# 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): Entry:**tetragon** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**quadrilateral**