Definition:Quadrilateral
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Definition
A quadrilateral (or tetragon) is a polygon with four sides.
As Euclid defined it:
- 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 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, the angles and sides of a square are all right angles:
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.
That is, both squares and oblongs are types of rectangle.
The word oblong is rarely seen nowadays; rectangle is the term usually used instead.
Containment
As Euclid defined it:
- Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.
(The Elements: Book II: Definition $1$)
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.
It can be shown that Opposite Sides and Angles of Parallelogram are Equal.
Thus a rectangle is a parallelogram all of whose angles are equal to a right angle.
Euclid, in Book II Definition 1: Containment of Rectangle, refers to this as a rectangular parallelogram.
Base
For a given parallelogram, one of the sides is distinguished as being the base. It is immaterial which is so chosen, but usual practice is that it is one of the two longer sides.
In the parallelogram above, line $AB$ is considered to be the base.
Altitude
An altitude of a parallelogram is a line drawn perpendicular to its base, through one of its vertices to the side opposite the base (which is extended if necessary).
In the diagram above, line $DE$ is an altitude of the parallelogram $ABCD$.
The term is also used for the length of such a line.
It follows that the altitude of a rectangle is equal to one of its sides adjacent to its base.
Rhombus
A rhombus or rhomb is a parallelogram whose sides are all the same length.
Its angles may or may not all be equal.
Thus a square is a rhombus all of whose angles are equal to a right angle.
Rhomboid
A rhomboid is a parallelogram whose sides are not all the same length.
Its angles may or may not all be equal.
Thus an oblong is a rhomboid all of whose angles are equal to a right angle.
Trapezoid
A trapezoid is a quadrilateral which has one pair of sides parallel.
Outside the US (one of a few countries that use this definition), a trapezoid is a quadrilateral with no parallel sides, that is, what the US defines as a trapezium.
Trapezium
A trapezium (plural: trapezia), otherwise known as an irregular quadrilateral, is a quadrilateral with no parallel sides.
Outside the US (one of a few countries that use this definition), a trapezium is a quadrilateral which has one pair of sides parallel, that is, what the US defines as a trapezoid.
Thus when such a quadrilateral is intended, it is probably better to use the term irregular quadrilateral instead.
Euclid, in his definitions, did not distinguish between trapezia and trapezoids.
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.
Some sources prefer to reserve the term dart for a reflex-angled trapezium which has both pairs of adjacent sides equal, e.g. $EFGH$ in the diagram above.
Euclid's Definitions
As Euclid defined it:
- 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.
