Deleng sumberé Quadrilateral

{{Infobox Polygon
| name        = Quadrilateral
| image       = Six Quadrilaterals.svg
| caption     = Enem wangun quadrilateral
| edges       = 4
| schläfli    = {4} (for square)
| area        = various methods;<br />[[#Area of a convex quadrilateral|see below]]
| angle       = 90° (for square)
}}
Sajeroning [[géomètri]] ([[Géomètri Euclidean|Euclidean]], '''quadrilateral''' utawa '''quarangle''' utawa '''tetragon''' iku sawijining wangun segi akèh [[polygon]] kanthi papat '''sisi''' (utawa 'rusuk') lan papat '''pojok'''.  Tembung '''quadrilateral''' asal saka tembung '''quad''' (tegesé "papat") lan '''lateral''' (tegesé "sisih").

Quadrilateral iku poligon sing bisa awujud prasaja utawa komplèks. Quadrilateral bisa awujud [[convex polygon|convex]] utawa [[concave polygon|concave]].
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The [[Internal and external angle|interior angles]] of a simple quadrilateral add up to 360 [[degrees of arc]]. This is a special case of the ''n''-gon interior angle sum formula (''n'' - 2)×180°. In a crossed quadrilateral, the interior angles on either side of the crossing add up to 720°.<ref>[http://mysite.mweb.co.za/residents/profmd/stars.pdf Stars: A Second Look]</ref>

All convex quadrilaterals [[tessellation|tile the plane]] by repeated rotation around the midpoints of their edges.

==Convex quadrilaterals - parallelograms==
A [[parallelogram]] is 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 also include the square, rectangle, rhombus and rhomboid.

* [[Rhombus]] or rhomb: all four sides are of equal length. Equivalent conditions are that opposite sides are parallel and opposite angles are equal, or that the diagonals perpendicularly bisect each other. An informal description is "a pushed-over square" (including a square).
* [[Rhomboid]]: a parallelogram in which adjacent sides are of unequal lengths and angles are [[Angle#Types of angles|oblique]] (not right angles). Informally: "a pushed-over rectangle with no right angles."
* [[Rectangle]]: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Informally: "a box or oblong" (including a square).
* [[Square (geometry)|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).
* [[rectangle|Oblong]]: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).

==Convex quadrilaterals - other ==

* [[Kite (geometry)|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. (It is common, especially in the discussions on plane [[tessellations]], to refer to the concave quadrilateral with these properties as a ''dart'' or ''arrowhead'', with term kite being restricted to the convex shape.)
* [[Orthodiagonal quadrilateral]]: the diagonals cross at [[right angle]]s.
* [[Trapezoid|Trapezium]] ([[British English]]) or [[trapezoid]] ([[American English]]): one pair of [[opposite]] sides are [[parallel (geometry)|parallel]].
* [[Isosceles trapezium]] (Brit.) or [[isosceles trapezoid]] (NAm.): one pair of opposite sides are parallel and the base [[angle]]s are equal in measure. 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.  A quadrilateral is cyclic if and only if opposite angles sum to 180°.
* [[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.
[[Image:Quadrilaterals.svg]]

==More quadrilaterals==

*A geometric chevron (dart or arrowhead) is a [[Concave polygon|concave]] quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex.
*A self-intersecting quadrilateral is called variously a '''cross-quadrilateral''', '''crossed quadrilateral''', '''[[butterfly]] quadrilateral''' or '''[[bow-tie]] quadrilateral'''.
*A non-planar quadrilateral is called a '''skew quadrilateral'''. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as [[cyclobutane]] that contain a "puckered" ring of four atoms.<ref>M.P. Barnett and J.F. Capitani, ''Modular chemical geometry and symbolic calculation'',
International Journal of Quantum Chemistry, 106 (1) 215--227, 2006.</ref> See [[skew polygon]] for more.

==Area of a convex quadrilateral==

There are various general formulas for the area of a convex quadrilateral.

The area of a quadrilateral ''ABCD'' can be calculated using [[Vector (geometric)|vectors]]. Let vectors ''AC'' and ''BD'' form the diagonals from ''A'' to ''C'' and from ''B'' to ''D''. The area of the quadrilateral is then
:<math>Area = \frac{1}{2} |{AC}\times{BD}|,</math>

which is the magnitude of the [[cross product]] of vectors ''AC'' and ''BD''. In two-dimensional Euclidean space, expressing vector ''AC'' as a [[Euclidean vector#In Cartesian space|free vector in Cartesian space]] equal to (''x''<sub>1</sub>,''y''<sub>1</sub>) and ''BD'' as (''x''<sub>2</sub>,''y''<sub>2</sub>), this can be rewritten as:
:<math>Area = \frac{1}{2} |x_1 y_2 - x_2 y_1|.</math>

The area can be expressed in trigonometric terms as
:<math>Area = \frac{1}{2} pq \cdot \sin \theta,</math>

where the lengths of the diagonals are p and q and the angle between them is <math>\theta</math>.<ref>Harries, J. "Area of a quadrilateral," ''Mathematical Gazette'' 86, July 2002, 310-311.</ref> In the case of an orthodiagonal quadrilateral e.g. rhombus, square, and kite, this formula reduces to <math>\tfrac{1}{2}pq</math> since <math>\theta</math> is 90°.

