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

1 Classification

Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave. Convex quadrilaterals are further classified as follows:

• Trapezoid ( British English) or trapezium ( American English): no sides are parallel.
• Trapezium (Brit.) or trapezoid (Amer.): one pair of opposite sides is parallel.
• Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): 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.
• KiteA separate article is about kite flying. In geometry, a kite is a type of quadrilateral, a four-sided polygon. It has two pairs of equal sides, and the two sides in each pair are adjacent. contrast with the parallelogram, where the sides of each pair are: 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.
• RhombusIn geometry, a rhombus (also known as a rhomb is a parallelogram in which all of the sides are of equal length. More colloquially it may be described as a diamond or lozenge shape. In any rhombus, opposite sides will be parallel. Thus, the rhombus is a sp: four sides have equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
• RectangleIn geometry, a rectangle is a defined as a quadrilateral polygon in which all four angles are right angles. From this definition, it follows that a rectangle has two pairs of opposite sides of equal length; that is, a rectangle is a parallelogram. A squar: 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.
• SquareIn plane geometry, a square is a polygon with four equal sides and equal angles. Those angles are then necessarily right angles. Squares are regular quadrilaterals, rectangles, rhombi, kites, parallelograms, and isosceles trapezoids/isosceles trapezia. (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.

2 Is a square a rectangle?

Some people define categories exclusively, so that a rectangle is a quadrilateral with four right angles that is not a square. This is appropriate for everyday use of the words, as people typically use the less specific word only when the more specific word will not do.

But in mathematics, it is important to define categories inclusively, so that a square is a rectangle. Inclusive categories make statements of theorems shorter, by eliminating the need for tedious listing of cases. For example, the visual proof that vector addition is commutative is known as the "parallelogram diagram". If categories were exclusive it would have to be known as the "parallelogram (or rectangle or rhombus or square) diagram"!