Trapezoid

Trapezoid (AmE) Trapezium (BrE)
Trapezoid or trapezium
Type	quadrilateral
Edges and vertices	4
Area	${\tfrac {1}{2}}(a+b)h$
Properties	convex

In geometry, a trapezoid (/ˈtræpəzɔɪd/) in North American English, or trapezium (/trəˈpiːziəm/) in British English, is a quadrilateral that has at least one pair of parallel sides.

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 quadrilateral in Euclidean geometry, but there are also crossed cases. If ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.

Definitions

Trapezoid can be defined exclusively or inclusively. Under an exclusive definition a trapezoid is a quadrilateral having exactly one pair of parallel sides, with the other pair of opposite sides non-parallel. Parallelograms including rhombi, rectangles, and squares are then not considered to be trapezoids. Under an inclusive definition, a trapezoid is any quadrilateral with at least one pair of parallel sides. In an inclusive classification scheme, definitions are hierarchical: a square is a type of rectangle, a rectangle or rhombus is a type of parallelogram, and every parallelogram is a type of trapezoid.

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. Quadrilateral § Taxonomy.)

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.

Etymology

In the ancient Greek geometry of Euclid's Elements (c. 300 BC), quadrilaterals were classified into exclusive categories: square; oblong (non-square rectangle); (non-square) rhombus; rhomboid, meaning a non-rhombus non-rectangle parallelogram; or trapezium (τραπέζιον, literally "table"), meaning any quadrilateral not already included in the previous categories.

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 (c. 100 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.

Hutton's definitions in 1795

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. Hobbs (1899) and Hopkins (1891) exemplify the American conventions by defining 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 are today sometimes called irregular quadrilaterals.

Special cases

Trapezoid special cases. The orange figures also qualify as parallelograms.

An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry. This is possible for acute trapezoids or right trapezoids as rectangles. An acute trapezoid is a trapezoid with two adjacent acute angles on its longer base, and the isosceles trapezoid is an example of an acute trapezoid. The isosceles trapezoid has a special case known as a three-sided trapezoid, meaning it is a trapezoid wherein two trapezoid's legs have equal lengths as the trapezoid's base at the top. The isosceles trapezoid is the convex hull of an antiparallelogram, a type of crossed quadrilateral. Every antiparallelogram is formed with such a trapezoid by replacing two parallel sides by the two diagonals.

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 right triangles, which was used by James Garfield to prove the Pythagorean theorem.

A tangential trapezoid is a trapezoid that has an incircle.

Condition of existence

Four lengths a, c, b, d can constitute the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when

$\displaystyle |d-c|<|b-a|<d+c.$

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

Characterizations

general trapezoid/trapezium:
parallel sides: $a,\,b$ with $a<b$
legs: $c,\,d$
diagonals: $q,\,p$
midsegment: $m$
height/altitude: $h$

trapezoid/trapezium with opposing triangles $S,\,T$ formed by the diagonals

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:

• It has two adjacent angles 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 triangles 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

${\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},$

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 $\sin A\sin C=\sin B\sin D.$
• The cosines of two adjacent angles sum 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

$v={\frac {|a-b|}{2}}.$

Properties

Midsegment and height

The midsegment or median of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m is equal to the average of the lengths of the bases a and b of the trapezoid,

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

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

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

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

Area

The area $K$ of a trapezoid is given by the product of the midsegment (the average of the two bases) and the height: $$K={\tfrac {1}{2}}(a+b)h$$ where $a$ and $b$ are the lengths of the bases, and $h$ is the height (the perpendicular distance between these sides). This method has been used in Aryabhata's Aryabhatiya in section 2.8 in the classical age of Indian, yielding as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

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

$K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.$

When one of the parallel sides has shrunk to a point (say a = 0), this formula reduces to Heron's formula for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is

$K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}}$

where $s={\tfrac {1}{2}}(a+b+c+d)$ is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).

From Bretschneider's formula, it follows that

$K={\sqrt {{\frac {\left(ab^{2}-a^{2}b-ad^{2}+bc^{2}\right)\left(ab^{2}-a^{2}b-ac^{2}+bd^{2}\right)}{4(b-a)^{2}}}-\left({\frac {c^{2}+d^{2}-a^{2}-b^{2}}{4}}\right)^{2}}}.$

The bimedian connecting the parallel sides bisects the area. More generally, any line drawn through the midpoint of the median parallel to the bases, that intersects the bases, bisects the area. Any triangle connecting the two ends of one leg to the midpoint of the other leg is also half of the area.

