Matemáticas
Trapezium: One Pair of Parallel Sides, and the American Name Clash
A trapezium has one pair of parallel sides and area ½(a+b)h. Worked in full, plus why Americans call this shape a trapezoid and mean something else.
Trapezium
A trapezium is a quadrilateral with one pair of parallel sides; the remaining two sides need be neither parallel, equal, nor at any particular angle.
- También llamado
- Trapezoid (in American English)
- Dónde lo encuentra el alumnado
- Grade 6 mathematics, and again at GCSE and IGCSE where the trapezium area formula turns up inside compound-shape and prism-volume questions.
La respuesta corta
A trapezium is a quadrilateral with one pair of parallel sides. The other two sides may be any length and any angle. Its area is half the sum of the two parallel sides multiplied by the perpendicular distance between them, so parallel sides of 6 cm and 10 cm that are 4 cm apart enclose 32 cm².
Un ejemplo
Area = ½(6 + 10) × 4 = 32 cm²
The parallel sides measure 6 cm and 10 cm, and the perpendicular gap between them is 4 cm. Averaging the parallel sides gives 8 cm, which is the width of a rectangle of the same area, and 8 multiplied by 4 gives 32 cm². Reading the formula as "average the parallel sides, then treat it as a rectangle" makes it far harder to misremember than reading it as a string of symbols.
One pair of parallel sides, and nothing else promised
The parallel pair is the entire definition. The two slant sides carry no guarantee at all — they can differ in length, lean in the same direction, and produce a shape that looks nothing like the symmetrical picture most textbooks print.
One special case has its own name. An isosceles trapezium has slant sides of equal length, which gives it equal base angles, equal diagonals and a single line of symmetry down the middle. A right trapezium has two right angles, so one slant side is itself perpendicular to the parallel pair and doubles as the height.
Why the area formula is an average in disguise
Take two identical trapezia, turn one upside down, and push them together along a slant side. The result is a parallelogram whose base is a + b and whose height is h. One trapezium is half of that, which is where the half comes from.
Two things trip people up in the arithmetic. The height must be the perpendicular distance between the parallel sides, never a slant side, exactly as in a parallelogram. And a and b must be the parallel pair specifically — adding the wrong two sides produces a number that means nothing.
Trapezium in Britain, trapezoid in America
The two words have swapped meanings across the Atlantic. In British English, and in Pakistani, Gulf and Cambridge International classrooms that follow it, a trapezium has one pair of parallel sides. In American English that same shape is called a trapezoid, and trapezium means a quadrilateral with no parallel sides at all.
This matters when revision comes from the internet rather than the textbook. A student on Edexcel, AQA, Cambridge OCR or Cambridge International writes trapezium; a student working through Digital SAT material will see trapezoid on the screen and should read it as the same shape.
A separate and quieter disagreement is whether "one pair" means exactly one or at least one. Under the second reading a parallelogram counts as a trapezium. British exam papers describe the shape as having one pair of parallel sides and do not set questions that turn on the distinction, so it is not something to lose sleep over.
Preguntas frecuentes
What is the difference between a trapezium and a trapezoid?
It is a difference of language, not of shape. British English calls a quadrilateral with one pair of parallel sides a trapezium; American English calls it a trapezoid. American usage then applies trapezium to a quadrilateral with no parallel sides. Pakistani, UK and Cambridge International papers all use the British meaning.
Is a parallelogram a trapezium?
Under the definition British schools teach — one pair of parallel sides — a parallelogram is treated as its own shape rather than as a trapezium. Some mathematicians prefer an inclusive definition where it does count. No GCSE or IGCSE mark has ever depended on the answer, so use the trapezium formula only when just one pair is parallel.
How do you find the height of a trapezium from its area?
Rearrange rather than guess. Double the area, then divide by the sum of the parallel sides. An area of 32 cm² with parallel sides of 6 cm and 10 cm gives 64 divided by 16, so the height is 4 cm. Doubling first avoids fractions inside the division.
Does a trapezium have a line of symmetry?
A general one does not. Only the isosceles trapezium, where the two slant sides are equal, has a single line of symmetry running perpendicular to the parallel sides. That version also has equal base angles and equal diagonals, which a lopsided trapezium does not.
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