How to find the area of a trapezium

The trapezium formula is short, and students still lose marks on it. Not because of the arithmetic, but because the diagram hands them the wrong number and they use it.

The formula

Area = half of (a + b) times h

Where a and b are the two parallel sides, and h is the perpendicular height between them.

In words: add the two parallel sides, halve the answer, multiply by the height.

Why it works

Here is the picture that makes it obvious.

Take your trapezium. Make an identical copy, turn the copy upside down, and push the two together along a slanted side.

What you get is a parallelogram. Its base is a plus b, because the short side of one is now next to the long side of the other. Its height is still h.

Area of a parallelogram is base times height, so the two trapeziums together cover (a + b) times h. One trapezium is half of that.

That is where the half comes from. It is not an arbitrary rule to memorise; it is there because you doubled the shape.

Another way to see the same thing: half of (a + b) is the average of the two parallel sides. So a trapezium has the same area as a rectangle whose width is the average of its two parallel sides. That is a useful sanity check on any answer.

Mistake 1: adding the wrong sides

Only the parallel sides go into the bracket.

In a diagram the parallel sides are marked with matching arrowheads. If the arrows are missing, the parallel sides are usually the top and the bottom, and the slanted ones are the left and right. Check before you add.

The slanted sides never appear in the area formula at all. They matter for perimeter, not area.

Mistake 2: using a slanted side as the height

This is the one that costs the most marks, and exam questions are built to cause it.

The height must be measured at a right angle to the parallel sides. A good diagram shows it as a dashed line inside the shape with a small square where it meets the base.

Exam diagrams often also label a slanted side, because that is the measurement a ruler would naturally give. It is there to tempt you. A slanted side is always longer than the perpendicular height, so using it makes the answer too big.

Look for the dashed line and the right-angle square. If the question gives you a slanted side and an angle instead, you have to find the height with trigonometry first.

Mistake 3: mixed units

If one parallel side is in centimetres and the other is in millimetres, convert before you add.

Check the units on your answer too. An area is always in squared units: cm squared, m squared. If your answer has plain cm, something went wrong.

A worked example

A trapezium has parallel sides of 8 cm and 14 cm. The perpendicular height is 5 cm. One slanted side is marked 6 cm. Find the area.

1. The parallel sides are 8 and 14. Add them: 22.
2. Halve it: 11. That is the average width.
3. Multiply by the height, 5: 55 square centimetres.

The 6 cm slanted side was not used. It was in the question to see whether you would reach for it.

Sanity check: 11 cm is between 8 and 14, which is what an average should be. And a rectangle 11 cm by 5 cm is 55 square centimetres, which matches.

Working backwards

Questions often give you the area and ask for a missing side. Do not try to rearrange the formula in your head.

If a trapezium has an area of 300 square metres, parallel sides of 20 m and 30 m, and you need the height:

1. Add the parallel sides: 50. Halve it: 25.
2. So 25 times h equals 300.
3. h equals 300 divided by 25, which is 12 metres.

Build the equation from the formula, then solve it. One step at a time beats one clever step.

How to practise it

Draw six trapeziums. On each one, write only two things: the two parallel sides, and which measurement is the height. Do not work out any areas.

Then check them. Reading the diagram correctly is the skill being tested; the arithmetic afterwards is the easy half.

FutureSchool teaches area of a trapezium in its Maths programme as a video lesson, then a worksheet, then a test, with the “which one is the height” step on its own. It sits in the lower secondary years and comes back in GCSE revision.

If you want to know which shape and measure topics your child has secure, the free assessment takes about twenty minutes.

Questions parents ask

What is the formula for the area of a trapezium?

Add the two parallel sides, divide by two, then multiply by the perpendicular height. Written out: area equals half of (a plus b) times h.

Which sides do I add?

Only the two parallel sides, the ones marked with matching arrows. The two slanted sides never go into the formula.

Which measurement is the height?

The perpendicular distance between the two parallel sides, measured at a right angle. It is not the length of a slanted side, even when the diagram puts a number on one.

Is a trapezium the same as a trapezoid?

In British and Commonwealth maths a trapezium has exactly one pair of parallel sides. In American maths that shape is called a trapezoid. The formula is the same.