Quadrilateral

In geometry a quadrilateral is a four‑sided polygon , having four edges (sides) and four corners (vertices). The term is derived from the Latin words quadri (a variant of four) and latus (meaning “side”). It is also called a tetragon, from the Greek tetra (“four”) and gon (“corner” or “angle”), following the pattern of other polygons such as pentagon . Because gon denotes “angle”, the figure is sometimes referred to as a quadrangle or “4‑angle”. A quadrilateral with vertices

◻
A
◻
B
◻
C
◻
D

is sometimes denoted as

◻ A
◻ B
◻ C
◻ D

◻ABCD◻.[1]

Quadrilaterals may be either simple (not self‑intersecting) or complex (self‑intersecting, or crossed). Simple quadrilaterals are further classified as either convex or concave.

The sum of the interior angles of a simple (and planar ) quadrilateral ABCD equals 360 degrees , i.e.

∠ A + ∠ B + ∠ C + ∠ D = 360∘

∘

∠ A + ∠ B + ∠ C + ∠ D = 360∘

∘

∠ A + ∠ B + ∠ C + ∠ D = 360∘

∘

This relationship is a special case of the general formula for the interior‑angle sum of an n-gon:

S = (n − 2) × 180°,

where n = 4.[2]

All non‑self‑crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.[3]

Simple quadrilaterals

Any quadrilateral that is not self‑intersecting is termed a simple quadrilateral.

Convex quadrilateral

In a convex quadrilateral every interior angle is less than 180°, and both diagonals lie entirely inside the figure. The accompanying Euler diagram illustrates several common types of simple quadrilaterals (British English uses “trapezium”; American English uses “trapezoid”).

A hierarchical Hasse diagram visualizes the relationships among the various classes of convex quadrilaterals.

Classification

• Irregular quadrilateral – no sides are parallel.
• Trapezoid (UK) or trapezoid (US): at least one pair of opposite sides are parallel . All trapezia/trapezoids include parallelograms as a special case.
• Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel and the base angles are congruent. Equivalent characterizations include: an axis of symmetry bisecting a pair of opposite sides, or a trapezoid whose diagonals are equal in length.
• Parallelogram : a quadrilateral with two pairs of parallel sides. Equivalent properties are: opposite sides equal in length, opposite angles equal, or diagonals that bisect each other. Parallelograms encompass all rhombi (including squares) and all rhomboids (including oblongs). In other words, every rhombus and every rhomboid is a parallelogram, and every rectangle is a parallelogram.
• Rhombus , or rhomb: all four sides have equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other. Informally, a rhombus is a “pushed‑over square,” though it formally includes the square itself.
• Rhomboid : a parallelogram where adjacent sides are of unequal length and some angles are oblique (i.e., not right angles). Informally, it is a “pushed‑over oblong.” Some sources restrict the term to parallelograms that are not rhombi.[4]
• Rectangle : all four interior angles are right angles (equiangular). Equivalently, the diagonals bisect each other and are equal in length. Rectangles include squares and oblongs. Informally, a rectangle is a “box” or “oblong,” encompassing the square.
• Square (regular quadrilateral): all sides are equal and all angles are right angles. Equivalently, it is both a rhombus and a rectangle; its diagonals perpendicularly bisect each other and are equal. A quadrilateral is a square iff it satisfies both the rhombus and rectangle conditions.
• Oblong: a rectangle whose length exceeds its width (or vice‑versa).[5]
• Kite : two distinct pairs of adjacent sides are equal. This forces one diagonal to bisect the other at right angles and creates two congruent triangles. All rhombi are kites, but not all kites are rhombi.

