Parallelograms

Parallelogram
This parallelogram is a rhomboid as it has no right angles and unequal sides.
Type	quadrilateral
Edges and vertices	4
Symmetry group	C2, [2]+, (22)
Area	b × h (base × height); ab sin θ (product of adjacent sides and sine of any vertex angle)
Properties	convex

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.

The three-dimensional counterpart of a parallelogram is a parallelepiped.

The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.

A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:

Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.

All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms:

A parallelogram with base b and height h can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in the figure to the left. This means that the area of a parallelogram is the same as that of a rectangle with the same base and height: