Quadrilateral

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Quadrilateral

In geometryGeometry

Geometry arose as the field of knowledge dealing with spatial relationships....
, a quadrilateral is a polygonPolygon

A polygon is a closed planar path composed of a finite number of sequential line segments....
with four sides or edges and four vertices or corners. Sometimes, the term quadrangle is used, for etymological symmetry with triangleTriangle

A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line seg...
, and sometimes tetragon for consistency with pentagonPentagon

In geometry, a pentagon is any five-sided polygon....
(5 sided), hexagonHexagon

In geometry, a hexagon is a polygon with six edges and six vertices....
(6 sided) and so on. The interior angles of a quadrilateral add up to 360 degrees.

Quadrilaterals are either simpleSimple polygon

In geometry, two edges of a polygon may cross or even overlap in general....
(not self-intersecting) or complexComplex polygon

A complex polygon is a polygon which intersects itself....
(self-intersecting). Simple quadrilaterals are either convexConvex polygon

In geometry, a convex polygon is a simple polygon whose interior is a convex set....
or concave.
Convex quadrilateralsConvex quadrilaterals are further classified as follows:

More quadrilaterals
Taxonomy A taxonomyTaxonomy

Taxonomy was once only the science of classifying living organisms , but later the word was applied in a wider sense, and m...
of quadrilaterals is illustrated by the following graph.

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Convex quadrilaterals

Convex quadrilaterals are further classified as follows:

TrapeziumTrapezium

The term trapezium can mean more than one thing:...
or trapezoidTrapezoid

A trapezoid or trapezium is a quadrilateral two of whose sides are parallel to each other....
(Amer.): two oppositeOpposite

Opposite can mean:* An antonym* An inverse, usually the additive inverse....
sides are parallelParallel (geometry)

Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or pla...
.
Isosceles trapezium (Brit.) or isosceles trapezoidIsosceles trapezoid

An isosceles trapezoid is a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatica...
(Amer.): two opposite sides are parallel, the two other sides are of equal length, and the two ends of each parallel side have equal angleAngle

An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle....
s. This implies that the diagonals are of equal length.
TrapeziumTrapezium

The term trapezium can mean more than one thing:...
(Amer.): no sides are parallel.
ParallelogramParallelogram Overview

A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides....
: both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other.
KiteKite (geometry)

In geometry, a kite, or deltoid, is a quadrilateral with two pairs of equal adjacent sides....
: two adjacent sides are of equal length and the other two sides also of equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart or arrowhead.)
RhombusFacts About Rhombus

In geometry, a rhombus is a quadrilateral in which all of the sides are of equal length, i.e., it is an equilateral quadrang...
or rhomb: all four sides are of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
RhomboidRhomboid

In geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are oblique....
: a parallelogram in which adjacent sides are of unequal lengths and angles are obliqueOblique

Oblique can mean one of several things:...
(not right angles).
RectangleRectangle

In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles....
(or OblongOblong

Oblong may refer to:*A geometrical figure longer than wide, such as a non-square rectangle; more specifically, a figure whi...
): all four angles are right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length.
SquareSquare (geometry)

In plane geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides....
(regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are equal (equiangular), with each angle a right angle. This implies that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
Cyclic quadrilateralCyclic quadrilateral

External links* of Euclid's Elements* by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"....
: the four vertices lie on a circumscribed circle.
Tangential quadrilateralTangential quadrilateral

In geometry, a tangential quadrilateral is a convex quadrilateral whose sides all lie tangent to a single circle inscribed w...
: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
Bicentric quadrilateral: both cyclic and tangential.

More quadrilaterals

An geometric chevron [arrowhead] has bilateral symmetry like a kite, but the top concaves inwards.
A self-intersecting quadrilateral is called variously a cross-quadrilateral, butterfly quadrilateral or bow-tie quadrilateral.
An equiangular quadrilateral is a rectangleRectangle

In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles....
if convex, and an "angular eight" with corners on a rectangle if non-convex.
In solid geometrySolid geometry

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for pr...
, a quadrilateral whose vertices do not all lie in a flat planePlane (mathematics)

In mathematics, a plane is a fundamental two-dimensional object....
is a skew quadrilateral. Opposite sides in a skew quadrilateral are (segments of) skew linesSkew lines

In geometry, skew lines are two lines that do not intersect but are not parallel....
.

Taxonomy

A taxonomyTaxonomy

Taxonomy was once only the science of classifying living organisms , but later the word was applied in a wider sense, and m...
of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezium" here is referring to the British definition (the American equivalent is a trapezoid).

External links

at Geometry Atlas website.
and by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Analytic Geometry of Quadrilaterals
, and of Quadrilaterals from cut-the-knotCut-the-knot

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variet...
and from Mathopenref