By definition, a polygon is a two-dimensional figure with three or more straight sides. Under that definition, triangles are a type of polygon. However, since triangles are such an important part of the SAT, we gave them their own section. This section will deal with polygons of four sides or more.

There are a number of properties common to all polygons.

As with triangles, the perimeter of a polygon is equal to the sum of the length of its sides. The SAT occasionally makes up fancy perimeter questions in which they create diagrams such as the following:

Such a figure looks more complicated than it is. It’s actually just a little puzzle where all the information is given to you. You can figure out that the lengths of AD and BC are 9 since the dotted lines of 4 and 5 span the same distance as AD and BC.

Another good rule to remember for perimeter problems on the SAT is that a straight line will always be shorter than a curved or otherwise non-straight line.

Just as the angles of triangles always add up to 180 degrees, the angles in various polygons will also add up to the same number. The formula that defines the sum of the angles of all polygons is where n equals the number of sides of the polygon. Take a look at the table below.

If you come upon any questions on the SAT in which you are shown a polygon and the angle values of all of the vertices but one are given, you can always find the lone missing value by using this formula to calculate the total angle value and subtracting the value of each angle you know.

The word “parallelogram” refers to a great number of different geometric figures. The parallelogram is the most general; the rectangle, rhombus, and square are all parallelograms with certain special features.

A parallelogram is a four sided figure (a quadrilateral) whose opposite sides are parallel.

In a parallelogram:

A rectangle is a parallelogram whose angles all equal 90º. All of the rules that hold for a parallelogram also hold for a rectangle. A rectangle has further properties, however, that you should also know.

In a rectangle:

A rhombus is a specialized parallelogram in which all four sides are of equal length.

In a rhombus:

A square combines the special features of the rectangle and rhombus: all its angles are 90º, and all four sides are equal in length.

The square has many, many additional special qualities. In a square:

To calculate the area of a parallelogram, we must introduce a new term: altitude. The altitude of a parallelogram is the line segment perpendicular to a pair of opposite sides with one endpoint on each. The dotted lines show the altitudes of various parallelograms.

The area of a parallelogram is the product of the length of its altitude and the length of a side that contains an 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 opposite side. A common way to describe the area of a parallelogram is the base times the 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 lengths of any two adjacent sides. Because the sides of a square are equal, the area of a square is the length of any one side squared. The area of a rhombus is equal to one-half the product of its diagonals.

Trapezoids are four-sided figures but not parallelograms. In a trapezoid, one pair of sides is parallel while the other is not.

The area of a trapezoid equals the product of half the sum of the length of its bases and the height of the altitude: A = ¹/₂ The altitude of a trapezoid is a segment perpendicular to the bases (the pair of parallel lines) with one endpoint on each base. In the images, below, the lines marked by an a are the altitudes of the trapezoids:

If you were presented with the trapezoid pictured below, you can just plug the numbers from the trapezoid into the trapezoid area formula.

It is also sometimes possible to solve for the area of a trapezoid by transforming the trapezoid into a rectangle and a triangle. In the picture above, the area of the trapezoid ABCD is equal to the area of the rectangle plus the area of the triangle (¹/ ₂), which means the total area is 32 + 4 = 36.

The SAT will sometimes include polygons that have more than four sides. The most important rule to remember for dealing with these many-sided polygons is one we already covered: the sum of the angles of a many-sided polygon is equal to 180º (n – 2), where n is the number of sides the polygon has.

Many of the polygons you encounter on the SAT will be regular polygons, meaning that the angles and sides of the polygon will be equal. Since all the angles of a regular polygon are equal, you can easily calculate the value of every angle within that polygon. All you have to do is calculate the sum of the angles and divide by the number of angles.

Therefore, the value of an angle in a regular polygon is ^180º(n–2)/ _n.

If you are working with polygons, especially if you are trying to find the area of an irregular polygon, it is often a good idea to try to cut the polygon down into more manageable parts. For example, you might be able to cut a trapezoid into two triangles and a square rather than have to deal with it as a trapezoid at all.

The ability to see simpler shapes within more complicated ones can be a powerful tool on the SAT. A part of this skill rests on being able to deal with shapes in reference to each other. For example, let’s say you are shown the following diagram:

This figure asks you to determine the value of x based only on the information that DE equals 1. The question might seem impossible, but if you have some knowledge of squares and triangles, it’s extremely easy. If you remember, the diagonals of a square bisect each other, so just as ED = 1, so too do EB and EC.

You should also know that two diagonals cut a square into four 45-45-90 triangles. DC is the hypotenuse of one of these triangles, since it is opposite the right angle. And, as you should know from the section on triangles, the ratio of sides to hypotenuse in a 45-45-90 triangle is always 1: .

We solved this problem simply by knowing about the attributes of squares and 45-45-90 triangles and without having to do any time-consuming math.