(Redirected from Parallelograms)


Typequadrilateral, trapezium
Edges and vertices4
Symmetry groupC2, [2]+,
Areab × h (base × height);
ab sin θ (product of adjacent sides and sine of the vertex angle determined by them)


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

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

Special casesEdit

  • Rectangle – A parallelogram with four angles of equal size (right angles).
  • Rhombus – A parallelogram with four sides of equal length. Any parallelogram that is neither a rectangle nor a rhombus was traditionally called a rhomboid but this term is not used in modern mathematics.[1]



  • 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]


Other propertiesEdit

  • Any line through the midpoint of a parallelogram bisects the area.[6]

Area formulaEdit

Animation for the area formula  .






and the area of a single orange triangle is


Therefore, the area of the parallelogram is








where   and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into two congruent triangles.

Area in terms of Cartesian coordinates of verticesEdit





Proof that diagonals bisect each otherEdit


  (alternate interior angles are equal in measure)
  (alternate interior angles are equal in measure).



Therefore, triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).





Lattice of parallelogramsEdit

Parallelograms can tile the plane by translation. If edges are equal, or angles are right, the symmetry of the lattice is higher. These represent the four Bravais lattices in 2 dimensions.

Form Square Rectangle Rhombus Parallelogram
System Square
Centered rectangular
Constraints α=90°, a=b α=90° a=b None
Symmetry p4m, [4,4], order 8n pmm, [∞,2,∞], order 4n p1, [∞+,2,∞+], order 2n

Parallelograms arising from other figuresEdit

Proof without words of Varignon's theorem:
  1. An arbitrary quadrilateral and its diagonals.
  2. Bases of similar triangles are parallel to the blue diagonal.
  3. Ditto for the red diagonal.
  4. The base pairs form a parallelogram with half the area of the quadrilateral, Aq, as the sum of the areas of the four large triangles, Al is 2 Aq (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, As is a quarter of Al (half linear dimensions yields quarter area), and the area of the parallelogram is Aq minus As.

Automedian triangleEdit


Varignon parallelogramEdit


Tangent parallelogram of an ellipseEdit



Faces of a parallelepipedEdit


See alsoEdit


  1. ^ "CIMT - Page no longer available at Plymouth University servers" (PDF). Archived from the original (PDF) on 2014-05-14.
  5. ^ Problem 5, 2006 British Mathematical Olympiad, [1].
  7. ^ Weisstein, Eric W. "Triangle Circumscribing". Wolfram Math World.

External linksEdit