Quadrilateral

A quadrilateral is a four-sided polygon. This article describes its types, core properties, historical names, common theorems and practical uses in geometry and design.

Overview

A quadrilateral is a polygon with four sides and four vertices. In the context of Euclidean plane geometry it is one of the simplest closed shapes after triangles. Other historic names include quadrangle and tetragon; these terms emphasize its four-sided nature in the same family as triangles, pentagons and hexagons. A quadrilateral with vertices labeled A, B, C and D is often denoted by the symbol ◻ ABCD. $A$

Basic properties and classification

All quadrilaterals made of straight sides satisfy a few elementary properties. The internal angles of any non-self-intersecting quadrilateral sum to 360°: ∠A + ∠B + ∠C + ∠D = 360°. A quadrilateral may be simple (its edges meet only at their endpoints) or complex (self-intersecting, also called crossed). Simple quadrilaterals are further split into convex and concave depending on whether every internal angle is less than 180°.

Important structural elements include the four sides, the four interior angles and the two diagonals. The diagonals and their lengths and intersection point determine many special properties: for example, whether opposite angles sum to 180° or whether diagonals bisect each other.

Common types and special cases

Some named classes of quadrilaterals arise by imposing symmetry or parallelism conditions. Notable examples are listed below.

Square — four equal sides and four right angles.
Rectangle — opposite sides equal and four right angles.
Rhombus — four equal sides; angles need not be right angles.
Parallelogram — opposite sides are parallel and equal in length.
Kite — two distinct pairs of adjacent equal sides; one diagonal often acts as an axis of symmetry.
Trapezoid (or trapezium in some dialects) — at least one pair of parallel sides.

Technically, squares, rectangles and rhombi are all specialised parallelograms. $B$ $C$

Theorems and notable facts

Several classical results characterize important quadrilateral subclasses. For a cyclic quadrilateral (one inscribed in a circle) opposite angles are supplementary; such figures obey Ptolemy's relation between side lengths and diagonals and their area can be computed by Brahmagupta's formula when side lengths are known. For any quadrilateral, Varignon's theorem states that the midpoints of the four sides form a parallelogram. Many of these properties are central to geometry problems and constructions. $D$

History, terminology and notation

The word quadrilateral derives from Latin roots meaning "four" and "side." Over time different cultures and mathematical traditions emphasized particular subclasses: Euclid studied parallelograms and rectangles, while later work on circle geometry and cyclic quadrilaterals produced specialized formulas. Notationally, the order of vertices matters: the sequence A–B–C–D specifies which sides connect consecutive vertices.

Uses, examples and applications

Quadrilaterals appear widely in architecture, engineering, computer graphics and tiling. Every simple quadrilateral can tile the plane by repeated rotations about edge midpoints, a property exploited in decorative patterns and tessellation design. In computer graphics, quadrilaterals (often split into triangles) are basic elements for mesh modeling and surface approximation. Practical problems frequently reduce to identifying whether a shape is convex, cyclic, tangential (has an inscribed circle), or satisfies the parallelism conditions of parallelograms. $\square ABCD$ $\angle A+\angle B+\angle C+\angle D=360^{\circ }.$

For further reading on the geometry and classification of four-sided polygons consult introductory geometry texts or the linked topics: polygon, tiling and tessellation, and specialized entries on each named quadrilateral class above.