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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.

## Special cases

• Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles[1]
• Rectangle – A parallelogram with four angles of equal size (right angles).
• Rhombus – A parallelogram with four sides of equal length.
• Square – A parallelogram with four sides of equal length and angles of equal size (right angles).

## Characterizations

A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]

• Two pairs of opposite sides are parallel (by definition).
• Two pairs of opposite sides are equal in length.
• Two pairs of opposite angles are equal in measure.
• The diagonals bisect each other.
• One pair of opposite sides is parallel and equal in length.
• Each diagonal divides the quadrilateral into two congruent triangles.
• The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
• It has rotational symmetry of order 2.
• The sum of the distances from any interior point to the sides is independent of the location of the point.[4] (This is an extension of Viviani's theorem.)
• There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.[5]

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

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:[2][3]

• Two pairs of opposite sides are parallel (by definition).
• Two pairs of opposite sides are equal in length.
• Two pairs of opposite angles are equal in measure.
• The diagonals bisect each other.
• One pair of opposite sides is parallel and equal in length.
• Each diagonal divides the quadrilateral into two congruent triangles.
• The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
• It has rotational symmetry of order 2.
• The sum of the distances from any interior point to the sides is independent of the location of the point.[4] (This is an extension of Viviani's theorem.)
• There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.[5]

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.

## Other properties

• Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• Any line through the midpoint of a parallelogram bisects the area.[6]
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
• A parallelogram has rotational symmetry of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a square.
• The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides.
• Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.[7]
• The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.[8]
• If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area.[8]
• The diagonals of a parallelogram divide it into four triangles of equal area.

## Area formula

conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.

## Other properties

• Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• Any line through the midpoint of a parallelogram bisects the area.[6]
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
• A parallelogram has rotational symmetry of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a square.
• The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent

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:

${\displaystyle K=bh.}$
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram

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

${\displaystyle K_{\text{rect}}=(B+A)\times H\,}$

and the area of a single orange triangle is

${\displaystyle K_{\text{tri}}={\frac {1}{2}}A\times H.\,}$

Therefore, the area of the parallelogram is