Additional notable subclasses include:

• Tangential quadrilateral : the four sides are tangent to an inscribed circle. A convex quadrilateral is tangential iff opposite sides have equal sums.
• Tangential trapezoid : a trapezoid that is also tangential.
• Cyclic quadrilateral : the four vertices lie on a common circumscribed circle . A convex quadrilateral is cyclic iff opposite angles sum to 180°.
• Right kite : a kite that possesses two opposite right angles; it is automatically cyclic.
• Harmonic quadrilateral : a cyclic quadrilateral for which the product of the lengths of opposite sides are equal.
• Bicentric quadrilateral : a quadrilateral that is simultaneously tangential and cyclic.
• Orthodiagonal quadrilateral : the diagonals intersect at right angles.
• Equidiagonal quadrilateral : the diagonals are of equal length.
• Bisect‑diagonal quadrilateral: one diagonal bisects the other into equal segments; every dart and kite possesses this property. When both diagonals bisect each other, the figure is a parallelogram.
• Ex‑tangential quadrilateral : the extensions of the four sides are tangent to an excircle.
• Equilic quadrilateral: two opposite sides are equal, and when extended they meet at a 60° angle.
• Watt quadrilateral: a quadrilateral possessing a pair of opposite sides of equal length.[6]
• Quadric quadrilateral: a convex quadrilateral whose four vertices all lie on the perimeter of a square.[7]
• Diametric quadrilateral: a cyclic quadrilateral that has one side serving as a diameter of its circumcircle.[8]
• Hjelmslev quadrilateral: a quadrilateral with two right angles at opposite vertices.[9]

Concave quadrilaterals

In a concave quadrilateral one interior angle exceeds 180°, and exactly one of the two diagonals lies outside the figure.

• Dart (or arrowhead): a concave quadrilateral with bilateral symmetry akin to a kite, but featuring a reflex interior angle. See the entry on Kite .

Complex quadrilaterals

A self‑intersecting quadrilateral is variously called a cross‑quadrilateral, crossed quadrilateral, butterfly quadrilateral, or bow‑tie quadrilateral. In such a figure the four “interior” angles on either side of the crossing (two acute and two reflex) together sum to 720°.[10]

• Crossed trapezoid (US) or trapezium (Commonwealth): a crossed quadrilateral with one pair of nonadjacent sides parallel, analogous to a conventional trapezoid.[11]
• Antiparallelogram : a crossed quadrilateral where each pair of nonadjacent sides are equal in length, mirroring the property of a parallelogram.
• Crossed rectangle : a special case of an antiparallelogram formed from two opposite sides and the two diagonals of a rectangle, thereby possessing one pair of parallel opposite sides.
• Crossed square : a particular instance of a crossed rectangle where two sides intersect at right angles.

Special line segments

• The diagonals of a convex quadrilateral are the line segments that join opposite vertices.
• The bimedians are the segments that connect the midpoints of opposite sides; they intersect at the vertex centroid of the quadrilateral.[12]
• The maltitudes are the perpendiculars drawn to a side through the midpoint of the opposite side.[13]

Area of a convex quadrilateral

Several formulas exist for the area K of a convex quadrilateral ABCD with side lengths

a = AB, b = BC, c = CD, d = DA.

Trigonometric formulas

The area can be expressed using the diagonals p and q and the angle θ between them:

K = ½ p q sin θ.[15]

When the quadrilateral is orthodiagonal (θ = 90°), this reduces to

K = ½ p q.

The area may also be written in terms of the bimedians m and n and the angle φ between them:

K = m n sin φ.[16]

Bretschneider’s formula provides a general expression in terms of the side lengths and two opposite angles (A and C):

K = √{(s − a)(s − b)(s − c)(s − d) − ½ a b c d [1 + cos(A + C)]},

where s is the semiperimeter. This simplifies to Brahmagupta’s formula for cyclic quadrilaterals when A + C = 180°.[17]

An alternative formula, using sides a, d and angle A, and sides b, c and angle C, is

K = ½ a d sin A + ½ b c sin C.[14]

For a cyclic quadrilateral this becomes

K = ½ (ad + bc) sin A.

In a parallelogram, where opposite sides and angles are equal, the formula collapses to

K = a b sin A.

Another expression, valid when the diagonals are not perpendicular, relates the area to the intersection angle θ of the diagonals:

K = ¼ |tan θ| |a² + c² − b² − d²|.[18]

For a parallelogram this reduces to

K = ½ |tan θ| |a² − b²|.

