Vocabulary
Diagonal- a line segment that connects two non-adjacent vertices of a polygon.
Examples
Sum
of Interior Angles
of a Polygon |
=
180(n - 2) (where n = number  of
sides) |
 |
Let's investigate why this formula is true.
 |
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 = 540Notice 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
. |
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°, and solve for
n.
|
||
180(n
- 2)
= 720
n - 2 = 4 n = 6 |
Set
the formula = 720° Divide both sides by 180 Add 2 to both sides | |
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