Diagonals

The lengths of the diagonals are $${\begin{aligned}p&={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},\\q&={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}},\end{aligned}}$$ where $a$ is the short base, $b$ is the long base, and $c$ and $d$ are the trapezoid legs.

If the trapezoid is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O, then the area of $\triangle$ AOD is equal to that of $\triangle$ BOC, and the product of the areas of $\triangle$ AOD and $\triangle$ BOC is equal to that of $\triangle$ AOB and $\triangle$ COD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.

Let the trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and DC. Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC:

${\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).$

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.

Other properties

The center of area (center of mass for a uniform lamina) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by

$x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).$

The center of area divides this segment in the ratio (when taken from the short to the long side)

${\frac {a+2b}{2a+b}}.$

If the angle bisectors to angles A and B intersect at P, and the angle bisectors to angles C and D intersect at Q, then

$PQ={\frac {|AD+BC-AB-CD|}{2}}.$

Applications

The trapezoidal rule for numerical integration

The Temple of Dendur in the Metropolitan Museum of Art in New York City

Example of a trapeziform pronotum outlined on a spurge bug

Ontario Highway 502

In calculus, the definite integral of an arbitrary function $f(x)$ can be numerically approximated as a discrete sum by partitioning the interval of integration into small uniform intervals and approximating the function's value on each interval as the average of the values at its endpoints: $$\int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\tfrac {1}{2}}{\bigl (}f(x_{k-1})+f(x_{k}){\bigr )}\Delta x,$$ where $N$ is the number of intervals, $x_{0}=a$, $x_{N}=b$, and $\Delta x=(b-a)/N$. Graphically, this amounts to approximating the region under the graph of the function by a collection of trapezoids, so this method is called the trapezoidal rule.

Trapezoids, especially isosceles trapezoids, are common in architecture, in the shapes of doors, windows, and whole buildings. This was the standard style for the doors and windows of the Inca.

Canals and drainage ditches commonly have a trapezoidal cross-section.

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

Trapezoids may also be found in the highway signs, as in Ontario Highway 502 with an isosceles trapezoidal shaped sign.

Non-Euclidean geometry

In spherical or hyperbolic geometry, the internal angles of a quadrilateral do not sum to 360°, but quadrilaterals analogous to trapezoids, parallelograms, and rectangles can still be defined, and additionally there are a few new types of quadrilaterals not distinguished in the Euclidean case.

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.

Related topics

An example of trapezoidal number: 15 = 4 + 5 + 6

The trapezoidal number is a set of positive integer obtained by summing consecutively two or more positive integers greater than one, forming a trapezoidal pattern.

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.

See also

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

Bibligraphy

• Alsina, Claudi; Nelsen, Roger (2020). A Cornucopia of Quadrilaterals. Mathematical Association of America.
• Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2016). The Symmetries of Things. CRC Press. ISBN 978-1-4398-6489-0.
• Gamer, Carlton; Roeder, David W.; Watkins, John J. (1985). "Trapezoidal numbers". Mathematics Magazine. 58 (2): 108–110. doi:10.2307/2689901. JSTOR 2689901.
• Hobbs, Charles Austin (1899). The Elements of Plane Geometry. A. Lovell & Company.
• Garfield, James (1876). "Pons Asinorum". New England Journal of Education. 3 (14): 161. ISSN 2578-4145. JSTOR 44764657.
• Murray, James (1926). A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society. Vol. X. Clarendon Press at Oxford.
• Usiskin, Zalman; Griffin, Jennifer (2008). The Classification of Quadrilaterals: A Study of Definition. Information Age Publishing. pp. 49–52, 63–67.
• Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007). Calculus (9th ed.). Pearson Prentice Hall. ISBN 978-0131469686.
• Posamentier, Alfred S.; Bannister, Robert L. (2014). "Quadrilaterals". Geometry, Its Elements and Structure: Second Edition. Dover Books on Mathematics (2nd ed.). Courier Corporation. ISBN 9780486782164.
• Dodge, Clayton W. (2012). Euclidean Geometry and Transformations. Dover Books on Mathematics. Courier Corporation. ISBN 9780486138428.
• Ringenberg, Lawrence A. (1977). "Coordinates in a Plane". College Geometry. R. E. Krieger Publishing Company. ISBN 9780882755458.
• Hopkins, George Irving (1891). Manual of Plane Geometry. D.C. Heath & Company.

Further reading

• D. Fraivert, A. Sigler and M. Stupel : Common properties of trapezoids and convex quadrilaterals

External links

• "Trapezium" at the Encyclopedia of Mathematics
• Weisstein, Eric W. "Right trapezoid". MathWorld.
• 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)