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Math
A |
Sum
of Interior Angles of a Polygon |
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A
Polygon
is a
many sided figure. It has the same number of angles as
sides. |
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The formula we
use to find the
sum
of the
interior angles
of any
polygon comes from the following idea:
Suppose
you start with a pentagon. If you pick any vertex (the
point where any 2 sides meet)
of that figure, and connect it to all the other vertices, how many
triangles can you form?
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If
you start with vertex
A
and
connect it to all other vertices (it's already connected to
B
and E
by sides) you form three
triangles. Each triangle contains
1800.
So the total number of degrees in the interior angles of a
pentagon is:
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3
 1800 = 5400
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Using the pentagon
example, we can come up with a formula that works for all polygons.
Notice that a pentagon has 5
sides, and that you can form 3
triangles by connecting the vertices. That's 2
less than the number of sides. It's the same principle for all
polygons. If we represent the number of sides of a polygon as n,
then the number of triangles you can form is (n-2).
Since each triangle contains 1800,
that gives us the formula:
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Sum
of Interior Angles |
=
180(n-2) |
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There are two
types of problems in which we can use 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. |
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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 formula, such as the words sum,
interior, and
angles. |
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 our formula from above, that gives us 180(8-2)
= 180(6) = 10800. |
Example
2:
How
many sides does a polygon have if the sum of its interior angles
is 7200 ?
| Since,
this time, we know the number of degrees,
we set the formula
above equal to 7200,
and solve for n. |
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180(n-2)
= 720
n-2 = 4
n= 6 |
Set
the formula = 7200
Divide both sides by 180
Add 2 to both sides |
Listed below are
some of the more common polygons whose names you should know:
| Triangle |
3
sides |
| Quadrilateral |
4
sides |
| Pentagon |
5
sides |
| Hexagon |
6
sides |
| Heptagon
or Septagon |
7
sides |
| Octagon |
8
sides |
| Nonagon
or Novagon |
9
sides |
| Decagon |
10
sides |
| Dodecagon |
12
sides |
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