The area of a parallelogram is given by the formula
where b is the length of the base, and h is the height.
Rectangles
A rectangle is a quadrilateral in which the opposite sides are parallel and the interior angles are all right angles. Another way to look at rectangles is as parallelograms in which the angles are all right angles. As with 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.
The diagonals of a rectangle are always equal to each other. And one 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.
Since 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. Also, keep in mind that the diagonal might cut the rectangle into a 30-60-90 triangle. That would make your calculating job even easier.
Rhombus
A rhombus is a specialized parallelogram in which all four sides are of equal length.
In a rhombus,
- All four sides are equal: AD = DC = CB = BA
- The diagonals bisect each other and form perpendicular lines (but note that the diagonals are not equal in length)
- The diagonals bisect the vertex angles
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 on the SAT (you guessed it), you’ll probably have to split it into triangles:
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If ABCD is a rhombus, AC = 4, and ABD is an equilateral triangle, what is the area of the rhombus? |
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Since ABD is an equilateral triangle, the length of each side of the rhombus must be 4, and angles ADB and ABD are 60º. All you have to do is find the height of the rhombus. Draw an altitude from A to DC to create a 30-60-90 triangle.
Since the hypotenuse of the 30-60-90 triangle is 4, you can use the ratio 1:
:2 to calculate that the length of this altitude is 2
. The area formula for a rhombus is bh, so the area of this rhombus is 4 
2
= 8
.
Square
A square combines the special features of the rectangle and rhombus: All its angles are 90º, and all four of its sides are equal in length.
The square has two more crucial special qualities. In a square,
- Diagonals bisect each other at right angles and are equal in length.
- Diagonals bisect the vertex angles to create 45º angles. (This means that one diagonal will cut the square into two 45-45-90 triangles, while two diagonals break the square into four 45-45-90 triangles.)
The formula for the area of a square is
where s is the length of a side of the square.
