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.
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.
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
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
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
4 = 8
4 = 32. Check out the figure below,
which includes all the information we know about the trapezoid:
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.
angles are congruent.
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:
The area of a parallelogram is given by the formula:
where b is the length of the base, and h is
In area problems, you will likely have to find the height
using techniques similar to the one used in the previous example
problem with trapezoids.
The next three quadrilaterals that we’ll review—rectangles,
rhombuses, and squares—are all special types of parallelograms.
A rectangle is a quadrilateral in
which the opposite sides are parallel and the interior angles are all
right angles. A rectangle is essentially a parallelogram in which
the angles are all right angles. Also similar to 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.
A 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.
Because 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.
A rhombus is a quadrilateral in which the
opposite sides are parallel and the sides are of equal length.
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, use the same methods as
used to find the area of a parallelogram. For example:
||If ABCD is
a rhombus, AC = 4, and ABD is
an equilateral triangle, what is the area of the rhombus?
is an equilateral triangle, then
the length of a side of the rhombus is 4, and angles ADB
60º. Draw an altitude from a
create a 30-60-90 triangle, and you can calculate the length of
this altitude to be 2
. The area of a
rhombus is bh
, so the area of this rhombus is 4
A square is a quadrilateral in which all
the sides are equal and all the angles are right angles. A square
is a special type of rhombus, rectangle, and parallelogram:
The formula for the area of a square is:
where s is the length of a side of the
square. Because all the sides of a square are equal, it is also
possible to provide a simple formula for the perimeter: P = 4s,
where s is, once again, the length of a side.
A diagonal drawn into the square will always
form two congruent 45-45-90 triangles:
From the properties of a 45-45-90 triangle, we know that
words, if you know the length of one side of the square, you can
easily calculate the length of the diagonal. Similarly, if you know
the length of the diagonal, you can calculate the length of the sides
of the square.