A trapezium behaves like a rectangle whose width is the average of its parallel sides
Two parallel sides
A trapezium is a four-sided shape with one pair of parallel sides, and the two parallel sides are different lengths. In this one the top side is 6 and the bottom side is 10.
The height of a trapezium is the distance between its two parallel sides, measured at a right angle to them. It is not the length of a sloping side. Here the height is 4.
The parallel sides are 6 on top and 10 on the bottom. The height, 4, is measured straight across from one to the other.
Swing the corners up
A rectangle is easy: multiply its length by its width. So turn the trapezium into a rectangle without adding or losing any of it.
The bottom side is 4 longer than the top side, so it sticks out by 2 at each end. Cut a small triangle off each bottom corner: cut straight up, 1 unit in from the end, as far as the sloping side at half the height. Turn each triangle half a turn and it fits exactly into the gap beside the top side.
Now the bottom has lost 1 at each end, so it is 10 − 2 = 8. The top has gained 1 at each end, so it is 6 + 2 = 8. The two triangles only moved, so the area is the same as before.
The dashed triangles are where the two corners were cut from. Turned half a turn, they fill the gaps on top, and the trapezium becomes a rectangle 8 wide and 4 high.
The rectangle in the middle
The rectangle is 8 wide and 4 high, so its area is 8 × 4 = 32. The trapezium has the same area: 32 square units.
The 8 is halfway between 6 and 10. It is the average of the two parallel sides: (6 + 10) ÷ 2 = 16 ÷ 2 = 8. The long side gave up exactly what the short side gained, so they met in the middle.
4 rows of 8 unit squares: 8 × 4 = 32, the area of the trapezium.
The formula
Call the two parallel sides a and b, and the height h. Add the parallel sides, halve the total to find their average, and multiply by the height:
Area of a trapezium .
For the trapezium above: .
a and b are the two parallel sides, and h is the height between them.
Two copies make a parallelogram
There is a second way to see where the comes from. Take a copy of the trapezium, turn it upside down, and fit it against the first one. The two sloping sides that touch are the same length, so they meet along their whole length.
Together they make a parallelogram. Its base is the long side of one and the short side of the other: 10 + 6 = 16. Its height is still 4, so its area is 16 × 4 = 64. The parallelogram is two copies of the trapezium, so one trapezium is half of it: 64 ÷ 2 = 32.
The gold copy is upside down. Its short side of 6 sits beside the long side of 10, so the parallelogram has base 16 and height 4, and holds two trapeziums.
Three ways to go wrong
Forgetting the half: (6 + 10) × 4 = 64 is the area of the whole parallelogram, two trapeziums. Multiplying the two parallel sides, 6 × 10 = 60, uses no height at all. And a sloping side is never the height: the height is measured at a right angle to the parallel sides.
Composite areas · A Trapezium Cut and Rearranged into a Rectangle
QuestionA trapezium has parallel sides of 10 cm and 16 cm, and its height is 6 cm. Its two slanted sides are equal. A right-angled triangle is cut off one end of the trapezium and fitted onto the other end, making a rectangle. (a) How long is the rectangle? (b) What is the area of the trapezium?
1.The longer parallel side is 16 − 10 = 6 cm longer than the shorter one. The slanted sides are equal, so it sticks out by the same amount at each end: 6 ÷ 2 = 3 cm.
The longer side is 16 − 10 = 6 cm longer, so it sticks out 3 cm at each end. 2.Cut straight down from the right end of the shorter side. This cuts off a right-angled triangle 3 cm wide and 6 cm tall.
Cut straight down from the right end of the top: a right-angled triangle 3 cm wide and 6 cm tall. 3.Turn the triangle over and fit it into the gap beside the slanted side at the left end. It fits exactly, because that gap is also 3 cm wide and 6 cm tall.
Turn the triangle over and fit it into the gap at the left end, which is the same size. 4.(a) The shorter side has grown by 3 cm and the longer side has lost 3 cm, so both are 10 + 3 = 16 − 3 = 13 cm. The rectangle is 13 cm long and 6 cm high.
(a) Both sides are now 13 cm: a rectangle 13 cm long and 6 cm high. 5.(b) No piece was added or thrown away, so the trapezium has the same area as the rectangle: 13 × 6 = 78 cm2. Check: 12 × (10 + 16) × 6 = 78 cm2.
(b) The trapezium has the rectangle's area: 13 × 6 = 78 cm².
Answer: (a) 13 cm; (b) 78 cm2
Watch out for
- Multiplying the longer side by the height, 16 × 6 = 96 cm2. That is the rectangle drawn round the trapezium, and it includes the two corner triangles outside it. The rectangle made by moving the triangle is only 13 cm long, so the area is 78 cm2.
- Adding all 6 cm to the shorter side, because the longer side is 6 cm longer, and making the rectangle 16 cm long. The 6 cm sticks out in two parts, 3 cm at each end, and only one part is moved. So the shorter side grows by 3 cm, to 13 cm.
Your turn
1.A trapezium with parallel sides 7 and 11, height 7. What is its area?