Trapezoid

	Email		Print
	Bookmark		Link

Trapezoid

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a trapezoid or trapezium is a quadrilateral

Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
with two parallel

Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain. (To add to the confusion, the word “trapezium” is used in North America to refer to a quadrilateral with no parallel sides, while the word “trapezoid” has sometimes been used historically with this same meaning.)

In geometry, a trapezoid with vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
ABCD is denoted

Mathematical notation

A mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics and the physical sciences, engineering and economics....
.
reviously remarked, a figure with two parallel sides is referred to as trapezoid in North America, and as a trapezium in Britain.

In North America, the term trapezium is used to refer to a quadrilateral with no parallel sides.

Discussion

Ask a question about 'Trapezoid'

Start a new discussion about 'Trapezoid'

Answer questions from other users

Full Discussion Forum

Encyclopedia

In geometry

Geometry

Quadrilateral

Parallel (geometry)

Vertex (geometry)

Mathematical notation

Definition and terminology

As previously remarked, a figure with two parallel sides is referred to as trapezoid in North America, and as a trapezium in Britain.

In North America, the term trapezium is used to refer to a quadrilateral with no parallel sides. The term trapezoid has been defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage (the Oxford English Dictionary says “Often called by English writers in the 19th century”).

According to the Oxford English Dictionary

Oxford English Dictionary

The Oxford English Dictionary , published by the Oxford University Press , is a comprehensive dictionary of the English language. Two fully-bound print editions of the OED have been published under its current name, in 1928 and 1989; as of December 2008 the dictionary's current editors have completed a quarter of the third edition....
, the trapezoid as a figure with no sides parallel is the sense for which Proclus

Proclus

Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek philosophy Neoplatonist philosophy, one of the last major Classical philosophers ....
introduced the term; it is retained in the French "trapézoïde", German "trapezoïd", and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a parallelogram is the sense of the term in Euclid

Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
. The sense of a trapezium as an irregular quadrilateral having no sides parallel was the usual sense in England from c1800 to c1875, but is now rare. This sense is the one that is standard in the U.S., but in practice quadrilateral is used rather than trapezium.

There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelogram

Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
s, which have two pairs of parallel sides, should be counted as trapezoids. Some authors define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors define a trapezoid as a quadrilateral with at least one pair of parallel sides, making a parallelogram a special type of trapezoid.

Characteristics and properties

In an isosceles trapezoid

Isosceles trapezoid

An isosceles trapezoid is a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid. Two opposite sides are Parallel , the two other sides are of equal length....
, the base angles are equal, and so are the other pair of opposite sides AD and BC.

If sides AD and BC are also parallel, then they are equal, and the trapezoid is also a parallelogram

Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
. Otherwise, the other two opposite sides may be extended until they meet at a point, forming a triangle containing the trapezoid.

A quadrilateral is a trapezoid if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
it contains two adjacent angle

Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s that are supplementary

Supplementary angles

A pair of angles is supplementary if their measures add up to 180 degree s.If the two supplementary angles are adjacent angles , their non-shared sides form a straight line ....
, that is, they add up to one straight angle of 180 degree

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
s (p

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
radian

Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s). Another necessary and sufficient condition is that the diagonal

Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s cut each other in mutually the same ratio

Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
; this ratio is the same as that between the lengths of the parallel sides.

The mid-segment (occasionally referred to as the median) of a trapezoid is the segment that joins the midpoints of the other pair of opposite sides. It is parallel to the two parallel sides, and its length is the arithmetic mean

Arithmetic mean

In mathematics and statistics, the arithmetic mean of a list of numbers is the sum of all of the list divided by the number of items in the list....
of the lengths of those sides. The line joining the mid-points of the parallel sides (which could also be called the median) bisects the area. The area

Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
of a trapezoid can be computed as the length of the mid-segment, multiplied by the distance along a perpendicular

Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
line between the parallel sides. 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.

Thus, if a'' and b'' are the lengths of the two parallel sides and h'' is the distance (height) between the parallels, the area formula is as follows:

The quantity is the average

Average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "Expected value" value of the data set....
of the horizontal lengths of the trapezoid, so the area can be understood to be the product of the height and average length of the shape.

Another formula for the area can be used when all that is known are the lengths of the four sides. If the sides are a'', b'', c'' and d'', and a'' and c'' are parallel (where a'' is the longer parallel side), then:

This formula does not work when the parallel sides a'' and c'' are equal since we would have division by zero. In this case the trapezoid is necessarily a parallelogram (and so ) and the numerator of the formula would also equal zero. In fact, the sides of a parallelogram aren't enough to determine its shape or area, the area of a parallelogram with side lengths a'' and b'' can be any number from to 0.

When the smaller parallel side c'' is set to zero, this formula reduces to Heron's formula

Heron's formula

In geometry, Heron's formula states that the area of a triangle whose sides have lengths a, b, and c iswhere s is the semiperimeter of the triangle:...
.

If the trapezoid above is divided into 4 triangles by its diagonals AC'' and BD'', intersecting at O'', then the area of ΔAOD is equal to that of ΔBOC, and the product of the areas of ΔAOD and ΔBOC is equal to that of ΔAOB and ΔCOD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.

Architecture

In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering towards the top, in Egyptian style.

External links

on MathWorld

MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
With interactive animations at elsy.at: Animated course (construction, circumference, area) on Numerical Methods for Stem Undergraduate