


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 foursided 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 s_{1} and s_{2} 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:
 Opposite sides are equal.
 Opposite angles are congruent.
 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 454590
triangles.
Since the hypotenuse of a 454590 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.
