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The following formulas give the area of a **general quadrilateral**
(see Figure 1, left, for the notation).

**Figure 1:** Left: notation for a general quadrilateral; in addition
*s*=½(*a*+*b*+*c*+*d*). Right: a parallelogram.

Often, however, it is easiest to compute the area by dividing the quadrilateral into triangles. One can also divide into triangles to compute one side given the other sides and angles, etc.

More formulas can be given for
**special cases** of quadrilaterals.
In a **parallelogram** opposite sides are parallel
(Figure 1, right). It follows that
opposite sides have the same length and that two consecutive angles
add up to 180°. In the notation of the figure, we have

(All this follows from the triangle formulas applied to the triangles
*ABD* and *ABC*.)

Two particular cases of parallelograms are:

- the
**rectangle**, where all angles equal 90°. The diagonals of a rectangle have the same length. The general formulas for parallelograms reduce to*h*=*a*

area=*ab*

*p*=*q*=. - the
**rhombus**or**diamond**, where adjacent sides have the same length (*a=b*). The diagonals of a rhombus are perpendicular. In addition to the general formulas for parallelograms we have area=½*pq*and*p*+*q*=4*a*.

A quadrilateral is a **trapezoid** if two sides are parallel.
In the notation of the figure below we have

*A*+*D*=*B*+*C*=180°,

area=½(*AB*+*CD*)*h*.

A quadrilateral is **cyclic** if it can be inscribed in a circle,
that is, if its four vertices belong to a single, circumscribed,
circle. This is possible if and only if the sum of opposite angles is
180°. If *R* is the radius of the circumscribed circle, we
have (in the notation of Figure 1, left):

A quadrilateral is **circumscribable** if it has an inscribed circle
(that is, a circle tangent to all four sides). Its area is *rs*,
where *r* is the radius of the inscribed circle and *s* is as above.

For a quadrilateral that is both cyclic and circumscribable we have
the following additional equalities, where *m* is the distance between
the centers of the inscribed and circumscribed circles:

Additional fact about quadrilaterals:

- The diagonals of a quadrilateral with consecutive sides
*a*,*b*,*c*,*d*are perpendicular if and only if*a*+*c*=*b*+*d*.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.