A further formula involving the side lengths and the distance x between the midpoints of the diagonals is

K = ½ √{[(a² + c²) − 2x²][(b² + d²) − 2x²]} sin φ.[19]

Non‑trigonometric formulas

The area can also be written using the side lengths, the semiperimeter s, and the diagonals p and q:

K = √{(s − a)(s − b)(s − c)(s − d) − ¼ (ac + bd + pq)(ac + bd − pq)}.[20]

or

K = ¼ √{4p²q² − (a² + c² − b² − d²)²}.[21]

The first expression reduces to Brahmagupta’s formula for cyclic quadrilaterals because then pq = ac + bd.

The area may also be expressed via the bimedians m, n and the diagonals p, q:

K = ½ √{(m + n + p)(m + n − p)(m + n + q)(m + n − q)}.[22]

or

K = ½ √{p²q² − (m² − n²)²}.[23]

A related identity holds for any quadrilateral:

p² + q² = 2(m² + n²).[24]

Vector formulas

If vectors AC and BD represent the diagonals, the area equals half the magnitude of their cross product:

K = ½ |AC × BD|.[54]

In Cartesian coordinates, with AC = (x₁, y₁) and BD = (x₂, y₂), this becomes

K = ½ |x₁y₂ − x₂y₁|.

Diagonals

The lengths of the diagonals p and q of a convex quadrilateral ABCD can be computed using the law of cosines on the constituent triangles:

p = √{a² + b² − 2ab cos B} = √{c² + d² − 2cd cos D},

q = √{a² + d² − 2ad cos A} = √{b² + c² − 2bc cos C}.

More symmetric expressions are

p = √{[(ac + bd)(ad + bc) − 2abcd(cos B + cos D)] /(ab + cd)},

q = √{[(ab + cd)(ac + bd) − 2abcd(cos A + cos C)] /(ad + bc)}.

Other metric relations

If X and Y are the feet of the perpendiculars from B and D to diagonal AC = p in a convex quadrilateral ABCD with sides a, b, c, d, then

XY = |a² + c² − b² − d²| / (2p).

When the diagonals intersect at E, with AE = e, BE = f, CE = g, DE = h, the following identity holds:

efgh (a + c + b + d)(a + c − b − d)
= (agh + cef + beh + dfg)(agh + cef − beh − dfg).[30]

Angle bisectors

The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral (the four intersection points of adjacent bisectors are concyclic) or they are concurrent. In the concurrent case the quadrilateral is tangential.

If the bisectors of angles A and C meet on diagonal BD, then the bisectors of B and D meet on diagonal AC.[31]

Bimedians

The segment joining the midpoints of opposite sides is a bimedian. The two bimedians intersect at the centroid of the vertices.

• By Varignon’s theorem, the four side‑midpoints of any quadrilateral form a parallelogram (the Varignon parallelogram). Its opposite sides are parallel to the diagonals of the original quadrilateral, each side being half the length of that diagonal, and its area is exactly half the area of the original quadrilateral. The perimeter of the Varignon parallelogram equals the sum of the lengths of the two diagonals of the original figure.

• The lengths of the bimedians are

m = ½ √{−a² + b² − c² + d² + p² + q²},

n = ½ √{a² − b² + c² − d² + p² + q²},

where p and q are the diagonal lengths. Consequently

p² + q² = 2(m² + n²).

• The bimedians are equal in length iff the diagonals are perpendicular; they are perpendicular iff the diagonals are equal.

Trigonometric identities

For a simple quadrilateral ABCD with interior angles A, B, C, D the following identities hold:

sin A + sin B + sin C + sin D
= 4 sin½(A + B) sin½(A + C) sin½(A + D).

and

[(tan A tan B − tan C tan D) / (tan A tan C − tan B tan D)]
= tan(A + C) / tan(A + B).

Additionally,

(tan A + tan B + tan C + tan D) / (cot A + cot B + cot C + cot D)
= tan A tan B tan C tan D.

These formulas are valid provided no angle equals 90°, where the tangent or cotangent would be undefined.

