Chapter 5 of 8

All 8 chapters
  1. 1Changing metric units, readunderstood
  2. 2Perimeter, readunderstood
  3. 3Area of rectangles, readunderstood
  4. 4Area of triangles and parallelograms, readunderstood
  5. 5Trapeziums and joined shapes, readunderstood
  6. 6Circles, readunderstood
  7. 7Volume of cuboids, readunderstood
  8. 8Surface area, readunderstood

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

  • 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 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. A trapezium with parallel sides of 8 cm below and 4 cm above. Then a dotted copy of it, turned half way round, sits beside it. Together they make a parallelogram. A dashed upright line, 5 cm long with a small square for the right angle, shows the height. Below it says base: 8 + 4 = 12 cm.8 cm4 cm

    1 Draw the trapezium. Its parallel sides are 8 cm and 4 cm long.

  2. A trapezium with parallel sides of 8 cm below and 4 cm above. Then a dotted copy of it, turned half way round, sits beside it. Together they make a parallelogram. A dashed upright line, 5 cm long with a small square for the right angle, shows the height. Below it says base: 8 + 4 = 12 cm.5 cm8 cm4 cm

    2 Add a copy, turned half way round. The height is 5 cm.

  3. A trapezium with parallel sides of 8 cm below and 4 cm above. Then a dotted copy of it, turned half way round, sits beside it. Together they make a parallelogram. A dashed upright line, 5 cm long with a small square for the right angle, shows the height. Below it says base: 8 + 4 = 12 cm.5 cm8 cm4 cmbase: 8 + 4 = 12 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?

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

Answer: 30 cm²

A trapezium. The parallel sides are labelled 8 cm below and 4 cm above. A dashed upright line, 5 cm long with a small square for the right angle, shows the height.5 cm8 cm4 cm
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?

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

Answer: 32 cm²

A trapezium. The parallel sides are labelled 10 cm below and 6 cm above. A dashed upright line, 4 cm long with a small square for the right angle, shows the height.4 cm10 cm6 cm
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?

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

Answer: 33 m²

The end of a house. A dashed flat line splits it into a rectangle below, 24 m², and a triangle roof above, 9 m². The rectangle is 6 m wide and 4 m high. A dashed upright line, 3 m long with a small square, shows the height of the roof.3 m6 m4 m4 m24 m²9 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?

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

Answer: 37 m²

A shape made from a rectangle with the top left corner cut off. The bottom is 8 m, the right side is 5 m, the top is 5 m and the left side is 3 m. The cut away triangle is dotted.8 m5 m5 m3 m
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

A parallelogram made of two trapeziums. The base is labelled 12 cm. A dashed upright line, 5 cm long with a small square for the right angle, shows the height.5 cm12 cm

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.

  1. Draw a trapezium on the paper. Make the top side shorter than the bottom side. Cut it out.
  2. Trace round it and cut out a copy.
  3. 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.
  4. 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 trapezium with parallel sides of 8 cm below and 4 cm above. A dotted copy, turned half way round, sits beside it. Together they make a parallelogram. A dashed upright line shows the height, 5 cm.5 cm8 cm4 cm

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.

Check yourself

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

  1. What is the area of this trapezium?

    A trapezium. The parallel sides are labelled 9 cm below and 5 cm above. A dashed upright line, 6 cm long with a small square for the right angle, shows the height.6 cm9 cm5 cm
    Tap your answer.

    1. Yes, that is right. Add the parallel sides: 9 + 5 = 14 cm. Multiply by the height: 14 × 6 = 84 cm². Halve it: 84 ÷ 2 = 42 cm².

    2. Not this one. You multiplied the longer parallel side by the height, 9 × 6. The top side is shorter, so that gives too much. Add both parallel sides and halve. Go to M3.6.5. Try another.

    3. Not this one. You added the parallel sides and multiplied by the height, 14 × 6. That is the area of the parallelogram made of two trapeziums. Halve it. Go to M3.6.5. Try another.

    See the answer

    42 cm²

    Add the parallel sides: 9 + 5 = 14 cm. Multiply by the height: 14 × 6 = 84 cm². Halve it: 84 ÷ 2 = 42 cm².

    • If you chose B: You multiplied the longer parallel side by the height, 9 × 6. The top side is shorter, so that gives too much. Add both parallel sides and halve. Go to M3.6.5.
    • If you chose C: You added the parallel sides and multiplied by the height, 14 × 6. That is the area of the parallelogram made of two trapeziums. Halve it. Go to M3.6.5.

  2. The end of a shed is a rectangle 8 m wide and 3 m high. A triangle roof, 3 m high, sits on top. What is the area of the end?

    The end of a shed. A dashed flat line splits it into a rectangle below and a triangle roof above. The rectangle is 8 m wide and 3 m high. A dashed upright line, 3 m long with a small square, shows the height of the roof.3 m8 m3 m3 m
    Tap your answer.

    1. Not this one. You counted the roof as 8 × 3 = 24 m². A triangle is only half of base × height, so the roof is 12 m². Go to M3.6.4. Try another.

    2. Not this one. That is only the rectangle, 8 × 3. The roof triangle has area too. Add it. Go to M3.6.5. Try another.

    3. Yes, that is right. The rectangle is 8 × 3 = 24 m². The roof is half of 8 × 3, which is 12 m². Then 24 + 12 = 36 m².

    See the answer

    36 m²

    The rectangle is 8 × 3 = 24 m². The roof is half of 8 × 3, which is 12 m². Then 24 + 12 = 36 m².

