In geometry, a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .
Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees, that is
This is a special case of the n-gon interior angle sum formula: S = (n − 2) × 180° (here, n=4).
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 quadrilateral
In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral. Irregular quadrilateral: no sides are parallel. 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 called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles. Rhombus, rhomb: all four sides are of equal length (equilateral). 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 (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus. Rectangle: all four angles are right angles (equiangular). 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), 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 (i.e., four equal sides and four equal angles). Oblong: longer than wide, or wider than long (i.e., a rectangle that is not a square). 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. It is a type of tangential quadrilateral.
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 cyclic kite, and thus a type of bicentric quadrilateral. Harmonic quadrilateral: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal. Bicentric quadrilateral: it is both tangential and cyclic. Orthodiagonal quadrilateral: the diagonals cross at right angles. Equidiagonal quadrilateral: the diagonals are of equal length. Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram. 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. A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square. A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle. A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.
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 where one interior angle is reflex. See Kite.
Complex quadrilaterals
A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all on the right as the figure is traced out) add up to 720°.
Crossed trapezoid (US) or trapezium (Commonwealth): a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a trapezoid). Antiparallelogram: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a parallelogram). Crossed rectangle: an antiparallelogram whose sides are two opposite sides and the two diagonals of a rectangle, hence having one pair of parallel opposite sides. Crossed square: a special case of a crossed rectangle where two of the sides intersect at right angles.
Special line segments The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices.
The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral (see below).
The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.
Area of a convex quadrilateral There are various general formulas for the area of a convex quadrilateral ABCD with sides .
Trigonometric formulas The area can be expressed in trigonometric terms as
where the lengths of the diagonals are and and the angle between them is .Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311. In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to since is .
The area can be also expressed in terms of bimedians as
where the lengths of the bimedians are and and the angle between them is .
Bretschneider's formulaR. A. Johnson, Advanced Euclidean Geometry, 2007, Dover Publ., p. 82. expresses the area in terms of the sides and two opposite angles:
where the sides in sequence are , , , , where is the semiperimeter, and and are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral—when .
Another area formula in terms of the sides and angles, with angle being between sides and , and being between sides and , is
In the case of a cyclic quadrilateral, the latter formula becomes
In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to
Alternatively, we can write the area in terms of the sides and the intersection angle of the diagonals, as long as is not :Mitchell, Douglas W., "The area of a quadrilateral," Mathematical Gazette 93, July 2009, 306–309.
In the case of a parallelogram, the latter formula becomes
Another area formula including the sides , , , is.
where is the distance between the midpoints of the diagonals, and is the angle between the bimedians.
The last trigonometric area formula including the sides , , , and the angle (between and ) is:
which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle ), by just changing the first sign to .
Non-trigonometric formulas The following two formulas express the area in terms of the sides , , and , the semiperimeter , and the diagonals , :
J. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", American Mathematical Monthly, 46 (1939) 345–347.
The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then .
The area can also be expressed in terms of the bimedians , and the diagonals , :
Archibald, R. C., "The Area of a Quadrilateral", American Mathematical Monthly, 29 (1922) pp. 29–36.
.
In fact, any three of the four values , , , and suffice for determination of the area, since in any quadrilateral the four values are related by The corresponding expressions are:Josefsson, Martin (2016) '100.31 Heron-like formulas for quadrilaterals', The Mathematical Gazette, 100 (549), pp. 505–508.
if the lengths of two bimedians and one diagonal are given, and
if the lengths of two diagonals and one bimedian are given.
Vector formulas The area of a quadrilateral can be calculated using vectors. Let vectors and form the diagonals from to and from to . The area of the quadrilateral is then
which is half the magnitude of the cross product of vectors and . In two-dimensional Euclidean space, expressing vector as a free vector in Cartesian space equal to and as , this can be rewritten as:
Diagonals Properties of the diagonals in quadrilaterals In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length. The list applies to the most general cases, and excludes named subsets.
Quadrilateral Bisecting diagonals Perpendicular diagonals Equal diagonals Trapezoid See note 1 Isosceles trapezoid See note 1 Right trapezoid See note 3 See note 1 Parallelogram Kite See note 2 See note 2 Rectangle Rhombus Square
Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral. Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).
Lengths of the diagonals The lengths of the diagonals in a convex quadrilateral ABCD can be calculated using the law of cosines on each triangle formed by one diagonal and two sides of the quadrilateral. Thus
and
Other, more symmetric formulas for the lengths of the diagonals, areRashid, M. A. & Ajibade, A. O., "Two conditions for a quadrilateral to be cyclic expressed in terms of the lengths of its sides", Int. J. Math. Educ. Sci. Technol., vol. 34 (2003) no. 5, pp. 739–799.
and
Generalizations of the parallelogram law and Ptolemy's theorem