As we learned in the Polygons SparkNote, quadrilaterals can be classified into
three basic groups:

Those with zero pairs of parallel sides.

Trapezoids--those with one pair of parallel
sides.

Parallelograms--those with two pairs of
parallel sides.

When calculating the area of a quadrilateral with no parallel sides, one
must divide the quadrilateral into two triangles
and sum their areas. Techniques for calculating the area of a triangle will be
discussed in the next lesson. When
the quadrilateral is a parallelogram or a trapezoid, however, there are
shortcuts for calculating area.

The Area of a Parallelogram

To calculate the area of a parallelogram, we must introduce a new term: the
altitude of a parallelogram. The altitude of a parallelogram is the
line
segment
perpendicular to one pair of sides with one
endpoint on one of these sides of the parallelogram, and the other endpoint on
the line containing the opposite
side of the parallelogram. Below appear the
altitudes of various parallelograms.

As you can see, one endpoint of an altitude always lies on a side, and the other
endpoint sometimes lies on the opposite side, and sometimes doesn't.

The area of a parallelogram is the product of the length of its altitude and the
length of the side that contains the endpoint of the altitude. This side is
called the base of the parallelogram. Any side can become a base of a given
parallelogram: all you need to do is draw an altitude from it to the line that
contains the opposite side. A common way to describe the area of a
parallelogram is the base times the height (base x height), where the height is
the altitude.

The formulas for the area of various special parallelograms are even simpler.
For a rectangle, the area is the product of the
length of any two adjacent sides. In any such
case, one side is the base, and the other side is the altitude. The area of a
square is the length of any side (all four are
congruent) squared. The area of a
rhombus is equal to one-half the product of its
diagonals. Note that these special formulae are
simply modified versions of the original base-times-height formula given for the
area of a parallelogram.

The Area of a Trapezoid

The formula for the area of a trapezoid also
requires new vocabulary. The altitude of a trapezoid is the
segment
perpendicular to the bases with one endpoint on a base of the trapezoid, and the
other endpoint on the line that contains the other base. Here are various
trapezoids pictured with their altitudes:

The area of a trapezoid equals half the product of the sum of the length of its
bases and the length of its altitude((1/2(b_{1} + b_{2})) x A).
Half the sum of its bases is the average length of its bases, which is equal to
the length of its median. So another way to find
the area of a trapezoid is to calculate the product of the lengths of its
altitude and median.