[[Bretschneider's formula]]<ref>R. A. Johnson, ''Advanced Euclidean Geometry'', 2007, Dover Publ., p. 82.</ref> expresses the area in terms of the sides and angles:
:<math>
\begin{align}
Area &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{2} abcd \; [ 1 + \cos (\gamma + \lambda) ]} \\
&= \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd \left[ \cos^2 \left( \tfrac{\gamma + \lambda}{2} \right) \right]} \\
s &= \frac{1}{2} (a+b+c+d), \\
\end{align}
</math>

where the sides in sequence are <math>a,b,c,d</math>, <math>s</math> is the semiperimeter, and <math>\gamma</math> and <math>\lambda</math> are any two opposite angles.  This reduces to [[Brahmagupta's formula]] for the area of a cyclic quadrilateral when <math> \gamma + \lambda</math> = 180°.

Another area formula in terms of the sides and angles, with <math>\gamma</math> being between sides ''b'' and ''c'' and <math>\lambda</math> being between sides ''a'' and ''d'' (adjacent sides belonged to the angles), is
:<math>Area = \frac{1}{2}bc \cdot \sin \gamma + \frac{1}{2}ad \cdot \sin \lambda.</math>

In the case of a cyclic quadrilateral, the latter formula becomes
:<math>Area = \frac{1}{2}(bc+ad)\sin \gamma.</math>

And also for a parallelogram, since both the opposite sides and angles are equal, its final formula reduces to <math>ab \cdot \sin \gamma.</math>

Next,<ref>{{cite web |author=E. W. Weisstein |title=Bretschneider's formula |url= http://mathworld.wolfram.com/Bretschneider'sFormula.html |publisher=MathWorld -- A Wolfram Web Resource }}</ref> the following formula expresses the area in terms of the sides and diagonals:
:<math>
\begin{align}
Area &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)} \\
&= \frac{1}{4} \sqrt{4p^{2}q^{2}- \left( a^{2}+c^{2}-b^{2}-d^{2} \right) ^{2}}, \\
\end{align}
</math>

where ''p'' and ''q'' are the diagonals.  Again, this reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then <math>pq=ac+bd</math>.

Alternatively, we can write the area in terms of the sides and the intersection angle <math>\theta</math> of the diagonals, so long as this angle is not 90°:<ref name=Mitchell>Mitchell, Douglas W., "The area of a quadrilateral," ''Mathematical Gazette'' 93, July 2009, 306-309.</ref>
:<math>Area = \frac{|\tan \theta|}{4} \cdot \left| a^2 + c^2 - b^2 - d^2 \right|.</math>

In the case of a parallelogram, the latter formula becomes
:<math>Area = \frac{|\tan \theta|}{2} \cdot \left| a^2 - b^2 \right|.</math>

==Miscellaneous facts about special quadrilaterals==
* The diagonals of a crossed or concave quadrilateral do not intersect inside the shape.
* The diagonals of a rhombus bisect the angles.
* Let ''ABCD'' be a trapezoid (in the U.S. sense of having two parallel sides) with vertices ''A'', ''B'', ''C'', and ''D'' in sequence and with parallel sides ''AB'' and ''DC''.  Let ''E'' be the intersection of the diagonals, and let ''F'' be on side ''DA'' and ''G'' be on side ''BC'' such that ''FEG'' is parallel to ''AB'' and ''CD''.  Then ''FG'' is the [[harmonic mean]] of ''AB'' and ''DC'':
*: <math>\frac{1}{FG}=\frac{1}{2} \left( \frac{1}{AB}+ \frac{1}{DC} \right).</math>
* A parallelogram with equal diagonals is a rectangle.
* A cyclic quadrilateral with successive sides ''a, b, c, d'' and diagonals ''p, q'' has ''pq=ac+bd''.
* A cyclic quadrilateral with successive vertices ''A, B, C, D'' and successive sides ''a=AB, b=BC, c=CD,'' and ''d=DA'' and with diagonals ''p=AC'' and ''q=BD'' has:
*: <math>\frac {p}{q}= \frac{ad+cb}{ab+cd},</math>
*: <math>p^{2}= \frac{(ac+bd)(ad+bc)}{ab+cd},</math> and
*: <math>q^{2}= \frac{(ac+bd)(ab+dc)}{ad+bc}.</math>
* A cyclic quadrilateral with successive sides ''a, b, c, d'' and semiperimeter ''s'' has circumradius (the radius of the circumscribing circle) given by<ref>Hoehn, Larry, "Circumradius of a cyclic quadrilateral," ''Mathematical Gazette'' 84, March 2000, 69-70.</ref>
*: <math>\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.</math>
* A parallelogram with diagonals ''p, q'' and successive sides ''a, b, c,'' and ''d'' with ''d=b'' and ''c=a'' has
*: <math>p^{2}+q^{2} = a^{2}+b^{2}+c^{2}+d^{2}.</math>
* For any point ''P'' in the interior of a rectangle with successive vertices ''A, B, C, D,'' we have
*: <math>(AP)^{2}+(CP)^{2}=(BP)^{2}+(DP)^{2}.</math>
* Any line through the midpoint of a parallelogram bisects the area.
* An [[orthodiagonal quadrilateral]] (one with perpendicular diagonals) with sides ''a, b, c, d'' in sequence has<ref name=Mitchell/><ref name=Altshiller-Court/>{{rp|p.136}} <math>a^{2}+c^{2} = b^{2}+d^{2}.</math>
* There are no cyclic quadrilaterals with unequal rational sides in arithmetic progression and with rational area.<ref name=Buchholz>Buchholz, R. H., and MacDougall, J. A. "Heron quadrilaterals with sides in arithmetic or geometric progression", ''Bull. Austral. Math. Soc.'' 59 (1999), 263-269. http://journals.cambridge.org/article_S0004972700032883</ref>
* There are no cyclic quadrilaterals with unequal rational sides in geometric progression and with rational area.<ref name=Buchholz/>

