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A Quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a Rhombus, and a parallelogram whose Angles are all Right Angles is called a Rectangle.

A parallelogram of base $b$ and height $h$ has Area

A = bh = ab\sin A=ab\sin B.
\end{displaymath} (1)

The height of a parallelogram is
h=a\sin A=a\sin B,
\end{displaymath} (2)

and the Diagonals are
$\displaystyle p$ $\textstyle =$ $\displaystyle \sqrt{a^2+b^2-2ab\cos A}$ (3)
$\displaystyle q$ $\textstyle =$ $\displaystyle \sqrt{a^2+b^2-2ab\cos B}$ (4)
  $\textstyle =$ $\displaystyle \sqrt{a^2+b^2+2ab\cos A}$ (5)

(Beyer 1987).

The Area of the parallelogram with sides formed by the Vectors $(a,c)$ and $(b,d)$ is

A={\rm det}\left({\left[{\matrix{a & b\cr c & d\cr}}\right]}\right)= \vert ad-bc\vert.
\end{displaymath} (6)

Given a parallelogram $P$ with area $A(P)$ and linear transformation $T$, the Area of $T(P)$ is
A(T(P)) = \left\vert\matrix{a & b\cr c & d\cr}\right\vert A(P).
\end{displaymath} (7)

\begin{figure}\begin{center}\BoxedEPSF{ParallelogramTheorem.epsf scaled 800}\end{center}\end{figure}

As shown by Euclid, if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in Area (and conversely), so in the above figure, $A_1=A_2$ (Johnson 1929).

See also Diamond, Lozenge, Parallelogram Illusion, Rectangle, Rhombus, Varignon Parallelogram, Wittenbauer's Parallelogram


Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 61, 1929.

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© 1996-9 Eric W. Weisstein