Inequalities

Area bounds

For a convex quadrilateral with consecutive sides a, b, c, d and diagonals p, q the area K satisfies several classical inequalities:

• K ≤ ¼ (a + c)(b + d), with equality only for a rectangle.
• K ≤ ¼ (a² + b² + c² + d²), with equality only for a square.
• K ≤ ¼ (p² + q²), with equality only when the diagonals are equal and perpendicular.
• K ≤ ½ √{(a² + c²)(b² + d²)}, with equality only for a rectangle.

From Bretschneider’s formula it follows that

K ≤ √{(s − a)(s − b)(s − c)(s − d)},

with equality iff the quadrilateral is cyclic or degenerate (collapsed to a line segment).

Also

K ≤ √{abcd},

with equality for a bicentric or rectangular quadrilateral.

A more refined inequality, due to an 18‑th‑century result, states

K ≤ ½ √[3]{(ab + cd)(ac + bd)(ad + bc)}.

If the perimeter is L, then

K ≤ L² / 16,

with equality only for a square.

Finally, K ≤ ½ pq, with equality precisely when the diagonals are perpendicular.

Diagonal and side relations

• The sum of the squares of the four sides always exceeds one third the square of any single side: a² + b² + c² > ⅓ d².
• Likewise, a⁴ + b⁴ + c⁴ ≥ ⅟₂₇ d⁴.

Maximal and minimal properties

• Among all quadrilaterals with a given perimeter, the square maximizes the area (the isoperimetric theorem for quadrilaterals).
• For fixed side lengths, the cyclic quadrilateral attains the maximal possible area.
• Of all convex quadrilaterals sharing given diagonals, the orthodiagonal quadrilateral has the greatest area; this follows from K = ½ pq sin θ ≤ ½ pq.

If P is an interior point of a convex quadrilateral ABCD, then

AP + BP + CP + DP ≥ AC + BD.

Thus the point minimizing the sum of distances to the vertices is the intersection of the diagonals, i.e., the quadrilateral’s Fermat point.[44]

Remarkable points and lines in a convex quadrilateral

The centroid can be defined in three distinct ways:

• Vertex centroid – the intersection of the two bimedians, obtained by treating the vertices as point masses.
• Side centroid – the centroid of the side‑midpoints.
• Area centroid – the centre of mass of the quadrilateral’s interior, found by dividing it into triangles and averaging their centroids.

These three points generally differ.

The Euler line of a quadrilateral is defined analogously to that of a triangle: the quasiorthocenter H, the area centroid G, and the quasicircumcenter O are collinear with HG = 2 GO.

Other notable constructs include:

• The Newton line, which connects the midpoints of the two diagonals; the vertex centroid bisects this segment in a 3 : 1 ratio.
• The Miquel point, a common point of four circles each passing through three vertices of the quadrilateral.
• The Pascal points, intersections of certain lines derived from a circle passing through the diagonal intersection and the extensions of opposite sides.

Other properties of convex quadrilaterals

• When external squares are erected on the sides of any quadrilateral, the segments joining the centers of opposite squares are equal in length and perpendicular—a theorem of Van Aubel.
• For any set of side lengths there exists a cyclic quadrilateral possessing those lengths.
• The four triangles formed by the intersection of the diagonals satisfy the relation that the product of the areas of opposite triangles are equal.
• The angle θ between the diagonals satisfies

cos θ = (b² + d² − a² − c²) / (2pq).

Taxonomy

A hierarchical taxonomy of quadrilaterals can be visualized with a Hasse diagram, where lower classes represent special cases of higher ones. The classification uses inclusive definitions (e.g., a square is simultaneously a rectangle, a rhombus, and a parallelogram).

Skew quadrilaterals

A non‑planar quadrilateral, termed a skew quadrilateral, occurs when the four vertices do not lie in a single plane. Such figures arise, for instance, in the geometry of molecules like cyclobutane. Historically they were called gauche quadrilaterals. A skew quadrilateral together with its diagonals forms a (generally irregular) tetrahedron, and conversely any tetrahedron yields a skew quadrilateral upon removal of a pair of opposite edges.

See also

• Complete quadrangle
• Perpendicular bisector construction of a quadrilateral
• Saccheri quadrilateral
• Types of mesh § Quadrilateral
• Quadrangle (geography)
• Homography – any quadrilateral can be transformed into another by a projective transformation.