Quadrilateral: Difference between revisions

For other uses, see Quadrilateral (disambiguation)

In geometry, a quadrilateral is a polygon with four sides and four vertices. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistence with pentagon.

Classification

Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave. Convex quadrilaterals are further classified as follows:

• Trapezium (American English): no sides are parallel.
• Trapezium (British English) or trapezoid (Amer.): one pair of opposite sides is parallel.
• Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): two of the opposite sides are parallel, the two other sides are equal, and the two ends of each parallel side have equal angles. This implies that the diagonals have equal length.
• Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides have equal length, opposite angles are equal, and the diagonals bisect each other.
• Kite: two adjacent sides have equal length, the other two sides have 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.)
• Rhombus: four sides have equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
• Rhomboid: In geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles).
• Rectangle(or Oblong): each angle is a right angle. This implies that opposite sides are parallel and have equal length, and the diagonals bisect each other and have equal length.
• Square (regular quadrilateral): four sides have equal length, and each angle is a right angle. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length.
• Cyclic quadrilateral: the four vertices lie on a circumscribed circle.
• Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
• Bicentric quadrilateral: both cyclic and tangential.

One special kind of concave quadrilateral is an arrowhead, which is like a kite, but the top concaves inwards.

Is a square a rectangle?

Some people define categories exclusively, so that a rectangle is a quadrilateral with four right angles that is not a square. This is appropriate for everyday use of the words, as people typically use the less specific word only when the more specific word will not do. Generally a rectangle which isn't a square is an oblong.

But in mathematics, it is important to define categories inclusively, so that a square is a rectangle. Inclusive categories make statements of theorems shorter, by eliminating the need for tedious listing of cases. For example, the visual proof that vector addition is commutative is known as the "parallelogram diagram". If categories were exclusive it would have to be known as the "parallelogram (or rectangle or rhombus or square) diagram"!

Compare the question of whether a real number is a complex number.

Taxonomic classification

The taxonomic classification 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

• Varignon and Wittenbauer Parallelograms by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Van Aubel's theorem Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Compendium Geometry Analytic Geometry of Quadrilaterals
• Quadrilaterals Formed by Perpendicular Bisectors
• Projective Collinearity in a Quadrilateral
• Definitions and examples of quadrilaterals With interactive animations
• Definition and properties of tetragons With interactive animations
• Quadrilateral. by Eric W. Weisstein (MathWorld -- A Wolfram Web Resource.)
• Classification of Quadrilaterals (Interactive Classification from cut-the-knot)