Trapezium: the area formula (half the sum of the parallel sides times the height, ½(a + b)h) and where it comes from, the properties (one pair of parallel sides, co-interior angles that add to 180°), the types (isosceles, right-angled, scalene), trapezium against trapezoid and the US definitions, the midsegment, perimeter, finding a missing side or the height from the area, worked examples, and a printable sheet with practice

"Trapezium" is searched about 30,000 times a month in the UK and 21,000 in Australia, which makes it one of the most-searched shapes in both countries, mostly by Year 6 to GCSE students who need the area formula and by parents who learned a different word for it. This page gives the formula with its derivation, the properties and the three types, the trapezium-against-trapezoid story, the midsegment and perimeter, how to work backwards from the area, six worked examples, the mistakes, and a printable sheet with practice. US readers: everything here applies to the trapezoid.

Download the trapezium sheet with practice (PDF) (A4, the formula, derivation, properties and types on page 1, the worked examples and the practice set with answers on page 2, free).

The area formula

A = ½(a + b)h, read as "half the sum of the parallel sides, times the height":

• a and b: the lengths of the two parallel sides (the bases)
• h: the perpendicular height, the distance between the parallel sides measured at right angles to them

Parallel sides 8 cm and 12 cm, height 5 cm: A = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50 cm².

Two other ways to write the same thing: A = (a + b) ÷ 2 × h, the average of the parallel sides times the height (a trapezium is a rectangle whose width is the average of the two bases); and A = m × h, where m is the midsegment.

The GCSE formula sheet has included the trapezium formula in recent years; check whether your exam year provides it or expects it memorised. Either way, learn it.

Where the formula comes from

Two triangles. Draw a diagonal. It splits the trapezium into two triangles that share the height h: one with base a (area ½ah) and one with base b (area ½bh). Add them: ½ah + ½bh = ½(a + b)h.

Two trapeziums make a parallelogram. Take a second copy of the trapezium, turn it upside down and fit it beside the first. The result is a parallelogram with base (a + b) and height h, area (a + b)h. The trapezium is half of it.

Either derivation is a GCSE "show that" question.

The properties

The co-interior fact is the one exams use: because the bases are parallel and a leg is a transversal, the two angles at the ends of that leg add to 180° (see parallel lines and transversals). If one base angle is 65°, the angle above it on the same leg is 115°.

The types

• Isosceles trapezium: the legs are equal, the base angles are equal in pairs, the diagonals are equal, and it has one line of symmetry. It is the shape of a bridge truss end, a lampshade in profile and a child's drawing of a boat. Its height comes from Pythagoras on the overhang: with bases 10 and 16 and legs 5, each overhang is (16 − 10) ÷ 2 = 3, so h = √(5² − 3²) = 4.
• Right-angled trapezium (right trapezium): one leg is perpendicular to both bases, so there are two right angles and that leg is the height. The cross-section of a swimming pool with a sloping floor.
• Scalene trapezium: no equal sides, no symmetry; the height must be given or found from other information.

Trapezium against trapezoid

The words were swapped by a dictionary error in the 1790s and never swapped back. On a British, Australian or Indian paper, write trapezium; on an American one, trapezoid. There is a second disagreement inside the US: the exclusive definition (exactly one pair of parallel sides, so a parallelogram is not a trapezoid) is the traditional one, and the inclusive definition (at least one pair, so parallelograms, rectangles and squares are all special trapezoids) is used by many newer US textbooks and by the Common Core's hierarchy of quadrilaterals. UK exams use the exclusive definition.

The midsegment and the perimeter

The midsegment (median, mid-line) joins the midpoints of the two legs. It is parallel to the bases and its length is their average: m = (a + b) ÷ 2. Bases 8 and 12: midsegment 10. The area is then simply m × h.

The perimeter is the sum of all four sides; the height is not a side. For an isosceles trapezium with bases 10 and 16 and legs 5: P = 10 + 16 + 5 + 5 = 36. If a leg is unknown, find it by Pythagoras from the height and the overhang.

Working backwards from the area

• Find the height: h = 2A ÷ (a + b). Area 60 cm², bases 7 and 13: h = 120 ÷ 20 = 6 cm.
• Find a parallel side: a = 2A ÷ h − b. Area 45 cm², height 5, one base 11: a = 90 ÷ 5 − 11 = 18 − 11 = 7 cm.

Rearrange before substituting, or substitute and solve the one-step equation: ½(a + 11) × 5 = 45 → a + 11 = 18 → a = 7.

Six worked examples

1. Bases 6 cm and 10 cm, height 4 cm. A = ½ × 16 × 4 = 32 cm².
2. Bases 3.5 m and 6.5 m, height 2 m. A = ½ × 10 × 2 = 10 m².
3. A right-angled trapezium with bases 9 and 15 and the perpendicular leg 8. A = ½ × 24 × 8 = 96; the other leg is √(8² + 6²) = 10, perimeter 9 + 15 + 8 + 10 = 42.
4. Isosceles trapezium, bases 10 and 16, legs 5. Overhang 3, h = 4, A = ½ × 26 × 4 = 52.
5. Area 84 cm², height 7 cm, one base 9 cm. 2 × 84 ÷ 7 = 24 = 9 + b, so b = 15 cm.
6. A trapezium-shaped field has parallel sides 120 m and 80 m, 60 m apart. Area ½ × 200 × 60 = 6,000 m² = 0.6 hectares.

The mistakes that cost marks

• Using a slanting leg as the height. The height is perpendicular to the bases.
• Multiplying the two parallel sides instead of adding them.
• Forgetting the half (giving the area of the parallelogram made from two copies).
• Adding the height into the perimeter.
• Mixing units (bases in metres, height in centimetres).
• On a UK paper, calling it a trapezoid; on a US paper, a trapezium.
• Assuming a trapezium is isosceles when it is not drawn or stated so.

Where it sits in each curriculum

Practice (answers in the PDF)

1. Find the area of a trapezium with parallel sides 7 cm and 11 cm and height 6 cm.
2. A trapezium has area 72 m², parallel sides 10 m and 14 m. What is its height?
3. An isosceles trapezium has bases 12 cm and 20 cm and legs 5 cm. Find its height, area and perimeter.
4. One base angle of a trapezium is 72°. What is the angle at the other end of the same leg, and why?
5. A right-angled trapezium has bases 4 and 9 and height 12. Find its area and the length of the slanting leg.
6. Which of these is a trapezium (UK definition): a shape with sides 5, 8, 5, 8 and two pairs of parallel sides; a shape with exactly one pair of parallel sides and all sides different; a square?

Download the sheet and the practice set (PDF)

Where to get help

A 1-on-1 online maths class in UK and Australian after-school hours with a tutor who holds a master's in mathematics and teaches KS2, KS3 and GCSE, the Australian Years 6 to 10 and CBSE Class 8 and 9 every week: shapes taught with the derivation drawn, so the formula is remembered as a picture rather than a string of letters. The GCSE maths page explains how the classes run; the free 30-minute first class is a real lesson on this week's homework. Related pages: area of a sector and arc length, the area of a triangle and parallel lines and transversals.

Questions parents ask

See how we teach this, 1-on-1 online →