Transcript Polygon Interior Angles Sum Theorem

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What is a Polygon?

A polygon has at least three
or more sides to form a
convex or irregular shape.
-Therefore the interior
angles of a polygon
involve the angles
INSIDE the polygon!
What are Interior Angles?
Interior angles are the angles inside any shape
 In this situation the interior angles are applied to polygons
 The sum of the interior angles depends on the number of
sides of the polygon

What is the Polygon Interior Angles Sum
Theorem?
The PIAST is the sum of the interior
angles dependent on the number of sides
of the polygon.
 Use this formula: (n-2)180
 Follow the steps below:

 “n” stands for the number of sides of the
polygon
 Once you’ve counted the number of sides
subtract by 2
 Then, multiply 180 by (n-2)
Tips & Additional Info.
A nifty tip is to just remember the
formula: (n-2)180
 If you forget the multiplication of 180,
just know that the smallest number of
any interior angle sum is 180 (a triangle)

Triangle Sum Theorem
Aside from the PIAST, the Triangle Sum Theorem can help
when remembering sum angles and is a relatable theorem
 So…What is it them?

 All interior angles of a triangle will ALWAYS equal 180
 Equilateral, isosceles, or scalene all angles will add to 180

How is this Useful for the PIAST?
 This theorem is handy because if you forget the formula for the
PIAST then you can remember the Triangle Sum Theorem and
know that 180 is the smallest angle sum for any shape!
How does this Theorem apply?
The Polygon Interior Angles
Sum Theorem is useful when
trying to find a missing interior
angle.
 Once you find the amount of
sides subtract by 2 and then
multiply by 180, (n-2)180, you
then know the total amount of
angles the polygon will have.

Example No. 1

Figure out the following problem:
12-gons (7):
12-2=10x180=
1800
1800x7= 12600
 If Zheng He has 2 triangles, 3 octagons, and
7 12-gons, he wants to combine all the shapes
into one unison figure. Show your work
giving the exact number of interior angles. In
two forms, separate shapes and then together
as a whole.

Explanation:
 When you use the PIAST you will be able to
figure out the interior angles for the triangles,
octagons, and 12-gons, by using the formula
(n-2)180. Then once you have figured out the
sum angles for all shapes you then use
addition to find the interior angles for the
entire figure.
Octagons (3):
8-2=6x180= 1080
1080x3= 3240
Triangles (2):
3-2=1x180= 180
180x2= 360
The total number
of angles is: 16200
2 Triangles: 360
3 Octagons: 3240
7 12-gons: 12600
Example No.2

Below is a figure displaying interior angles. Find the value of x.
What do we know:
This a hexagon-6 sided polygon
Therefore use (n-2)180
(6-2)180= 720
Actual Work:
100+140+110+115+135+x=720
Combine like terms
600+x=720
-600 -600
X=120
Explanation:
When using the Polygon Interior
Angles Sum Theorem, you will
be able to find the total amount of
sides of the polygon. Thus you
will be able to find the total angle
sum, and find the value of x.
5 Practice Problems





1) If n=7 what is the sum of the
interior angles?
2) If n=54 what is the sum of the
interior angles?
3) What are the interior angles of
congruent decagons?
4) Find the interior angles of a
congruent 17-gon
5) If a hexagon has the following
angles: 100, 50, 75, 115, 185, and x
what is the value of x?
Answers to Practice Problems
1) 900
 2) 9360
 3) 1440
 4) 2700
 5) x= 195
