Parallelograms have three very important properties:

1. Opposite sides are equal.
2. Opposite angles are congruent.
3. Adjacent angles are supplementary (they add up to 180º).

    To visualize this last property, simply picture the opposite sides of the parallelogram as parallel lines and one of the other sides as a transversal:

    The area of a parallelogram is given by the formula:

where b is the length of the base, and h is the height.

    In area problems, you will likely have to find the height using techniques similar to the one used in the previous example problem with trapezoids.

 

    The next three quadrilaterals that we’ll review—rectangles, rhombuses, and squares—are all special types of parallelograms.

 

    Rectangles

 

    A rectangle is a quadrilateral in which the opposite sides are parallel and the interior angles are all right angles. A rectangle is essentially a parallelogram in which the angles are all right angles. Also similar to parallelograms, the opposite sides of a rectangle are equal.

    The formula for the area of a rectangle is:

    where b is the length of the base, and h is the height.

 

    A diagonal through the rectangle cuts the rectangle into two equal right triangles. In the figure below, the diagonal BD cuts rectangle ABCD into congruent right triangles ABD and BCD.

    Because the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, if you know two of these values, you can always calculate the third with the Pythagorean theorem. If you know the side lengths of the rectangle, you can calculate the diagonal; if you know the diagonal and one side length, you can calculate the other side.

 

    Rhombuses

 

    A rhombus is a quadrilateral in which the opposite sides are parallel and the sides are of equal length.

    The formula for the area of a rhombus is:

    where b is the length of the base, and h is the height.

 

    To find the area of a rhombus, use the same methods as used to find the area of a parallelogram. For example:

    If ABD is an equilateral triangle, then the length of a side of the rhombus is 4, and angles ADB and ABD are 60º. Draw an altitude from a to DC to create a 30-60-90 triangle, and you can calculate the length of this altitude to be 2. The area of a rhombus is bh, so the area of this rhombus is 4 2 = 8.