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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

Quadrilaterals by symmetry
  • 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.
  • Adjacent angles are supplementary.
  • 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.
  • Adjacent angles are supplementary.
  • 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:

    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

    and the area of a single orange triangle is

    Therefore, the area of the parallelogram is