Polygons are enclosed geometric shapes that cannot have fewer than three sides. As this definition suggests, triangles are actually a type of polygon, but they are so important on the Math IIC that we gave them their own section. Polygons are named according to the number of sides they have, as you can see in the chart below.
All polygons, no matter the number of sides they possess, share certain
characteristics:
- The sum of the interior angles of a polygon with n sides is (n – 2)
. So, for example, the sum of the interior angles of an octagon is (8 – 2)
= 6
= 
.
- The sum of the exterior angles of any polygon is
.
- The perimeter of a polygon is the sum of the lengths of its sides. The perimeter of the hexagon below, for example, is 35.
Regular Polygons
Most of the polygons with more than four sides that you’ll deal with on the Math IIC will be regular polygons—polygons whose sides are all of equal length and whose angles are all congruent (neither of these conditions can exist without the other). Below are diagrams, from left to right, of a regular pentagon, a regular octagon, and a square (also known as a regular quadrilateral):
Area of a Regular Polygon
There is one more
characteristic of polygons with which to become familiar. It has to do specifically with regular hexagons. A regular hexagon can be divided into six equilateral triangles, as the figure below shows:
If you know the length of just one side of a regular hexagon, you can use that
information to calculate the area of the equilateral triangle that uses the side. To find the area of the hexagon, simply multiply the area of that triangle by 6.
Quadrilaterals
The most frequently seen polygon on the Math IC is the quadrilateral, which is a general term for a four-sided polygon. In fact, there are five types of quadrilaterals that pop up on the test: trapezoids, parallelograms, rectangles, rhombuses, and squares. Each of these five quadrilaterals has special qualities, as shown in the sections below.
Trapezoids
A trapezoid is a quadrilateral with one pair of parallel sides and one pair of nonparallel sides. Below is an example of a trapezoid:
In the trapezoid pictured above, AB is parallel to CD (shown by the arrow marks), whereas AC and BD are not parallel.
The area of a trapezoid is:
where s1 and s2 are the lengths of the parallel sides (also called the bases of the trapezoid), and h is the height. In a trapezoid, the height is the perpendicular distance from one base to the other.
Try to find the area of the trapezoid pictured below:
To find the area, draw in the height of the trapezoid so that you create a 45-45-90 triangle. You know that the length of the leg of this triangle—and the height of the trapezoid—is 4. Thus, the area of the trapezoid is
6+10⁄
2 
4 = 8
4 = 32. Check out the figure below, which includes all the
information we know about the trapezoid:
