Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7 Unit 7, lesson 3 (triangles and parallelograms): a triangle has a base of 10 cm and a height of 7 cm at a right angle to it. What is its area?
Answer
35 cm². Half of 10 × 7 = 70 is 35. - From Year 7 Unit 7, lesson 4 (compound shapes): a rectangle is 9 m by 4 m. A rectangle 3 m by 2 m is cut out of one corner. What area is left?
Answer
30 m². 9 × 4 = 36, 3 × 2 = 6, and 36 − 6 = 30. - From Year 8 Unit 3, lesson 3 (adding and subtracting integers): what is 3 − (−4)?
Answer
7. Taking away a negative is the same as adding: 3 + 4 = 7.
Words used in this lesson
- trapezium
- a four-sided shape with one pair of parallel sides. For example: A garden bed with one straight side parallel to the opposite side.
- parallel sides
- two sides that are always the same distance apart. For example: The two rails of a railway track.
- height
- the distance between the parallel sides, measured at a right angle. For example: How far apart the two ends of a field are.
- sloping side
- a side that leans and is not parallel to the other one. For example: The edge of the field that goes at a slant.
- average
- the number halfway between two numbers. For example: The average of 2 and 8 is 5.
- right angle
- a square corner, 90°. For example: The corner of a page.
How much land is in a field with two different ends?
A farmer in Kerala has a terraced rice field. One end is 12 m wide and the other end is 20 m wide, and the two ends are 15 m apart. The two long sides slope, so the field is not a rectangle. Predict: should the area be nearer 12 × 15, nearer 20 × 15, or somewhere in between? If in between, where? Then use the sliders and press Add a turned copy.
The trapezium has parallel sides 4 cm and 10 cm and height 5 cm, so its area is ½ × (4 + 10) × 5 = 35 cm². A turned copy would make a parallelogram with base 4 + 10 = 14 cm and area 14 × 5 = 70 cm².
Show me the steps
- The trapezium has parallel sides 4 cm and 10 cm and height 5 cm. Its area is ½ × (4 + 10) × 5 = 35 cm².
- Press Add a turned copy. The copy joins the trapezium to make a parallelogram with base 4 + 10 = 14 cm and height 5 cm. Its area is 14 × 5 = 70 cm², and the trapezium is half of it: 35 cm².
- Make both parallel sides 7 cm and the height 6 cm. The trapezium is a rectangle 7 cm by 6 cm, area 42 cm², and the formula gives ½ × (7 + 7) × 6 = 42 cm². The same formula covers the rectangle.
- Make the parallel sides 3 cm and 11 cm and the height 7 cm. The area is ½ × (3 + 11) × 7 = 49 cm².
The area of a trapezium is ½ × (a + b) × h, where a and b are the two parallel sides and h is the distance between them, measured at a right angle.
- Why: a copy turned half a turn fits against the trapezium and makes a parallelogram with base a + b and height h. The trapezium is half of it.
- Another way to say it: ½(a + b) is the number halfway between a and b, so the area is the middle width times the height.
- The sloping sides are not in the formula. Lean the top edge and they change; the area does not.
Back to the farmer: the middle width is halfway between 12 m and 20 m, and the area is worked out below.
The middle line does the work
Now lean the top edge and show the equal rectangle. The slanted sides change length. What stays the same, and what is the rectangle's width?
The middle line is (4 + 10) ÷ 2 = 7 cm long. Area = 7 × 4 = 28 cm². The sloping sides are 4.1 cm and 6.4 cm and are not used.
Show me the steps
- Top 4 cm, bottom 10 cm, height 4 cm. The middle line is (4 + 10) ÷ 2 = 7 cm and the area is 7 × 4 = 28 cm². The sloping sides are 4.1 cm and 6.4 cm.
- Press Show the equal rectangle. It is 7 cm wide and 4 cm high. The trapezium sticks out of it at the top and the rectangle sticks out of the trapezium at the bottom, by triangles of the same size.
