Definition:Quadrilateral

Definition

A quadrilateral is a polygon with exactly four sides.

In the words of Euclid:

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multi-lateral those contained by more than four straight lines.

(The Elements: Book $\text{I}$: Definition $19$)

Because it is a polygon, it follows that it also has four vertices.

Square

A square is a regular quadrilateral.

That is, a regular polygon with $4$ sides.

That is, a square is a plane figure with four sides all the same length and whose angles are all equal.

Oblong

An oblong is a quadrilateral whose angles are all right angles, but whose sides are not all the same length:

Rectangle

A rectangle is a quadrilateral all of whose angles are equal to a right angle, and whose sides may or may not all be the same length.

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel to each other, and whose sides may or may not all be the same length.

Rhombus

A rhombus is a parallelogram whose sides are all the same length.

Its angles may or may not all be equal.

Rhomboid

A rhomboid is a parallelogram whose sides are not all the same length.

Its angles may or may not all be equal.

Trapezoid

A trapezoid is a quadrilateral which has exactly one pair of sides parallel:

Trapezium

A trapezium is a quadrilateral with no parallel sides.

Further subclassifications

Various breeds of irregular quadrilateral are unofficially and informally recognised:

Kite

A kite is an irregular quadrilateral which has both pairs of adjacent sides equal.

Dart

A dart is an irregular quadrilateral with a reflex angle.

Also known as

A quadrilateral can also (rarely) be found referred to as a tetragon.

Also see

• Results about quadrilaterals can be found here.

Euclid's Definitions

In the words of Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

(The Elements: Book $\text{I}$: Definition $22$)

Linguistic Note

The word quadrilateral derives from the Latin for four sides.

Similarly, the word tetragon derives from the Greek for four sides.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): tetragon
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quadrilateral