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Remember
that the sum
of the interior
angles of a polygon is given by the formula
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Sum of
interior angles = 180(n - 2) |
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where n = the number of sides in the
polygon.
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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. |
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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
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Each interior angle of a "regular"
polygon = |
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where n = the number of
sides in the polygon.
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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!! |
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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.
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1. |
Find
the number of degrees in each interior angle of a regular
dodecagon. |
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It
is
a
regular
polygon,
so we can use the formula.
In a dodecagon, n =
12. |
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2. |
Each
interior angle of a regular polygon measures 135°. How
many sides does the polygon have ? |
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- First,
set the formula (for each interior angle) equal to the number of degrees given.
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Cross multiply.
- Multiply 180 by (n
- 2).
- Subtract 135n from both sides of the equation.
- Divide both sides of the equation by 45.
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