==Miscellaneous facts about quadrilaterals in general==
* The length of the diagonal that is opposite to the adjacent sides ''a'' and ''b'' at angle θ is given by <math>\sqrt{ a^2 + b^2 - 2ab \cos \theta }</math> which is derived from the [[law of cosines]].
* The midpoints of the sides of a quadrilateral are the vertices of a parallelogram.  The area of this inner parallelogram equals one-half the area of the outer quadrilateral.  The perimeter of the inner parallelogram equals the sum of the diagonals of the outer quadrilateral.
* Let exterior squares be drawn on all sides of a quadrilateral. The segments connecting the centers of opposite squares are (a) equal in length, and (b) perpendicular.
* The line segment joining the midpoints of two opposite sides of any quadrilateral, the segment joining the midpoints of the other two sides, and the segment joining the midpoints of the diagonals are [[Concurrent lines|concurrent]] and are all bisected by their point of intersection.<ref name=Altshiller-Court>Altshiller-Court, Nathan, ''College Geometry'', Dover Publ., 2007.</ref>{{rp|p.125}}
* The sum of the squares of the diagonals of any quadrilateral equals twice the sum of the squares of the two line segments joining the midpoints of the two pairs of opposite sides.<ref name=Altshiller-Court/>{{rp|p.126}}
* The internal bisectors of the angles of a quadrilateral form a cyclic quadrilateral.<ref name=Altshiller-Court/>{{rp|p.127}}

==Taksonomi==
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'').  Inclusive definitions are used throughout.
-->
[[Gambar:Quadrilateral hierarchy.png|Taxonomy of quadrilaterals. Lower forms are special cases of higher forms.]]

== Rujukan ==
{{reflist|2}}

== Pranala njaba ==
{{Commons category|Tetragons}}
* {{MathWorld | urlname=Quadrilateral | title=Quadrilateral}}
*[http://www.vias.org/comp_geometry/geom_quad_general.html Compendium Geometry] Analytic Geometry of Quadrilaterals
*[http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml Quadrilaterals Formed by Perpendicular Bisectors], [http://www.cut-the-knot.org/Curriculum/Geometry/ProjectiveQuadri.shtml Projective Collinearity] and  [http://www.cut-the-knot.org/Curriculum/Geometry/Quadrilaterals.shtml Interactive Classification] of Quadrilaterals from [[cut-the-knot]]
*[http://www.mathopenref.com/tocs/quadrilateraltoc.html Definitions and examples of quadrilaterals] and [http://www.mathopenref.com/tetragon.html Definition and properties of tetragons] from Mathopenref
* [http://math.kennesaw.edu/~mdevilli/quad-tree-web.html A (dynamic) Hierarchical Quadrilateral Tree] {{Webarchive|url=https://web.archive.org/web/20110718202031/http://math.kennesaw.edu/~mdevilli/quad-tree-web.html |date=2011-07-18 }} at [http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm Dynamic Geometry Sketches] {{Webarchive|url=https://web.archive.org/web/20090321024112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm |date=2009-03-21 }}
* [http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf An extended classification of quadrilaterals] {{Webarchive|url=https://web.archive.org/web/20191230004754/http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf |date=2019-12-30 }} at [http://mysite.mweb.co.za/residents/profmd/homepage4.html Dynamic Math Learning Homepage] {{Webarchive|url=https://web.archive.org/web/20180825150046/http://mysite.mweb.co.za/residents/profmd/homepage4.html |date=2018-08-25 }}
*[http://mzone.mweb.co.za/residents/profmd/classify.pdf The role and function of a hierarchical classification of quadrilaterals] {{Webarchive|url=https://web.archive.org/web/20110719175018/http://mzone.mweb.co.za/residents/profmd/classify.pdf |date=2011-07-19 }} by Michael de Villiers

{{Poligon}}

[[Kategori:Quadrilateral| ]]
[[Kategori:Poligon]]