KS3 · Maths

Area of a trapezium

You never need the leaning sides to find a trapezium's area. Put two copies together, a shape you already know appears, and the formula falls straight out.

Maths · Area

Which lengths decide the area?

a (top, parallel) 4.0 cm. b (base, parallel) 8.0 cm. h (perpendicular height) 3.0 cm. slanted side AD 3.6 cm. slanted side BC 3.6 cm. area 18.0 cm². Classification: Trapezium: one pair of parallel sides. Relationship: Area = (a + b) × h ÷ 2 — only the two parallel sides and the height between them go in.

a (top, parallel)4.0 cmb (base, parallel)8.0 cmh (perpendicular height)3.0 cmslanted side AD3.6 cmslanted side BC3.6 cmarea18.0 cm²Drag D or C (keep D left of C)

Trapezium: one pair of parallel sides

Area = (a + b) × h ÷ 2 — only the two parallel sides and the height between them go in.

A trapezium is a four-sided shape with one pair of parallel sides (more than one are called trapezia, or trapeziums). Here the parallel sides are the top, a, and the base, b. The faint line is the perpendicular height, h: the distance straight across between the parallel sides, meeting them at right angles. Drag D or C along the top line. The slanted sides stretch and shrink, and the top can even hang out past the base — but whatever you do, the area always works out as (a + b) × h ÷ 2.

Watch out: The slanted sides AD and BC change on every drag, but they never appear in the area. The height is the faint line straight across, not a slanted side.

Where the formula comes from

trapezium + identical trapezium

Take any trapezium with parallel sides a and b and perpendicular height h. Now make an exact copy of it.

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Two ways to the same area

Split: rectangle + trianglevsHalve the parallelogram

Start with the first row — it's where the two methods part company.

Focus

When the shorter side overhangs

Split: rectangle + triangle

The shortcut breaks: part of the triangle's base now sits under the overhanging top, so 'longer side minus shorter side' no longer gives the base of the triangle you can cut off.

Halve the parallelogram

Still works. Two copies always make a parallelogram, so (a + b) × h ÷ 2 is true for every trapezium.

The insight

This is the difference that matters. When the top hangs over the edge, trust the formula.

The big idea

Split: rectangle + triangle

Cut the trapezium into a rectangle and a triangle. Add the rectangle's area to the triangle's area.

Halve the parallelogram

Join two identical trapezia into a parallelogram, then take half of its area.

The base you use

Split: rectangle + triangle

The triangle's base is found from the difference between the two parallel sides.

Halve the parallelogram

The parallelogram's base is the sum of the two parallel sides, a + b.

The height you use

Split: rectangle + triangle

The same perpendicular height for the rectangle and for the triangle.

Halve the parallelogram

The perpendicular height h, straight across between the parallel sides.

Using the formula, step by step

Problem

A trapezium has parallel sides of 5 cm and 9 cm. The perpendicular distance between them is 6 cm. Find its area.

Check your thinking

What's in your head right now?

You're about to work out the area of a trapezium.

Which of these is closest to what you'd do?

It's basically a triangle, so I'd do base × height ÷ 2.If the height isn't marked, I'd just use one of the slanted sides.I'd add the parallel sides and multiply by the height. Done.Add the two parallel sides, multiply by the perpendicular height, then halve.

How sure are you?

I'm sureI think soI'm guessing

It's basically a triangle, so I'd do base × height ÷ 2.

This is a common misconception

Close, but not quite. The triangle formula would only work if the top side had length zero — and then the shape would actually be a triangle. A trapezium's top has a real length, so it has to go in: add a and b before you multiply.

If the height isn't marked, I'd just use one of the slanted sides.

This is a common misconception

Tempting, but no. The slanted sides lean, so they don't measure the distance straight across between the parallel sides. The height has to meet the parallel sides at right angles. In fact the slanted sides don't appear in the area formula at all — you'd only need them to find the perimeter.

I'd add the parallel sides and multiply by the height. Done.

This is a common misconception

Nearly there — but that gives you (a + b) × h, which is the parallelogram made from TWO trapezia. You've found double the area. Divide by 2 to get one trapezium. Forgetting to halve is the most common slip with this formula.

Add the two parallel sides, multiply by the perpendicular height, then halve.

