Quadrilateral

Quadrilateral
Six Quadrilaterals.svg
Some types of quadrilaterals
Edges and vertices 4
Schläfli symbol {4} (for square)
Area various methods;
see below
Internal angle ( degrees) 90° (for square and rectangle)

In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.

The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side".

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

The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is

This is a special case of the n-gon interior angle sum formula (n − 2) × 180°.

All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

Simple quadrilaterals

Any quadrilateral that is not self-intersecting is a simple quadrilateral.

Convex quadrilaterals

Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.

In a convex quadrilateral, all interior angles are less than 180° and the two diagonals both lie inside the quadrilateral.

  • Irregular quadrilateral ( British English) or trapezium ( North American English): no sides are parallel. (In British English this was once called a trapezoid.)
  • Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel. Trapezia (UK) and trapezoids (US) include parallelograms.
  • Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
  • Parallelogram: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles we call squares) and rhomboids (including those rectangles we call oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
  • Rhombus or rhomb: all four sides are of equal length. An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).
  • Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and some angles are oblique (equivalently, having no right angles). Informally: "a pushed-over oblong". Not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. [1]
  • Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
  • Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent 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 (four equal sides and four equal angles).
  • Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square). [2]
  • Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.

Quadrilaterals.svg

  • Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
  • Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle.
  • Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
  • Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
  • Bicentric quadrilateral: it is both tangential and cyclic.
  • Orthodiagonal quadrilateral: the diagonals cross at right angles.
  • Equidiagonal quadrilateral: the diagonals are of equal length.
  • Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
  • An equilic quadrilateral has two opposite equal sides that, when extended, meet at 60°.
  • A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length. [3]
  • A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square. [4]
  • A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle. [5]
  • A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. [6]

Concave quadrilaterals

In a concave quadrilateral, one interior angle is bigger than 180° and one of the two diagonals lies outside the quadrilateral.

  • A dart (or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex. See (concave) kite.
Other Languages
Afrikaans: Vierhoek
অসমীয়া: চতুৰ্ভুজ
asturianu: Cuadriláteru
azərbaycanca: Dördbucaqlı
বাংলা: চতুর্ভুজ
Bân-lâm-gú: Sì-kak-hêng
беларуская: Чатырохвугольнік
беларуская (тарашкевіца)‎: Чатырохкутнік
български: Четириъгълник
bosanski: Četverougao
brezhoneg: Pevarc'hostezeg
català: Quadrilàter
čeština: Čtyřúhelník
chiShona: Gonyoina
dansk: Firkant
Deutsch: Viereck
eesti: Nelinurk
Ελληνικά: Τετράπλευρο
español: Cuadrilátero
Esperanto: Kvarlatero
euskara: Lauki
فارسی: چهارضلعی
français: Quadrilatère
한국어: 사각형
Հայերեն: Քառանկյուն
हिन्दी: चतुर्भुज
hrvatski: Četverokut
íslenska: Ferhyrningur
italiano: Quadrilatero
עברית: מרובע
Basa Jawa: Quadrilateral
ქართული: ოთხკუთხედი
қазақша: Төртбұрыш
Kiswahili: Pembenne
Kurdî: Çargoşe
latviešu: Četrstūris
lietuvių: Keturkampis
Limburgs: Veerhook
magyar: Négyszög
македонски: Четириаголник
മലയാളം: ചതുർഭുജം
मराठी: चौकोन
Bahasa Melayu: Segi empat
မြန်မာဘာသာ: စတုဂံ
Nederlands: Vierhoek
नेपाली: चतुर्भुज
नेपाल भाषा: प्यकुं
日本語: 四角形
norsk: Firkant
norsk nynorsk: Firkant
occitan: Quadrilatèr
ଓଡ଼ିଆ: ଚତୁର୍ଭୁଜ
oʻzbekcha/ўзбекча: Toʻrtburchak
ਪੰਜਾਬੀ: ਚੁਬਾਹੀਆ
ភាសាខ្មែរ: ចតុកោណ
polski: Czworokąt
português: Quadrilátero
română: Patrulater
Runa Simi: Tawak'uchu
sámegiella: Njealječiegat
Simple English: Quadrilateral
slovenčina: Štvoruholník
slovenščina: Štirikotnik
کوردی: چوارلا
српски / srpski: Четвороугао
srpskohrvatski / српскохрватски: Četverokut
suomi: Nelikulmio
svenska: Fyrhörning
தமிழ்: நாற்கரம்
తెలుగు: చతుర్భుజి
Türkçe: Dörtgen
українська: Чотирикутник
Tiếng Việt: Tứ giác
文言: 四邊形
West-Vlams: Vieroek
粵語: 四邊形
中文: 四邊形