Trapezoid

 ( Elementary Shapes ) 

In geometry, a trapezoid () in North American English, or trapezium () in British English,

The parallel sides are called the bases of the trapezoid. The other two sides are called the legs or lateral sides. If the trapezoid is a parallelogram, then the choice of bases and legs is arbitrary.

A trapezoid is usually considered to be a Convex polygon quadrilateral in Euclidean geometry, but there are also Crossed polygon cases. If shape ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.

Professional mathematicians and post-secondary geometry textbooks nearly always prefer inclusive definitions and classifications, because they simplify statements and proofs of geometric theorems. In primary and secondary education, definitions of rectangle and parallelogram are also nearly always inclusive, but an exclusive definition of trapezoid is commonly found. This article uses the inclusive definition and considers parallelograms to be special kinds of trapezoids. (Cf. .)

To avoid confusion, some sources use the term proper trapezoid to describe trapezoids with exactly one pair of parallel sides, analogous to uses of the word proper in some other mathematical objects.

The Neoplatonist philosopher Proclus (mid 5th century AD) wrote an influential commentary on Euclid with a richer set of categories, which he attributed to Posidonius ( BC). In this scheme, a quadrilateral can be a parallelogram or a non-parallelogram. A parallelogram can itself be a square, an oblong (non-square rectangle), a rhombus, or a rhomboid (non-rhombus non-rectangle). A non-parallelogram can be a trapezium with exactly one pair of parallel sides, which can be isosceles (with equal legs) or scalene (with unequal legs); or a trapezoid (τραπεζοειδή, literally "table-like") with no parallel sides.

All European languages except for English follow Proclus's meanings of trapezium and trapezoid, as did English until the late 18th century, when an influential mathematical dictionary published by Charles Hutton in 1795 transposed the two terms without explanation, leading to widespread inconsistency. Hutton's change was reversed in British English in about 1875, but it has been retained in American English to the present. Late 19th century American geometry textbooks define a trapezium as having no parallel sides, a trapezoid as having exactly one pair of parallel sides, and a parallelogram as having two sets of opposing parallel sides. To avoid confusion between contradictory British and American meanings of trapezium and trapezoid, quadrilaterals with no parallel sides have sometimes been called irregular quadrilaterals.

An obtuse trapezoid, on the other hand, has one acute and one obtuse angle on each base. An example is parallelogram with equal acute angles.

A right trapezoid is a trapezoid with two adjacent right angle. One special type of right trapezoid is by forming three , which was used by James Garfield to prove the Pythagorean theorem.

A tangential trapezoid is a trapezoid that has an incircle.

The quadrilateral is a parallelogram when $d-c\; =\; b-a\; =\; 0$, but it is an ex-tangential quadrilateral (which is not a trapezoid) when $|d-c|\; =\; |b-a|\; \backslash neq\; 0$.

• It has two adjacent that are supplementary, that is, they add up to 180 degrees.
• The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
• The diagonals cut each other in mutually the same ratio (this ratio is the same as that between the lengths of the parallel sides).
• The diagonals cut the quadrilateral into four Triangle of which one opposite pair have equal areas.
• The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.
• The areas S and T of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation

:$\backslash sqrt\{K\}\; =\; \backslash sqrt\{S\}+\backslash sqrt\{T\},$

where K is the area of the quadrilateral.

• The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals are collinear.
• The angles in the quadrilateral ABCD satisfy $\backslash sin\; A\backslash sin\; C=\backslash sin\; B\backslash sin\; D.$
• The cosines of two adjacent angles Addition to 0, as do the cosines of the other two angles.
• The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.
• One bimedian divides the quadrilateral into two quadrilaterals of equal areas.
• Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.

Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel:

• The consecutive sides a, c, b, d and the diagonals p, q satisfy the equation

:$p^2+q^2=c^2+d^2+2ab.$

• The distance v between the midpoints of the diagonals satisfies the equation

:$v=\backslash frac${2}.

$m\; =\; \backslash frac\{a\; +\; b\}\{2\}.$

The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides the trapezoid into equal areas).

The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths ( a ≠ b), the height of a trapezoid h can be determined by the length of its four sides using the formula

$h=\; \backslash frac\{\backslash sqrt\{(p-2a)(p-2b)(p-2b-2d)(p-2b-2c)\}\}\{2|b-a|\}$

where c and d are the lengths of the legs and $p\; =\; a+b+c+d$.

The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides $a$, $c$, $b$, $d$:: $$K\; =\; \backslash frac\{1\}\{2\}(a+b)\backslash sqrt\{c^2-\backslash frac\{1\}\{4\}\backslash left((b-a)+\backslash frac\{c^2-d^2\}\{b-a\}\backslash right)^2\}$$ where $a$ and $b$ are parallel and $b\; >\; a$.

p &= \sqrt{\frac{ab^2-a^2b-ac^2+bd^2}{b-a}}, \\
q &= \sqrt{\frac{ab^2-a^2b-ad^2+bc^2}{b-a}},
     

In biology, especially morphology and taxonomy, terms such as trapezoidal or trapeziform commonly are useful in descriptions of particular organs or forms.

Trapezoids are sometimes used as a graphical symbol. In , a trapezoid is the symbol for a multiplexer.

A spherical or hyperbolic trapezoid is a quadrilateral with two opposite sides, the legs, each of whose two adjacent angles sum to the same quantity; the other two sides are the bases. As in Euclidean geometry, special cases include isosceles trapezoids whose legs are equal (as are the angles adjacent to each base), parallelograms with two pairs of opposite equal angles and two pairs of opposite equal sides, rhombuses with two pairs of opposite equal angles and four equal sides, rectangles with four equal (non-right) angles and two pairs of opposite equal sides, and squares with four equal (non-right) angles and four equal sides.

When a rectangle is cut in half along the line through the midpoints of two opposite sides, each of the resulting two pieces is an isosceles trapezoid with two right angles, called a Saccheri quadrilateral. When a rectangle is cut into quarters by the two lines through pairs of opposite midpoints, each of the resulting four pieces is a quadrilateral with three right angles called a Lambert quadrilateral. In Euclidean geometry Saccheri and Lambert quadrilaterals are merely rectangles.

The crossed ladders problem is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection.

• Frustum, a solid having trapezoidal faces
• Wedge, a polyhedron defined by two triangles and three trapezoid faces.

• (2025). 9781470453121, Mathematical Association of America. . ISBN 9781470453121
• (2025). 9781439864890, CRC Press. . ISBN 9781439864890
• (1999). 9780198606789, Oxford University Press. ISBN 9780198606789
• (2025). 9780486138428, Courier Corporation. ISBN 9780486138428
• (2025). 9780786469307, McFarland. ISBN 9780786469307

• "Trapezium" at the Encyclopedia of Mathematics
• Trapezoid definition, Area of a trapezoid, Median of a trapezoid (with interactive animations)
• Trapezoid (North America) at elsy.at: Animated course (construction, circumference, area)
• Trapezoidal Rule on Numerical Methods for Stem Undergraduate
• Autar Kaw and E. Eric Kalu, Numerical Methods with Applications (2008)