Wiki article on this topic: In Euclidean geometry, a parallelogram is a simple 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 reflects the definition.

A simple 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:

The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

and the area of a single orange triangle is

Therefore, the area of the parallelogram is

Another area formula, for two sides B and C and angle θ, is

The area of a parallelogram with sides B and C and angle γ {\displaystyle \gamma } at the intersection of the diagonals is given by

When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D1 of either diagonal, then the area can be found from Heron's formula. Specifically it is

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 reflects the definition.

A simple 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:

The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

and the area of a single orange triangle is

Therefore, the area of the parallelogram is

Another area formula, for two sides B and C and angle θ, is

The area of a parallelogram with sides B and C and angle γ {\displaystyle \gamma } at the intersection of the diagonals is given by

When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D1 of either diagonal, then the area can be found from Heron's formula. Specifically it is

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