is the whole edge, the distance all the way round.

The diameter is always twice the radius. So a radius of 3 cm means a diameter of 6 cm. If you know the diameter, halve it to find the radius.

Now a big question. How does the circumference compare with the diameter? Imagine the edge of a circle cut open and laid out straight.

The circumference is three diameters and a little more.

This is true for every circle, big or small. The circumference divided by the diameter is always the same number. That number is called pi, written π.

Pi is about 3.14. We use 3.14 because it is a rounded value of π. The digits of π go on for ever, so no number written in digits is exactly π.

So the circumference is π × the diameter. With 3.14 for π, circumference ≈ 3.14 × diameter.

Worked example

A circle has a diameter of 10 cm. What is its circumference? Use 3.14 for π.

The circumference is π × the diameter.
The diameter is 10 cm.
3.14 × 10 = 31.4 cm.

Answer: 31.4 cm

The diameter is 10 cm. Using 3.14, the circumference is about 31.4 cm.

Worked example

A circle has a radius of 7 m. What is its circumference? Use 3.14 for π.

The radius is 7 m. The diameter is double that: 2 × 7 = 14 m.
The circumference is π × the diameter.
3.14 × 14 = 43.96 m.

Answer: 43.96 m

The radius is 7 m. The diameter is 14 m. The circumference is about 43.96 m, from 3.14 × 14.

Now the area, the space inside a circle. Put the circle inside a square. The square has sides of two radii, so it is made of four small squares. Each small square is r by r.

The circle fills a little less than the four r × r squares.

The circle covers a little more than three of the four small squares. In fact it covers π of them, which is about 3.14. So the area of a circle is π × r × r. We write r × r as r².

Work out r × r first. Then multiply by 3.14. Use the radius, not the diameter.

Worked example

A circle has a radius of 5 cm. What is its area? Use 3.14 for π.

The area is π × r × r.
The radius is 5 cm, so r × r = 5 × 5 = 25 cm².
3.14 × 25 = 78.5 cm².
Check: the circle is smaller than the square round it. That square is 10 cm by 10 cm, which is 100 cm². And 78.5 is less than 100.

Answer: 78.5 cm²

Each small square is 5 × 5 = 25 cm². The circle is 3.14 of them: 78.5 cm².

See it three ways

With things

Wrap a string once round a tin. Straighten the string and measure it. Measure across the tin. Divide the first length by the second.

As a picture

In symbols

Circumference ÷ diameter ≈ 3.14. So circumference ≈ 3.14 × diameter, and area ≈ 3.14 × r × r.

Try it

Measure round tins

You need: three round tins or cups of different sizes, a piece of string, a ruler, a pencil.

Wrap the string once round the middle of a tin. Pinch the string where it meets itself.
Straighten the string and measure it with the ruler. This is the circumference.
Measure across the top of the tin, through the middle. This is the diameter.
Divide the circumference by the diameter. Do this for each tin and compare.

What you will see: Every answer is close to 3.1 or 3.2. It is not exactly 3.14, because string stretches and a ruler is read by eye. The answers are close to each other because the number is the same for every circle.

If you cannot try it, this is what happens.

A wrong idea many people have

To square the radius, you double it.

Why it feels right: For a radius of 2, both r × r and 2 × r give 4, so they look the same.

The test that breaks it: Try a radius of 5 cm. Then r × r is 5 × 5 = 25, but 2 × r is 10. The circle holds about 78.5 cm², not 3.14 × 10 = 31.4 cm².

What is true: r² means r × r. Multiply the radius by itself, then multiply by π.

Key words

radius: the distance from the centre of a circle to its edge.
diameter: the distance across a circle, through its centre. It is twice the radius.
circumference: the distance all the way round a circle.
pi: the number you get from circumference ÷ diameter. It is about 3.14.

Back to your guess

The answer is C: A little more than three times the distance across.

Go round any circle with a string, then lay the string along a line across the circle. The string is a little more than three times as long. The number is a little more than 3.14, and it is the same for every circle.

Keep it

Copy this into your exercise book.

Diameter = 2 × radius. Circumference = π × diameter, about 3.14 × diameter. Area = π × r × r. π is about 3.14, a rounded value.

Draw: A circle with its centre, a radius and a diameter drawn and named. Mark the edge thick and write circumference.

M3.6.7 Chapter 7

Volume of cuboids

How much water fits in this tank?

Your guess

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

Your guess

A Multiply the length, the width and the height.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Add the length, the width and the height.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Multiply the length by the width only.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Count cubes to find the volume of a box
Find the volume of a cuboid as length × width × height
Change between cm³, ml and litres

A tank, a box and a brick are all solid shapes. A solid shape with six rectangle faces is a cuboid. How much space is inside one? That space is its volume.

