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The sum of the measures
of the interior angles of a triangle is 180 degrees.
Since all the sides
of equilateral triangles are the same length, all the angles
are the same...
The interior angles
of an equilateral triangle are all 60 degrees.
What about a square?
That's easy! By definition, all
the interior angles of a square are right angles -- That means
that they are all 90 degrees.
What about other regular polygons?
To figure out the measure of the
interior angles of a regular pentagon, hexagon, heptagon,
etc, we need more than just a protractor! What if we needed
to find the interior angle of a regular polygon with 100 sides?
That might be a little difficult to draw!
Here are two methods to find the
measure of the interior angles of a regular polygon:
For both methods, we will use
the fact that the sum of the measures of the interior angles
of a triangle is 180 degrees!
METHOD 1:
Let's divide some regular polygons
into triangles by connecting one vertex to all of the others...
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A
square has 4 sides and we made 2 triangles.
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A
pentagon has 5 sides and we made 3 triangles.
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A
hexagon has 6 sides and we made 4 triangles.
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Do you see the pattern?
A heptagon has 7 sides... so we'd
be able to make 5 triangles.
If we had polygon with n sides...
we'd be able to make (n - 2) triangles.
Let's start with the square...
We made 2 triangles. Notice that all of the interior angles
of the 2 triangles make up the interior angles of the square.
The sum of the 2 triangle's angles
is 
There are 4 equal
angles in a square,
so 
gives us that one angle of a square is
!
Just what we expected.
Now for the pentagon.
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We made 3
triangles.
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So the interior angles
of a regular pentagon are each 108 degrees.
Can you figure out the hexagon?
How about a 100-gon? (That's a
regular polygon with 100 sides.) There would be 98 triangles...
So, in general, the measure of
an interior angle of a regular n-gon is
METHOD 2:
This method will be very similar
to that of the first method. Except that we will draw our
triangles using a point drawn inside the polygon.
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4
sides, 4 triangles
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5
sides, 5 triangles
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6
sides, 6 triangles
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Notice that not all of the angles
of the triangles are involved with the interior angles of
the polygons. We'll need to figure out how to deal with that.
Starting with the square:
4 triangles... 
At this point in method 1, we
had 360... So we are off by 360. But we haven't dealt with
the fact that those middle angles are not involved with the
interior angles of the square. It turns out that the sum of
the angles around that middle point is 360!
So 
and 
Let's try the pentagon...
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5 triangles...
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Can you figure out the hexagon?
In general, the measure of an
interior angle of a regular n-gon is
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