Quadrilaterals include a square [A] with right-angles and all its sides equal and parallel; a rectangle [B] with only opposite sides equal; a trapezium [C] with only two opposite sides parallel; a rhombus [D] and a parallelogram [E], both with opposite sides parallel and no right-angles.
A quadrilateral is a polygon that has four sides and four vertices (corners). Quadrilaterals, and polygons in general, may be convex or concave. A convex quadrilateral may be further classified as a trapezoid or a British trapezium (one pair of opposite sides are parallel), a trapezium (no sides parallel); an isosceles trapezoid (U.S.) or an isosceles trapezium (U.K.) (two of the opposite sides parallel, the two other sides equal, and the two ends of each parallel side of equal angles); a parallelogram (opposite sides are parallel); a kite (two adjacent sides of equal length, the other two sides of equal length); a rhombus (four sides of equal length); a rectangle (each angle is a right angle); or a square (four sides of equal length, each angle a right angle). Quadrangular prisms and quadrangular pyramids are ones whose bases are quadrilateral.
Ptolemy's theorem states that the sum of the products of the two pairs of opposite sides of a convex cyclic quadrilateral is equal to the product of the lengths of the diagonals. It is named after the mathematician, astronomer, and geographer Ptolemy of Alexandria.