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