NationMaster - Encyclopedia: Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped. Image File history File links Parallelogram. ... For other uses, see Geometry (disambiguation). ... This article is about the geometric shape. ... Parallel may refer to: Parallel (geometry) Parallel (latitude), an imaginary east-west line circling a globe Parallelism (grammar), a balance of two or more similar words, phrases, or clauses Parallel (manga), a shÅnen manga by Toshihiko Kobayashi Parallel (video), a video album by R.E.M. The Parallel, an... ... An example of congruence. ... In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek Ï€Î±ÏÎ±Î»Î»Î·Î»-ÎµÏ€Î¯Ï€ÎµÎ´Î¿Î½, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

1 Properties
2 Computing the area of a parallelogram
3 Proof that diagonals bisect each other
4 Derivation of the area formula
- 4.1 Alternate method
5 See also
6 External links

Properties

The two parallel sides are of equal length.
The area, $A$ , of a parallelogram is $A = B H$ , where $B$ is the base of the parallelogram and $H$ is its height.
The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
The area ia;sldks also equal to the magnitude of the vector cross product of two adjacent sides.
The diagonals of a parallelogram bisect each other.
It is possible to create a tessellation of a plane with any parallelogram.
The parallelogram is a special case of the trapezoid.
The rectangle is a special case of the parallelogram.
The rhombus is a special case of the parallelogram.

In mathematics, the cross product is a binary operation on vectors in three dimensions. ... Look up adjacent in Wiktionary, the free dictionary. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... For the bisection theorem, see ham sandwich theorem. ... A tessellated plane seen in street pavement. ... This article is about the geometric figure. ... A 5 by 4 rectangle In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... For other uses, see Rhombus (disambiguation). ...

Computing the area of a parallelogram

Let $a,binR^2$ and let $V=[a b]inR^{2times2}$ 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 $a,binR^n$ and let $V=[a b]inR^{ntimes2}$ . Then the area of the parallelogram generated by $a$ and $b$ is equal to $sqrt{det(V^T V)}$

Let $a,b,cinR^2$ , and let $V=[a b c]inR^{2times2}$ . Then the area of the parallelogram 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:

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

Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, first note a few pairs of equivalent angles: Image File history File links Parallelogram1. ...

$angle ABE cong angle CDE$

$angle BAE cong angle DCE$

Since they are angles that a transversal makes with parallel lines $A B$ and $D C$ . Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ...

Also, $angle AEB cong angle CED$ since they are a pair of vertical angles. Two lines intersect to create two pairs of vertical angles. ...

Therefore, $triangle ABE sim triangle CDE$ since they have the same angles.

From this similarity, we have the ratios // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...

${AB over CD} = {AE over CE} = {BE over DE}$

Since $A B = D C$ , we have

${AB over CD} = 1$ .

Therefore,

A E = C E

B E = D E

$E$ bisects the diagonals $A C$ and $B D$ . For the bisection theorem, see ham sandwich theorem. ...

Derivation of the area formula

Area of the parallelogram is in blue

The area formula, Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

$A_text{parallelogram} = B times H,,$

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

$A_text{tri} = frac{1}{2} A times H,$

Therefore, the area of the parallelogram is

$A_text{parallelogram} = A_text{rect} - 2 times A_text{tri} = left( (B+A) times H right) - left( A times H right) = B times H,$

Alternate method

Step one: ends of parallelogram are chopped off

Step two: pieces are rearranged

An alternative, less mathematically sophisticated method, to show the area is by rearrangement of the area. First, take the two ends of the parallelogram and chop them off to form two more triangles. Each of these two new triangles are equal in every way with the orange triangles. This first step is shown to the right. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

The second step is merely swap the left orange triangle with the right blue triangle. Clearly, the two blue triangles plus the blue rectangle have an area equivalent to $B H$ .

To further demonstrate this, the first image on the right could be printed off and cut up along the lines:

Cut along the lines between the orange triangles and the blue parallelogram
Cut along the vertical lines on the end to form the two blue triangles and the blue rectangle
Rearrange all five pieces as shown in the second image

External links

Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
Eric W. Weisstein, Parallelogram at 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
Varignon and Wittenbauer Parallelograms by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Van Aubel's theorem Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Parallelogram Quiz
Definition and properties of a parallelogram with animated applet
Interactive applet showing parallelogram area calculation interactive applet

Categories: Quadrilaterals

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

Results from FactBites:

Area of a Parallelogram (326 words)
	A parallelogram is a 4-sided shape formed by two pairs of parallel lines.
	The area of a parallelogram is 24 square centimeters and the base is 4 centimeters.
	The area of a parallelogram is 64 square inches and the height is 16 inches.

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