geometry, a parallelogram is a quadrilateralwith two sets of parallelsides. The oppositesides 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.
* The area, , of a parallelogram is , where is the base of the parallelogram and is its height.
* 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 productof two adjacentsides.
diagonals of a parallelogram bisect each other.
* Opposite sides of a parallelogram are equal.
* Opposite angles of a parallelogram are equal.
* Each diagonal bisects the parallelogram into two congruent triangles.
* It is possible to create a
tessellationof a plane with any parallelogram.
The properties of having equal opposite sides and opposite angles are shared with the
antiparallelogram, a type of non-convex quadrilateral in which the two longer edges cross each other.
Computing the area of a parallelogram
Let and let denote the matrix with columns and . Then the area of the parallelogram generated by and is equal to
Let and let . Then the area of the parallelogram generated by and is equal to
Let . 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::
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:
Since they are angles that a transversal makes with parallel lines and .
Also, since they are a pair of vertical angles.
Therefore, since they have the same angles.
From this similarity, we have the ratios
Since , we have:.
bisects the diagonals and .
You can also prove that the diagonals bisect each other, by placing the parallelogram on a coordinate grid, and assign variables to the vertexes, you can show that the diagonals have the same midpoint.
Derivation of the area formula
The area formula,
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
and the area of a single orange triangle is
Therefore, the area of the parallelogram is
An alternative, less mathematically sophisticated method, to show the area is by rearrangement of the perimeter.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.
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 .
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
Parallelogram of force
* [http://www.elsy.at/kurse/index.php?kurs=Parallelogram+and+Rhombus&status=public Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)]
* [http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/index.php Interactive Parallelogram --sides, angles and slope]
* [http://www.cut-the-knot.org/Curriculum/Geometry/AreaOfParallelogram.shtml Area of Parallelogram] at
* [http://www.cut-the-knot.org/Curriculum/Geometry/EquiTriOnPara.shtml Equilateral Triangles On Sides of a Parallelogram] at
* [http://agutie.homestead.com/files/VarigWitten.htm Varignon and Wittenbauer Parallelograms] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
* [http://agutie.homestead.com/files/vanaubel.html Van Aubel's theorem] Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
* [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=2623&CurriculumID=24 Parallelogram Quiz]
* [http://www.mathopenref.com/parallelogram.html Definition and properties of a parallelogram] with animated applet
* [http://www.mathopenref.com/parallelogramarea.html Interactive applet showing parallelogram area calculation] interactive applet
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