    • If you chose A: You counted the roof as 8 × 3 = 24 m². A triangle is only half of base × height, so the roof is 12 m². Go to M3.6.4.
    • If you chose B: That is only the rectangle, 8 × 3. The roof triangle has area too. Add it. Go to M3.6.5.

  3. A field is a rectangle 10 m by 6 m with a corner cut off. Find its area.

    A shape made from a rectangle with the top left corner cut off. The bottom is 10 m, the right side is 6 m, the top is 6 m and the left side is 3 m. The cut away triangle is dotted.10 m6 m6 m3 m
    Tap your answer.

    1. Not this one. That is the whole rectangle, 10 × 6. The corner has been cut off, so take its area away. Go to M3.6.5. Try another.

    2. Yes, that is right. The whole rectangle is 10 × 6 = 60 m². The cut corner is 10 − 6 = 4 m across and 6 − 3 = 3 m down. Its area is half of 4 × 3, which is 6 m². Then 60 − 6 = 54 m².

    3. Not this one. You added the corner. The corner is cut off, so it is missing from the shape. Take it away. Go to M3.6.5. Try another.

    See the answer

    54 m²

    The whole rectangle is 10 × 6 = 60 m². The cut corner is 10 − 6 = 4 m across and 6 − 3 = 3 m down. Its area is half of 4 × 3, which is 6 m². Then 60 − 6 = 54 m².

    • If you chose A: That is the whole rectangle, 10 × 6. The corner has been cut off, so take its area away. Go to M3.6.5.
    • If you chose C: You added the corner. The corner is cut off, so it is missing from the shape. Take it away. Go to M3.6.5.

  4. A trapezium has parallel sides of 7 cm and 5 cm and an area of 36 cm². How long is its height?

    A trapezium. The parallel sides are labelled 7 cm below and 5 cm above. A dashed upright line shows the height, and it is marked ? cm.? cm7 cm5 cm
    Tap your answer.

    1. Not this one. You divided 36 by 12. But 36 is only half of (7 + 5) × height. Double it first: 72, then divide by 12. Go to M3.6.5. Try another.

    2. Not this one. You halved 36. The trapezium is half of the parallelogram, so double the area first, then divide by the base, 12. Go to M3.6.5. Try another.

    3. Yes, that is right. Two copies make a parallelogram with an area of 2 × 36 = 72 cm². Its base is 7 + 5 = 12 cm. So the height is 72 ÷ 12 = 6 cm.

    See the answer

    6 cm

    Two copies make a parallelogram with an area of 2 × 36 = 72 cm². Its base is 7 + 5 = 12 cm. So the height is 72 ÷ 12 = 6 cm.

    • If you chose A: You divided 36 by 12. But 36 is only half of (7 + 5) × height. Double it first: 72, then divide by 12. Go to M3.6.5.
    • If you chose B: You halved 36. The trapezium is half of the parallelogram, so double the area first, then divide by the base, 12. Go to M3.6.5.

  5. Why do we halve at the end when we find the area of a trapezium?

    Tap your answer.

    1. Yes, that is right. The parallelogram has an area of (the two parallel sides added) × height. It holds two equal trapeziums. So one trapezium is half.

    2. Not this one. The number of parallel sides tells you the name of the shape. It is not the reason for halving. Go to M3.6.5. Try another.

    3. Not this one. Take the trapezium with sides 8 cm and 4 cm and height 5 cm. The rectangle round it is 8 × 5 = 40 cm². Half of that is 20 cm², but the trapezium is 30 cm². We halve because two copies make a parallelogram. Go to M3.6.5. Try another.

    See the answer

    Two copies make a parallelogram, and one trapezium is half of it.

    The parallelogram has an area of (the two parallel sides added) × height. It holds two equal trapeziums. So one trapezium is half.

    • If you chose B: The number of parallel sides tells you the name of the shape. It is not the reason for halving. Go to M3.6.5.
    • If you chose C: Take the trapezium with sides 8 cm and 4 cm and height 5 cm. The rectangle round it is 8 × 5 = 40 cm². Half of that is 20 cm², but the trapezium is 30 cm². We halve because two copies make a parallelogram. Go to M3.6.5.

  6. A trapezium has parallel sides of 7 cm and 11 cm and a height of 4 cm. What is its area?

    See the answer

    36 cm²

    Add the parallel sides: 7 + 11 = 18 cm. Multiply by the height: 18 × 4 = 72 cm². Halve it: 72 ÷ 2 = 36 cm².

  7. A shape is a rectangle 6 m wide and 3 m high, with a triangle on top. The triangle has a base of 6 m and a height of 4 m. What is the area?

    See the answer

    30 m²

    The rectangle is 6 × 3 = 18 m². The triangle is half of 6 × 4, which is 12 m². Then 18 + 12 = 30 m².

  8. From an earlier chapter

    A parallelogram has a base of 12 cm and a height of 5 cm. What is its area?

    See the answer

    60 cm²

    The area of a parallelogram is base × height: 12 × 5 = 60 cm².

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.

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

No internet tomorrow? Save the whole book as one file (825 KB). It opens on any phone.