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Equiangular Polygon Sums - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The sum of the angles in a polygon is always equal to the number of sides in a polygon minus two, all multiplied by 180. Since the angles in an equiangular polygon are equal, the measure of one angle in any equiangular or regular polygon is simply the sum of polygon angles divided by the number of angles in the polygon. Knowing this information allows us to solve polygon problems with missing angle measurements.

Let's say I tell you, you have ten-sided
figure it's eqiangular.
What's going to be the measure
of one of those angles.

To find that out we have
to first back up.
There's two key terms.
equiangular and regular that are going
to apply to what we're talking about here.

The first one is equiangular which means
that all the angles in the polygon
are congruent.
It doesn't have to do anything
with the sides. It just means the angles.
You can remind yourself of that because
it look like we have the word angle

Regular combines equilateral
and equiangular.
It says that all angles and all sides
must be congruent to each other.

So we look at these two
polygons right here.
We have two pentagons, only one
of these are equiangular.
Well, here we have all sides
marked as congruent. So this is not equiangular.
If we look at this pentagon right here, all
of your angles are marked as congruent
to each other.
So, yes, this one would be equiangular.
If you put these together, you have a regular
polygon, which is like this hexagon
right here.

Notice that all the sides are congruent
to each other, and all the angles are
congruent to each other.
But how do we calculate the measure
of one of those angles?
Well, to do that we need to look
at our polygon angle sum.
We said that the sum of the angles in
a polygon is equal to N minus 2 times
180. To find that, we said how many triangles
can we draw in a polygon.
And that was always the number of sides
minus 2. The sum of the angles
in the triangle is 180.

So if you want to find just one angle
in a regular polygon, you're going to
take this formula, which is the quantity
of N minus 2, times 180, and
since all the angles are the same, you
can just divide by the number of
angles that you have.

So you're taking this formula and you're
dividing by the number of sides that
you have in your equiangular polygon
or in your regular polygon.

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