Quadrilateral

(Redirected from Trapezium (shape))
Jump to: navigation, search

In geometry, a quadrilateral is a polygon with four sides or edges and four vertices or corners. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistency with pentagon (5 sided), hexagon (6 sided) and so on. The interior angles of a quadrilateral add up to 360 degrees...

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

Convex quadrilaterals

Convex quadrilaterals are further classified as follows:

  • Trapezium (British English) or trapezoid (Amer.): two opposite sides are parallel.
  • Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): two opposite sides are parallel, the two other sides are of equal length, and the two ends of each parallel side have equal angles. This implies that the diagonals are of equal length.
  • Trapezium (Amer.): no sides are parallel.
  • Parallelogram: 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.
  • Kite: 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.)
  • Rhombus 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.
  • Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles).
  • Rectangle (or Oblong): 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.
  • Square (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 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.

File:Quadrilateral.png

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 rectangle if convex, and an "angular eight" with corners on a rectangle if non-convex.
  • In solid geometry, a quadrilateral whose vertices do not all lie in a flat plane is a skew quadrilateral. Opposite sides in a skew quadrilateral are (segments of) skew lines.

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 American equivalent is a trapezoid).

Taxonomy of quadrilaterals. Lower forms are special cases of higher forms.

External links

Template:Polygons2

ar:رباعي الأضلاع ast:Cuadriláteru bn:চতুর্ভুজ zh-min-nan:Sì-kak-hêng bs:Četverougao bg:Четириъгълник ca:Quadrilàter cv:Тăват кĕтеслĕх cs:Čtyřúhelník da:Firkant de:Viereck el:Τετράπλευρο eo:Kvarlatero gd:Ceithir-cheàrnach gl:Cuadrilátero zh-classical:四邊形 ko:사각형 hi:चतुर्भुज hr:Četverokut io:Quadrilatero it:Quadrilatero he:מרובע ka:ოთხკუთხედი ku:Çargoşe lv:Četrstūris lt:Keturkampis li:Veerhook hu:Négyszög mk:Четириаголник nl:Vierhoek no:Firkant qu:Tawak'uchu sl:Štirikotnik sr:Четвороугао sh:Četverokut fi:Nelikulmio sv:Fyrhörning ta:நாற்கரம் te:చతుర్భుజి th:รูปสี่เหลี่ยม uk:Чотирикутник vls:Vieroek


Linked-in.jpg