parallelogram

Dictionary: par·al·lel·o·gram (păr'ə-lĕl'ə-grăm') pronunciation

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parallelogram

(Academy Artworks)

n.
A four-sided plane figure with opposite sides parallel.

[Late Latin parallēlogrammum, from Greek parallēlogrammon, from neuter sing. of parallēlogrammos, bounded by parallel lines : parallēlos, parallel; see parallel + grammē, line.]

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Sports Science and Medicine: parallelogram

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A plane four-sided rectilinear figure with opposite sides parallel. The opposite sides and angles of a parallelogram are equal; the diagonals bisect each other and the parallelogram itself.

Columbia Encyclopedia: parallelogram

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parallelogram, closed plane figure bounded by four line segments, or sides, with opposite pairs of sides parallel and equal in length. The rhombus, rectangle, and square are special types of parallelograms. Any side of a parallelogram is a base; an altitude is the perpendicular distance from a base to the opposite parallel side. The area of a parallelogram is equal to the product of the lengths of its base and altitude. The diagonals of a parallelogram, connecting opposite vertices, bisect one another; either diagonal divides the parallelogram into two congruent triangles.

Word Tutor: parallelogram

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IN BRIEF: A figure having four sides, with the opposite sides set apart at an equal distance and of equal length.

The student drew a parallelogram on the chalkboard when asked to show a shape that is a quadrilateral.

Wikipedia: Parallelogram

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This article is about the quadrilateral shape. For the album by Linda Perhacs, see Parallelograms (album).

Parallelogram
This parallelogram is a rhomboid as its angles are oblique.
Type	Quadrilateral
Edges and vertices	4
Symmetry group	C₂ (2)

In geometry, a parallelogram is a 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 equal. The three-dimensional counterpart of a parallelogram is a parallelepiped.

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

1 Properties
2 Types of parallelograms
3 Proof that diagonals bisect each other
4 The area formula
5 Computing the area of a parallelogram
6 See also
7 External links

Properties

Opposite sides of a parallelogram are equal in length.
Opposite angles of a parallelogram are equal in measure.
The area, $A$ , of a parallelogram is $A = b h$ , where $b$ is the base of the parallelogram and $h$ is its height.
Opposite sides of a parallelogram 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.
The diagonals of a parallelogram bisect each other.
Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
There is an infinite number of affine transformations which take any given parallelogram to a square.'

Types of parallelograms

Rhomboid - A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles
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 four angles of equal size (right angles).

Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:

$\angle ABE \cong \angle CDE$ (alternate interior angles are equal in measure)

$\angle BAE \cong \angle DCE$ (alternate interior angles are equal in measure).

(since these are angles that a transversal makes with parallel lines $A B$ and $D C$ ).

Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.

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

Therefore,

A E = C E

B E = D E .

Since the diagonals $A C$ and $B D$ divide each other into segments of equal length, the diagonals bisect each other.

In addition, the diagonals $A C$ and $B D$ are each bisected by point $E$ . Therefore, point $E$ is the midpoint of each diagonal.

The area formula

Area of the parallelogram is in blue

The area formula,

$A = B \times H,\,$

can be derived as follows:

The area 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

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

and the area of a single orange triangle is

$A_\text{tri} = \frac{1}{2} A \times H\,$ or $S_\text{tri} = \frac{1}{2} bh.$

Therefore, the area of the parallelogram is

$A = A_\text{rect} - 2 \times A_\text{tri} = \left( (B+A) \times H \right) - \left( A \times H \right) = B \times H.\,$

Computing the area of a parallelogram

Let $a,b\in\R^2$ and let $V=[a\ b]\in\R^{2\times2}$ denote the matrix with columns $a$ and $b$ . Then the area of the parallelogram generated by $a$ and $b$ is equal to $| det(V) |$

Let $a,b\in\R^n$ and let $V=[a\ b]\in\R^{n\times2}$ Then the area of the parallelogram generated by $a$ and $b$ is equal to $\sqrt{\det(V^T V)}$

Let $a,b,c\in\R^2$ . Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

$V = \left| \det \begin{bmatrix} a_1 & a_2 & 1 \\ b_1 & b_2 & 1 \\ c_1 & c_2 & 1 \end{bmatrix} \right|.$

External links

Wikimedia Commons has media related to category:
Parallelogram

Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
Weisstein, Eric W., "Parallelogram" from MathWorld.
Interactive Parallelogram --sides, angles and slope
Area of Parallelogram at cut-the-knot
Equilateral Triangles On Sides of a Parallelogram at cut-the-knot
Definition and properties of a parallelogram with animated applet
Interactive applet showing parallelogram area calculation interactive applet

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Translations: Parallelogram

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Dansk (Danish)
n. - parallellogram

Nederlands (Dutch)
parallellogram

Français (French)
n. - parallélogramme

Deutsch (German)
n. - Parallelogramm

Ελληνική (Greek)
n. - παραλληλόγραμμο

Italiano (Italian)
parallelogramma

Português (Portuguese)
n. - paralelogramo (m)

Русский (Russian)
параллелограмм

Español (Spanish)
n. - paralelogramo

Svenska (Swedish)
n. - parallellogram

中文（简体）(Chinese (Simplified))
平行四边形

中文（繁體）(Chinese (Traditional))
n. - 平行四邊形

한국어 (Korean)
n. - 평행사변형

日本語 (Japanese)
n. - 平行四辺形

العربيه (Arabic)
‏(الاسم) متوازي أضلاع‏

עברית (Hebrew)
n. - ‮מקבילית‬

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Best of the Web: parallelogram

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Some good "parallelogram" pages on the web:

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mathworld.wolfram.com

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	parallelogram condenser
	rhomboid (mathematics)

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Sports Science and Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved. Read more
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parallelogram

Contents

Properties

Types of parallelograms

Proof that diagonals bisect each other

The area formula

Computing the area of a parallelogram

See also

External links