We count volume in cubes. A cubic centimetre is a cube with edges of 1 cm. We write it 1 cm³. A box with a volume of 24 cm³ holds 24 of these small cubes.

One cubic centimetre, 1 cm³.

Draw it first

Fill a box layer by layer

1 Cover the bottom of the box with cubes. One layer holds 4 × 3 = 12 cubes.
2 Add a second layer. Now there are 2 × 12 = 24 cubes.
3 Add a third layer. The box is full, with 3 × 12 = 36 cubes. Its volume is 36 cm³.

Look at the bottom layer. Its cubes are length × width. That is the same as the area of the bottom. Then the height says how many layers there are. So the volume of a cuboid is length × width × height.

Worked example

A box is 5 cm long, 3 cm wide and 2 cm high. What is its volume?

One layer has 5 × 3 = 15 cubes.
The box is 2 cm high, so there are 2 layers.
The volume is 15 × 2 = 30 cm³.

Answer: 30 cm³

Two layers of 15 cubes make 30 cm³.

The unit is cm³ because you multiply three lengths in cm. A cube has the same length, width and height. A cube with edges of 5 cm has a volume of 5 × 5 × 5 = 125 cm³.

Liquids are measured in ml and litres. The two measures fit together. A cube with edges of 1 cm holds 1 ml of water. So 1 cm³ is 1 ml.

A bigger cube has edges of 10 cm. Its volume is 10 × 10 × 10 = 1 000 cm³, which is 1 000 ml. We call 1 000 ml one litre.

Worked example

A box is 10 cm long, 10 cm wide and 10 cm high. How much water does it hold?

The volume is length × width × height.
10 × 10 × 10 = 1 000 cm³.
1 cm³ holds 1 ml, so the box holds 1 000 ml.
And 1 000 ml is 1 litre.

Answer: 1 litre

This cube is 10 cm each way. It holds 1 000 cm³, which is 1 litre. Not to scale.

Worked example

A tank is 40 cm long, 25 cm wide and 20 cm high. How many litres of water fill it?

The base is 40 cm by 25 cm. Its area is 40 × 25 = 1 000 cm².
The tank is 20 cm high, so the volume is 1 000 × 20 = 20 000 cm³.
1 litre is 1 000 cm³, so divide by 1 000.
20 000 ÷ 1 000 = 20 litres.

Answer: 20 litres

The tank holds 20 000 cm³, which is 20 litres. Not to scale.

See it three ways

With things

Stack small cubes, such as dice, into a box shape. Make it 4 across, 3 deep and 2 high. Count them all.

As a picture

In symbols

Volume = length × width × height = 4 × 3 × 2 = 24 cm³.

Try it

Build a box from cubes

You need: 12 small cubes of the same size, such as dice or sugar lumps, a flat table.

Build a box that is 3 cubes long, 2 cubes wide and 1 cube high. Count the cubes.
Put a second layer on top. Count again.
Work out 3 × 2 × 2 and compare it with your count.

What you will see: The count and the product are the same, 12. If each cube has edges of 1 cm, the volume of the box is 12 cm³.

If you cannot try it, this is what happens.

A wrong idea many people have

To find the volume of a box, add its length, width and height.

Why it feels right: You add lengths to find a perimeter, so adding seems natural.

The test that breaks it: Take a box 4 cm by 3 cm by 2 cm. Adding gives 4 + 3 + 2 = 9. But the bottom layer alone has 4 × 3 = 12 cubes, so the box holds more than 9.

What is true: Volume counts cubes in layers. Multiply the length, the width and the height.

Key words

cuboid: a solid shape with six rectangle faces, such as a brick.
volume: the space inside a solid shape, counted in cubes.
cubic centimetre: a cube with edges of 1 cm. Written cm³.
litre: a unit for liquids. 1 litre is 1 000 ml, which is 1 000 cm³.

Back to your guess

The answer is A: Multiply the length, the width and the height.

A tank fills in layers. One layer has length × width cubes of water. The height tells you how many layers there are. So you multiply all three numbers.

Keep it

Copy this into your exercise book.

Volume = length × width × height, in cm³. 1 cm³ = 1 ml. 1 litre = 1 000 cm³ = 1 000 ml. A cube of 10 cm holds 1 litre.

Draw: A box of 1 cm cubes, 4 by 3 by 2, with its edges labelled. Write 4 × 3 × 2 = 24 cm³.

M3.6.8 Chapter 8

Surface area

How much paper do I need to wrap this box?

