Geometry Concepts 6.1 Polygon Angle-Sum Theorem Name___________________________________
Sides
Type
Sides
Type
4
quadrilateral
8
octagon
5
pentagon
9
nonagon
6
hexagon
10
decagon
7
heptagon
n
n-gon
--------------------------------------------------------------------------------------------------------------------The sum of the measures of the interior angles for any polygon with n sides is equal to
(n  2)  180 .
The whole concept of this formula is this: the "n - 2" portion of the formula finds how many
triangles exist inside of the polygon. When this is found, the formula then multiplies the number
of triangles by 180 degrees.
--------------------------------------------------------------------------------------------------------------------What is the sum of the measures of the interior angles of an octagon?
An octagon has 8 sides, so the sum of the measures of the interior angles is
(8  2)  180  1080 .
--------------------------------------------------------------------------------------------------------------------What is the sum of the measures of a 21-gon?
--------------------------------------------------------------------------------------------------------------------What if you are told that each interior angle of the pentagon has the same measure?
To find the measure of each one of them, you would simply need to take the sum of all of them,
and divide by how many angles there are.
For a pentagon, the sum is 540 . Since this is a pentagon, there are five interior angles.
Therefore, the measure of each interior angle (if they are the same) would be:
540  5  108 .
--------------------------------------------------------------------------------------------------------------------Polygons are given a special name if all of their sides are congruent, and, more importantly, all
of their interior angles are congruent. They are called regular polygons.
For example, a square is an example of a regular polygon.
1
What is the measure of each interior angle in a regular decagon (10 sides)?
First, find the sum of the measures of the interior angles:
(10  2)  180  1440 .
Since it is a regular decagon, all interior angles must have the same measure. So, divide the sum
by 10 to find the measure of each one of them:
1440 10  144 .
--------------------------------------------------------------------------------------------------------------------It is important to note that this only works for polygons in which all interior angles are
congruent.
What is the measure of each interior angle in a regular 15-gon?
--------------------------------------------------------------------------------------------------------------------Here is a formula which encompasses everything taught on the front:
( n  2)  180
the measure of each interior angle of a regular n-gon =
n
--------------------------------------------------------------------------------------------------------------------As with triangles, exterior angles can be drawn for any polygon by extending one of the sides.
In the figure below, an exterior angle (angle 1)
is formed by extending one of the sides of
the hexagon.
In the figure below, an exterior angle has
been drawn at every vertex.
2
3
1
1
4
6
5
If each of the exterior angles were cut out of the figure to the right above, and each of the
vertices of these angles were placed together, the angles would form a circle. This forms the
basis for the following property:
The sum of the measures of the exterior angles, one at each vertex,
for any convex polygon is 360 .
2
What is the sum of the measures of the exterior angles, one at each vertex, of a convex heptagon?
--------------------------------------------------------------------------------------------------------------------360
The measure of each exterior angle for a regular n-gon is
.
n
--------------------------------------------------------------------------------------------------------------------What is the measure of each exterior angle of a regular octagon (8 sides)?
Using the formula above: 360  8  45 .
What is the measure of each exterior angle of a regular 20-gon?
--------------------------------------------------------------------------------------------------------------------Find the value of x.
--------------------------------------------------------------------------------------------------------------------Classwork on Angles & Polygons
For Questions 1-2, consider the polygon to the right.
1.
What is the name of the polygon is pictured to the right?
2.
What is the sum of the measures of its interior angles?
--------------------------------------------------------------------------------------------------------------------3.
What is the name for a polygon with ten sides?
4.
What is the sum of the measures of its interior angles?
--------------------------------------------------------------------------------------------------------------------For Questions 5-7, find the sum of the measures of the interior angles of the polygon with the
given number of sides.
5.
17 sides
6.
42 sides
7.
100 sides
3
For Questions 8-15, provide the requested information for the given type of polygon. Round all
decimals to the nearest tenth. Assume all polygons are convex.
8. pentagon - sum of the measures of
the exterior angles, one at each vertex
9. nonagon - sum of the measures of
its interior angles
10. regular 20-gon - measure of each of
its interior angles
11. regular hexagon - measure of
each of its exterior angles
12. regular heptagon - measure of each of
its interior angles
13. hexagon - sum of the measures
of its interior angles
14. 100-gon - sum of the measures of the
exterior angles, one at each vertex
15. regular 40-gon - measure of each
of its exterior angles
--------------------------------------------------------------------------------------------------------------------For Questions 16-19, consider a regular 18-gon.
16.
What is the sum of the measures of its interior angles?
17.
What is the measure of each of its interior angles?
18.
What is the sum of the measures of its exterior angles, one at each vertex?
19.
What is the measure of each exterior angle?
--------------------------------------------------------------------------------------------------------------------20. Find x.
21. Find the measure of each interior angle.
4