Quadrilateral
From Wikipedia, the free encyclopedia
- 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.
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: 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.
[edit] The square as a rectangle
Some have defined categories exclusively, so that a rectangle is a quadrilateral with four right angles that is not a square.[citation needed] 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. A rectangle that is not a square is an oblong. In mathematics, it is important to define categories inclusively so that a square is special case of a rectangle. Inclusive categories make statements of theorems shorter by eliminating the need for tedious listing of cases
[edit] 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).
[edit] External links
- Eric W. Weisstein, Quadrilateral at MathWorld.
- Van Aubel's theorem and Varignon and Wittenbauer Parallelograms 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 and Interactive Classification of Quadrilaterals from cut-the-knot
- Definitions and examples of quadrilaterals and Definition and properties of tetragons from Mathopenref
- Interactive Quadrilaterals from Math Is Fun
| Polygons |
|---|
| Triangle • Quadrilateral • Pentagon •Hexagon • Heptagon • Octagon • Enneagon (Nonagon) • Decagon • Hendecagon • Dodecagon • Triskaidecagon • Pentadecagon • Hexadecagon • Heptadecagon • Enneadecagon • Icosagon • Chiliagon • Myriagon |