Your guess

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

Your guess

A The volume of the box, length × width × height.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The area of all six faces of the box added together.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The area of one face of the box.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the surface area of a cuboid from its net
Find how much paper covers a box with no gaps

You want to wrap a box in paper with no gaps and no overlaps. The paper must cover the whole outside. The total area of all the faces of a solid is its surface area.

A cuboid has six faces, and every face is a rectangle. They come in three pairs of equal faces. The top and bottom are a pair. The front and back are a pair. The left and right are a pair.

Cut a box along some edges and lay it flat. The flat shape is a net. Its six rectangles are the six faces. Find the area of each rectangle, then add them.

Draw it first

Add the six faces of a net

1 Draw the net of the box. It has six rectangles. Matching faces are shaded the same way.
2 Write the lengths on it: 5 cm, 3 cm and 2 cm.
3 Find the area of each face and add them. The total is 62 cm².

Faces come in pairs, so you only need three areas. Work out one face from each pair. Add the three areas. Then double the total.

Worked example

A box is 5 cm long, 3 cm wide and 2 cm high. What is its surface area?

The top and bottom are each 5 cm by 3 cm: 5 × 3 = 15 cm².
The front and back are each 5 cm by 2 cm: 5 × 2 = 10 cm².
The left and right are each 3 cm by 2 cm: 3 × 2 = 6 cm².
Add one of each pair: 15 + 10 + 6 = 31 cm². Then double it: 2 × 31 = 62 cm².

Answer: 62 cm²

Each rectangle appears twice on the box.

A cube is quicker, because all six faces are the same square.

Worked example

A cube has edges of 4 cm. What is its surface area?

Every face is a square, 4 cm by 4 cm.
One face has an area of 4 × 4 = 16 cm².
A cube has 6 faces: 6 × 16 = 96 cm².

Answer: 96 cm²

Six squares of 16 cm² each make 96 cm².

Surface area answers real questions. Paper that covers a box with no gaps is exactly its surface area. A box with no lid has one face fewer, so you take that face away.

Worked example

A gift box is 10 cm long, 6 cm wide and 5 cm high. How much paper covers it with no gaps?

The top and bottom are 10 × 6 = 60 cm² each.
The front and back are 10 × 5 = 50 cm² each.
The left and right are 6 × 5 = 30 cm² each.
One of each pair adds to 60 + 50 + 30 = 140 cm². Double it: 2 × 140 = 280 cm².

Answer: 280 cm²

The paper for the gift box is 280 cm². Not to scale.

See it three ways

With things

Wrap a small box in paper. Cut the paper along the edges so that it lies flat. You get six rectangles.

As a picture

In symbols

Surface area = 2 × (5 × 3) + 2 × (5 × 2) + 2 × (3 × 2) = 30 + 20 + 12 = 62 cm².

Try it

Cover a box with paper

You need: a small box, such as a matchbox or a tea box, a ruler, paper, scissors, a pencil.

Measure the length, the width and the height of the box in cm.
Work out the area of one face from each pair. Add them and double the total.
Cover the box with paper, with no gaps and no overlaps. Cut the paper into the six faces.
Lay out the six pieces. Check that matching faces are the same size.

What you will see: The six pieces make three matching pairs. Their areas add up to the total you worked out. A box 5 cm by 3 cm by 2 cm needs 62 cm² of paper.

If you cannot try it, this is what happens.

A wrong idea many people have

Surface area and volume are the same thing.

Why it feels right: Both are about a box, and both come from its length, width and height.

The test that breaks it: A box 4 cm by 3 cm by 2 cm holds 24 cm³. Its surface area is 52 cm². The numbers are different, and so are the jobs: one counts cubes inside, the other counts squares outside.

What is true: Volume is the space inside, in cm³. Surface area is the area of the outside, in cm².

Key words

surface area: the total area of all the faces of a solid shape.
faces: the flat sides of a solid shape.
net: a solid shape cut along some edges and laid flat.

Back to your guess

The answer is B: The area of all six faces of the box added together.

Wrapping paper must cover every face of the box. So you add the areas of all six faces. The volume counts the space inside, which is a different job.

Keep it

Copy this into your exercise book.

A cuboid has 6 faces in 3 pairs. Surface area = 2 × (l × w + l × h + w × h). It is the paper that covers the box with no gaps, in cm². Volume is a different measure, in cm³.

Draw: The net of a box with its six rectangles numbered, and the three pairs shaded alike.

Climb down the ladder

Measure in two units

Add every side

The string round a book

Count the squares in rows

Count the squares, then multiply

Turn a parallelogram into a rectangle

Two triangles make a parallelogram

Two trapeziums make a parallelogram

Make a parallelogram from two trapeziums

The parts of a circle

Measure round tins

Fill a box layer by layer

Build a box from cubes

Add the six faces of a net

Cover a box with paper

Key words, A to Z