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Polygons
Topic Index | Geometry Index | Regents Exam Prep Center
A polygon is a closed figure that is the union of line
segments in a plane. A polygon has three or more sides.
A polygon has the same number of angles as sides.
Polygons can be classified as either convex or concave.
A polygon is convex if no line that contains a side of the
polygon contains a point in the interior of the polygon. In
a convex polygon, each interior angle measures less than
180 degrees.
Concave polygons "cave-in" to their interiors, creating at
least one interior angle greater than 180 degrees (a reflex
angle).
Unless otherwise stated, we will be discussing convex
polygons.
Types of Polygons
Listed below are some of the more commonly used polygons.
(Do not assume that the diagrams under the "Graphic" column are "regular" polygons. Do not assume any
specific details about the diagrams such as the length of the sides or measures of the angles.)
Sides
Angles
Vertices
Diagonals
Number
Triangles
Triangle
3
3
3
0
1
Quadrilateral
4
4
4
2
2
Pentagon
5
5
5
5
3
Hexagon
6
6
6
9
4
Heptagon or
Septagon
7
7
7
14
5
Octagon
8
8
8
20
6
Nonagon or
Novagon
9
9
9
27
7
Decagon
10
10
10
35
8
Dodecagon
12
12
12
54
10
n
n
n
Polygon
n-gon
Graphic
---
(n - 2)
A polygon is equilateral if all of its sides are of the same length.
A polygon is equiangular if all of its angles are of equal measure.
A regular polygon is a polygon that is both equilateral and equiangular.
Sum of Interior Angles
of a Polygon
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Sum of Interior Angles
of a Polygon
Let's investigate why this formula is true.
= 180(n - 2)
(where n = number of sides)
Start with vertex A and connect it to all other vertices (it is already
connected to B and E by the sides of the figure). Three triangles are
formed. The sum of the angles in each triangle contains 180°. The total
number of degrees in all three triangles will be 3 times 180.
Consequently, the sum of the interior angles of a pentagon is:
3 180 = 540
Notice that a pentagon has 5 sides, and that 3 triangles were formed by
connecting the vertices. The number of triangles formed will be 2 less
than the number of sides.
This pattern is constant for all polygons. Representing the number of sides of a polygon as n,
the number of triangles formed is (n - 2). Since each triangle contains 180°, the sum of the
interior angles of a polygon is 180(n - 2).
Using the Formula
There are two types of problems that arise when using this formula:
1. Questions that ask you to find the number of degrees in the sum of the interior angles of a
polygon.
2. Questions that ask you to find the number of sides of a polygon.
Hint: When working with the angle formulas for polygons, be sure to read each question
carefully for clues as to which formula you will need to use to solve the problem. Look for
the words that describe each kind of formula, such as the words sum, interior, each, exterior
and degrees.
Example 1:
Find the number of degrees in the sum of the interior angles of an octagon.
An octagon has 8 sides. So n = 8. Using the formula from above,
180(n - 2) = 180(8 - 2) = 180(6) = 1080 degrees.
Example 2:
How many sides does a polygon have if the sum of its interior angles is 720°?
Since, the number of degrees is given, set the formula above equal to 720°,
180(n - 2) = 720
n-2=4
n=6
and solve for n.
Set the formula = 720°
Divide both sides by 180
Add 2 to both sides
Each Interior Angle
of a Regular Polygon
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Remember that the sum of the interior angles of a polygon is
given by the formula
Sum of interior angles = 180(n - 2)
where n = the number of sides in the polygon.
A polygon is called a REGULAR
polygon when all of its sides are of
the same length and all of its angles
are of the same measure.
A regular polygon is both
equilateral and equiangular.
Let's investigate the regular pentagon seen above.
To find the sum of its interior angles, substitute n = 5 into the formula 180(n - 2) and get
180(5 - 2) = 180(3) = 540°
Since the pentagon is a regular pentagon, the measure of each interior angle will be the same.
To find the size of each angle, divide the sum, 540º, by the number of angles in the pentagon.
(which is the same as the number of sides).
540°
5 = 108°
There are 108° in each interior angle of a regular pentagon.
This process can be generalized into a formula for finding each
interior angle of a REGULAR polygon ...
Each interior angle of a "regular" polygon =
where n = the number of sides in the polygon.
Be careful!!! If a polygon is NOT REGULAR (such as the one
seen at the right), you cannot use this formula. If the angles of a
polygon DO NOT all have the same measure, then you cannot find
the measure of any one of them just by knowing their sum.
NOT REGULAR = DO NOT USE FORMULA!!
Examples
Read these questions carefully! If the word "EACH" appears in the question, you will most likely
need the formula for "each interior angle" to solve the problem.
1. Find the number of degrees in each interior angle of a regular dodecagon.
It is a regular polygon, so we can use the
formula.
In a dodecagon, n = 12.
2. Each interior angle of a regular polygon measures 135°. How many sides does the polygon
have ?


First, set the formula (for each interior
angle) equal to the number of degrees
given.
 Cross multiply.
 Multiply 180 by (n - 2).
 Subtract 135n from both sides of the
equation.
Divide both sides of the equation by 45.
Exterior Angle
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An exterior angle of a polygon is an angle that forms a linear pair with
one of the angles of the polygon.
Two exterior angles can be formed at each vertex of a polygon.
The exterior angle is formed by one side of the polygon and the
extension of the adjacent side. For the hexagon shown at the left,
<1 and <2 are exterior angles for that vertex. Be careful, as <3 is
NOT an exterior angle.
Note: While it is possible to draw TWO (equal) exterior angles at
each vertex of a polygon, the sum of the exterior angles formula
uses only ONE exterior angle at each vertex.
Formula:
Sum exterior angles of any polygon = 360°
(using one exterior angle at a vertex)
Finding the sum of the exterior angles of a polygon is simple. No matter what type of polygon
you have, the sum of the exterior angles is ALWAYS equal to 360°.
If you are working with a regular polygon, you can determine the size of EACH exterior angle
by simply dividing the sum, 360, by the number of angles. Remember, the formula below will
ONLY work in a regular polygon.
Formula:
Each exterior angle (regular polygon) =
Examples
the sum of the exterior angles of:
1. Find
a) a pentagon
Answer: 360
b) a decagon
c) a 15 sided polygon
d) a 7 sided polygon
0
Answer: 3600
Answer: 3600
Answer: 3600
2. Find the measure of each exterior angle of a regular hexagon.
A hexagon has 6 sides, so n = 6
Substitute in the formula.
3. The measure of each exterior angle of a regular polygon is 45°. How many sides does the
polygon have ?
Set the formula equal to 450.
Cross multiply and solve for n.