# parallelogram

Parallelogram of forces. Two forces act simultaneously on an object in different directions. The resultant force, and the direction of any subsequent movement, is defined by the diagonal of a parallelogram whose sides are drawn parallel to the applied forces with lengths proportional to their magnitudes. This is an application of vector diagrams.

A parallelogram is a quadrilateral (four-sided figure)
whose opposite sides are parallel, and whose opposite angles, therefore,
are equal. The diagonals of a parallelogram bisect each other. A parallelogram
of base *b* and height *h* has an area:

*A* = *bh* = *ab* sin*A* = *ab* sin*B*.

The height of a parallelogram is *h* = *a* sin*A* = *a* sin*B*. The sides *a*, *b*, *c*, and *d* and diagonals *p* and *q* of a parallelogram satisfy the equality:

*p *^{2} + *q *^{2} = *a*^{ 2} + *b *^{2} + *c*^{ 2} + *d*^{ 2}

Special cases of a parallelogram are a rhombus, which has sides of equal length, a rectangle, which has two sets of parallel sides that are perpendicular to each other, and a square, which meets the conditions of both a rectangle and a rhombus.

## Parallelogram of forces

If a particle is under the action of two forces, which are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, the resultant of the two forces is represented by the diagonal of the parallelogram drawn from that point.

## Parallelogram of velocities

If a body has two component velocities, represented in magnitude and direction by two adjacent sides of a parallelogram drawn from a point, the resultant velocity of the body is represented by the diagonal of the parallelogram drawn from that point. The parallelogram of velocities, like the parallelogram of forces, is a particular case of the parallelogram of vectors.

A familiar application of it is to find the resultant velocity of an object, such as a boat, that is subject to a current acting in a different direction in the one in which it is trying to move.

## Zonohedron

A zonohedron is a polyhedron in which the faces are all parallelograms or parallel-sided.
The faces of a zonohedron can be grouped into **zones** –
encircling bands of faces which share a common edge direction (and length).