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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. |
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 |
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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