- Set the lean to 4. The shape leans further, one sloping side is now 5.7 cm, and the area is still 7 × 4 = 28 cm².
- Set the lean to −3, so the top edge leans the other way. The area is still 28 cm². The height now meets the line of the base outside the shape.
- Set the top to 8 cm, the bottom to 12 cm and the height to 5 cm. The middle line is 10 cm and the area is 10 × 5 = 50 cm².
A common mistake: “Half of 6, plus 10, times 5.” Multiplication goes before addition, so ½ × 6 + 10 × 5 = 3 + 50 = 53 cm², which is more than the whole rectangle 10 cm by 5 cm would hold. The half belongs to the two parallel sides added together: ½ × (6 + 10) × 5 = ½ × 16 × 5 = 40 cm². Add the parallel sides first, then take half, then multiply by the height.
Worked examples
A trapezium has parallel sides of 7 m and 11 m, and a height of 6 m. Find its area.
- The parallel sides are 7 m and 11 m. The height at a right angle to them is 6 m.
- Add the parallel sides: 7 + 11 = 18 m.
- Area = ½ × 18 × 6 = 9 × 6 = 54 m².
The rice field: parallel ends of 12 m and 20 m, 15 m apart. Find its area.
- The ends are the parallel sides and the distance between them is the height.
- The middle width is (12 + 20) ÷ 2 = 16 m.
- Area = 16 × 15 = 240 m². That is between 12 × 15 = 180 and 20 × 15 = 300.
A trapezium has an area of 60 cm². Its parallel sides are 7 cm and 13 cm. Find its height.
- Write the area as an equation: ½ × (7 + 13) × h = 60.
- That is 10 × h = 60, because ½ × 20 = 10.
- Divide both sides by 10: h = 6 cm.
Your go. The first step is given. A trapezium has parallel sides of 9 cm and 15 cm and a height of 6 cm. Find its area.
- Add the parallel sides: 9 + 15 = 24 cm.
- Now take half of that, multiply by the height, then check.
Check the full answer
½ × 24 = 12 and 12 × 6 = 72 cm². Check: the middle width is (9 + 15) ÷ 2 = 12 cm, and 12 × 6 = 72.
Check before you go on
1. Which calculation gives the area of this trapezium in square centimetres?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A trapezium has parallel sides of 6 cm and 10 cm, and a height of 4 cm at a right angle to them. What is its area in square centimetres?
3. A garden bed is a trapezium with parallel sides of 9 m and 5 m, and a height of 8 m between them. What is its area in square metres?
Level 2 · Building
4. Find the area of this trapezium in square centimetres.
5. A trapezium has an area of 90 cm², and its parallel sides are 11 cm and 19 cm. How high is it, in centimetres?
6. A trapezium has an area of 84 m² and a height of 7 m. One parallel side is 9 m. How long is the other parallel side, in metres?
Level 3 · Stretch
7. A rectangular tile is 10 cm long and 6 cm high. A straight cut runs from its bottom-left corner to a point on the top edge that is 4 cm from the top-right corner. The piece on the right is a trapezium. The other piece is a triangle. How many square centimetres is the triangle?
8. A trapezium has parallel sides of x cm and (x + 6) cm, and a height of 5 cm. Its area is 85 cm². What is x?
Something to think about
Keep the bottom side of a trapezium at 10 cm and the height at 4 cm, and shrink the top side towards 0 cm. What shape do you finish with, and does the trapezium formula still give its area?
One way to think about it
The top side shrinks to a point and the shape becomes a triangle with base 10 cm and height 4 cm. The formula gives ½ × (0 + 10) × 4 = 20 cm², which is exactly ½ × base × height. Make the top side equal to the bottom side instead and you get a rectangle, with ½ × (10 + 10) × 4 = 40 cm² = 10 × 4. One formula covers all three shapes.
Exit ticket
9. Trapezium P has parallel sides of 4 cm and 20 cm. Trapezium Q has parallel sides of 11 cm and 13 cm. Both have a height of 9 cm at a right angle to the parallel sides, and their sloping sides are different. Which is true?