Polygons
6.1 Lines and Angles
 
6.2 Triangles
 
6.3 Polygons
 
 
6.4 Circles
 
6.5 Key Formulas
 
Polygons
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 they merit their own section. Polygons are named according to the number of sides they have, as you can see in the following chart:
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)180º. So, for example, the sum of the interior angles of an octagon is (8 – 2)180º = 6(180º) = 1080º.
  • The sum of the exterior angles of any polygon is 360º.
  • 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
You should be familiar with one more characteristic of polygons, dealing 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 IIC 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 you’ll see 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.
Parallelogram
A parallelogram is a quadrilateral whose opposite sides are parallel. The figure below shows an example:
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. You should then be able to see why this would be true.
The area of a parallelogram is given by the formula:
where b is the length of the base and h is the height.
The next three quadrilaterals we’ll review—rectangles, rhombuses, and squares—are all special types of parallelograms.
Rectangles
A rectangle is a quadrilateral whose opposite sides are parallel and whose interior angles are all right angles. A rectangle is essentially a parallelogram whose 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.
Rhombuses
A rhombus is a quadrilateral whose opposite sides are parallel and whose sides are of equal length.
The area of a rhombus is:
where b is the length of the base and h is the height.
Squares
A square is a quadrilateral in which all the sides are equal and all the angles are right angles. It is a special type of rhombus, rectangle, and parallelogram.
The area of a square is:
where s is the length of a side of the square. The perimeter of a square is 4s.
Solving Polygons by Using Triangles
Polygons can often be cut into triangles, and if you can solve those triangles, then you can solve the entire polygon. For example, if you split a square on its diagonal, it forms two 45-45-90 triangles.
Since the hypotenuse of a 45-45-90 triangle always exists in the ratio of : 1 in relation to its sides, if you know the sides of a square, you therefore also always know the measure of its diagonal (d = s). This is simply one example of the value gained by thinking about a polygon in terms of the triangles that form it. There are many other ways to use triangles when thinking about polygons. You can divide a trapezoid or a parallelogram to make two triangles and a rectangle, or you can draw a diagonal through a rectangle to make two triangles. In short, as you deal with polygons, always remember that you can simplify the polygons by cutting them into triangles.
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