The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Parallelogram and Rhombus

A parallelogram is a quadrilateral with parallel opposite sides. As a consequence of the parallel sides the opposite angles are equal:

α = γ, and β = δ.

A parallelogram with equal sides (a = b) is called a rhombus, a parallelogram whose angles are all right angles is called a rectangle.
The following table lists the most important parameters of a parallelogram and a rhombus:

  Parallelogram Rhombus
Circumference 2(a+b) 4a
Area aha = absin α = bhb a2sinα = d1d2/2
Diagonal d1 = 2acos(α/2)
d2 = 2asin(α/2)
Circumcircle Radius - -
Incircle Radius - asin2(α/2)

The diagonals of a parallelogram bisect each other, the diagonals of a rhombus intersect at a right angle. The diagonals and the sides of a parallelogram are related by the following equation:

d12 + d22 = 2(a2 + b2)

If the sides a and b of a parallelogram are given as vectors u = AB and v = AD, then the area of the parallelogram can be calculated by the cross product of the two vectors:

A = uxv

Last Update: 2011-01-11