Quadrilateral

In geometry, a quadrilateral is a polygon with four sides and four vertices.

Classifications

Quadrilaterals, and polygons in general, are broadly divided into two groups: convex and concave polygons. (A quadrilaterial, like a general polygon can be self-intersecting, for example, an hour-glass figure, and then would be classified as complex and be neither convex nor concave)

Convex quadrilaterals are further classified as follows:

• Trapezoid (Amer. Eng.) or trapezium (Brit. Eng.): one pair of opposite sides is parallel.
• Trapezium (Amer. Eng.) or trapezoid (Brit. Eng.): no sides are parallel.
• Isosceles trapezium (Brit. Eng.) or isosceles trapezoid (Amer. Eng.): two of the opposite sides are parallel, the two other sides are equal, and the two ends of each parallel side have equal angles. This implies that the diagonals have equal length.
• Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides have equal length, opposite angles are equal, and the diagonals bisect each other.
• Kite: two adjacent sides have equal length, the other two sides have equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other.
• Rhombus: four sides have equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
• Rectangle: each angle is a right angle. This implies that opposite sides are parallel and have equal length, and the diagonals bisect each other and have equal length.
• Square (regular quadrilateral): four sides have equal length, and each angle is a right angle. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length.

Taxonomic classification

The taxonomic classification of quadrillaterals is illustrated by the following graph:

Lower forms are special cases of higher forms.

                          Polygon
                             |
                       Quadrilaterial
                         /       \
                      simple     complex
                       /  \
                  convex  concave 
                    /          
            Trapezium/oid [Am/Eng]
             /        \
            /          \
           /         Trapezoid/ium [Am/Eng]
          /            /     \
         Kite         /       \
          \    Parallelogram  Isosceles Trapezoid/ium [Am/Eng]
           \    /     \       /
            \  /       \     /
           Rhombus   Rectangle
              \        /
               \      /
                Square

This graph shows some of the shapes are special cases of more general shapes, with all the specific shapes also being either a trapezium or a trapezoid:

Square: Rectangle, rhombus, kite, parallelogram, isosceles trapezium/oid.

Rectangle: Parallelogram, isosceles trapezium/oid.

Rhombus: Kite, parallelogram.

Quadrilateral is also the name of a group of fortresses in Italy and of the southern part of Dobrudja, ceded by Romania to Bulgaria in the opening month of World War II (see Romania during World War II).