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 (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad and lateral. Quad means four and lateral means sides. The interior angles of a quadrilateral add up to 360 degrees of arc.

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave.

Convex quadrilaterals - parallelograms
A parallelogram is a quadrilateral with two sets of parallel sides. Equal conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.


 * Rhombus or rhomb: all four sides are of equal length. Equal conditions are that opposite sides are parallel and opposite angles are equal, or that the diagonals perpendicularly bisect each other. "A pushed-over square."
 * Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). "A pushed-over rectangle."
 * Rectangle: all four angles are right angles. Equal conditions are that opposite sides are parallel and of equal length, or that the diagonals bisect each other and are equal in length.
 * Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equal condition is that opposite sides are parallel (a square is a parallelogram), 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.
 * Oblong: a term sometimes used to denote a rectangle which has unequal sides (i.e. a rectangle that is not a square).


 * Rhombus (four equal sides) + Rectangle (four equal angles) = Square (four equal sides and four equal angles) → Parallelogram (opposite sides are parallel) → Quadrilateral (four-sided polygon)

Convex quadrilaterals - other

 * Kite: two adjacent sides are of equal length and the other two sides also of equal length. This implies that the angles between the two pairs of congruent sides are equal, and also implies that the diagonals are perpedicular. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart or arrowhead.)


 * Trapezium (British English) or trapezoid (NAm.): two opposite sides are parallel.
 * Isosceles trapezium (Brit.) or isosceles trapezoid (NAm.): two opposite sides are parallel and the base angles are congruent. This implies that the other two sides are of equal length, and that the diagonals are of equal length. An alternative definition is a quadrilateral with an axis of symmetry bisecting one pair of opposite sides.
 * Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
 * 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.

More quadrilaterals

 * A 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.
 * The area can be computed using Brahmagupta's formula.
 * A non-planar quadrilateral is khjkbjbjbhbjbhjhbjbjhbjbjhbmmcalled a skew quadrilateral.

Taxonomy
A taxonomy 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 North American equivalent is a trapezoid), and "kite" excludes the concave kite (arrowhead or dart).