This matches the evidence

Exactly right. Parallel sides in, perpendicular height in, halve at the end — and the slanted sides stay out of it.

Add the parallel sides, times the perpendicular height, then halve. The slanted sides never go in.

Maths · Algebra

Working backwards from the area

Sometimes you know the area and one length is missing. Put what you know into the formula, then undo it one step at a time.

GoalArea 45 cm², one parallel side 11 cm, height 5 cm. Find the other parallel side, a.

(a + 11) × 5 ÷ 2 = 45

Put everything you know into the formula.

Step 1 of 4

Put everything you know into the formula.

Watch out: Double the area first. If you skip it, every line after it is out by a factor of 2.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two copies make a parallelogram — so one trapezium is exactly half of it.

What you need to know

A trapezium has one pair of parallel sides.
Area of a trapezium = (a + b) × h ÷ 2, where a and b are the parallel sides and h is the perpendicular height between them.
Two identical trapezia make a parallelogram — that's why the formula halves.
Given the area, you can solve an equation to find a missing length.

The big picture

A trapezium has one pair of parallel sides. Two identical trapezia fit together to make a parallelogram with base a + b and height h, so one trapezium has area (a + b) × h ÷ 2. Only the parallel sides and the perpendicular height between them are used — never the slanted sides. You can also work backwards from an area to find a missing parallel side or the height.

Key points

1The formula uses three lengths: the two parallel sides and the perpendicular distance between them. The slanted sides never go in.

2Order matters: add the parallel sides first, multiply by the height, then halve.

3(a + b) × h on its own is a parallelogram of two trapezia — double the answer you want.

4Splitting into a rectangle plus a triangle also works, but the 'difference' shortcut for the triangle's base fails when the shorter side overhangs.

5To work backwards, double the area first, then divide and add or subtract to find the missing length.

6The slanted sides matter for the perimeter, not the area — so the area formula can't find them.

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Worked example

Problem

The end wall of a garden shed is a trapezium. Its two upright edges are parallel: one is 2.5 m tall and the other is 1.9 m tall. They stand 3 m apart, measured straight across along the floor. The sloping roof edge is about 3.1 m long. Find the area of the wall.

⚠ Watch out

Forgetting to halve. (a + b) × h is the area of the parallelogram made from two trapezia, so stopping there gives double the right answer.

🧠

Memory hook

Twin it, then halve it: two trapezia make a parallelogram, so one trapezium is half of (a + b) × h.

✓

Check yourself

Parallel sides 4 cm and 10 cm, perpendicular height 7 cm — what's the area? (Answer: (4 + 10) × 7 ÷ 2 = 49 cm².)

Flashcards

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What makes a shape a trapezium?

It is a quadrilateral with one pair of parallel sides.

What is the plural of trapezium?

Trapezia, or trapeziums.

What is the formula for the area of a trapezium?

Area = (a + b) × h ÷ 2, where a and b are the parallel sides and h is the perpendicular height.

In the trapezium formula, what exactly is h?

The perpendicular distance between the two parallel sides — it meets them at right angles.

Two identical trapezia are fitted together, one turned half a turn. What shape do they make?

A parallelogram.

What is the base of the parallelogram made from two identical trapezia?

The sum of the two parallel sides, a + b.

Why does the trapezium formula divide by 2?

(a + b) × h is the area of a parallelogram made of two identical trapezia, and one trapezium is half of it.

In what order do you work out (a + b) × h ÷ 2?

Add the parallel sides first, then multiply by the height, then halve.

Do the slanted sides of a trapezium go into the area formula?

No. They only matter if you need the perimeter.

Can the area formula find the length of a slanted side?

No — the slanted sides don't appear in the formula, so it can't tell you anything about them.

Why isn't the area of a trapezium base × height ÷ 2?

That would need one parallel side to have length zero, and then the shape would be a triangle, not a trapezium.

Describe the split (composite) method for a trapezium's area.

Cut it into a rectangle and a triangle, find both areas using the same perpendicular height, and add them.

In the split method, when does 'longer side minus shorter side' fail as the triangle's base?

When the shorter parallel side overhangs, so part of the triangle's base sits under it.

You know the area and need a missing length. What is usually the first move after substituting?

Double both sides, to undo the halving.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What's in your head right now?

Working backwards from the area

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