Trapezoid


In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium in English outside North America, but as a trapezoid in American and Canadian English. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the [|special cases] below.

Etymology

The term trapezium has been in use in English since 1570, from Late Latin trapezium, from Greek τραπέζιον, literally "a little table", a diminutive of τράπεζα, "a table", itself from τετράς, "four" + πέζα, "a foot; end, border, edge".
The first recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclid's Elements.
This article uses the term trapezoid in the sense that is current in the United States and Canada. In many languages also using a word derived from the Greek, the form used is the one closest to trapezium, not to trapezoid.

''Trapezium'' vs ''Trapezoid''

The term trapezoid was once defined as a quadrilateral without any parallel sides in Britain and elsewhere. The Oxford English Dictionary says "Often called by English writers in the 19th century". According to the OED, the sense of a figure with no sides parallel is the meaning for which Proclus introduced the term "trapezoid". This is retained in the French trapézoïde, German Trapezoid, and in other languages. However, this particular sense is considered obsolete.
A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in the 17th and 18th centuries, and again the prevalent one in recent use outside North America. A trapezium as any quadrilateral more general than a parallelogram is the sense of the term in Euclid.
Confusingly, the word trapezium was sometimes used in England from c. 1800 to c. 1875, to denote an irregular quadrilateral having no sides parallel. This is now obsolete in England, but continues in North America. However this shape is more usually just called an irregular quadrilateral.

Inclusive vs exclusive definition

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides, thereby excluding parallelograms. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.
Under the inclusive definition, all parallelograms are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.

Special cases

A right trapezoid has two adjacent right angles. Right trapezoids are used in the trapezoidal rule for estimating areas under a curve.
An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute and one obtuse angle on each base.
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.
A parallelogram is a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry. It is possible for obtuse trapezoids or right trapezoids.
A tangential trapezoid is a trapezoid that has an incircle.
A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the hyperbolic plane has 3 right angles.

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
The quadrilateral is a parallelogram when, but it is an ex-tangential quadrilateral when.

Characterizations

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel:
The midsegment 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,
The midsegment of a trapezoid is one of the two bimedians.
The height is the perpendicular distance between the bases. In the case that the two bases have different lengths, the height of a trapezoid h can be determined by the length of its four sides using the formula
where c and d are the lengths of the legs.

Area

The area K of a trapezoid is given by
where a and b are the lengths of the parallel sides, h is the height, and m is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya. This yields 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:
where a and b are parallel and b > a. This formula can be factored into a more symmetric version
When one of the parallel sides has shrunk to a point, 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
where is the semiperimeter of the trapezoid..
From Bretschneider's formula, it follows that
The line that joins the midpoints of the parallel sides, bisects the area.

Diagonals

The lengths of the diagonals are
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, intersecting at O, then the area of is equal to that of, and the product of the areas of and is equal to that of and. 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:
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 lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by
The center of area divides this segment in the ratio
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

Applications

Architecture

In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids. This was the standard style for the doors and windows of the Inca.

Geometry

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.

Biology

In morphology, taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as trapezoidal or trapeziform commonly are useful in descriptions of particular organs or forms.

Computer Engineering

In computer engineering, specifically digital logic and computer architecture, trapezoids are typically utilized to symbolize multiplexors. Multiplexors are logic elements that select between multiple elements and produce a single output based on a select signal. Typical designs will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent.