# Geometry: Measurements

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

1. Those with zero pairs of parallel sides.
2. Trapezoids--those with one pair of parallel sides.
3. 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.

Figure %: Various parallelograms and their altitudes
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:

Figure %: Various trapezoids and 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(b1 + b2)) 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.

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