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
In a polygon, an exterior angle is formed by a side and an extension of an adjacent side. Exterior angles of a polygon have several unique properties. The sum of exterior angles in a polygon is always equal to 360 degrees. Therefore, for all equiangular polygons, the measure of one exterior angle is equal to 360 divided by the number of sides in the polygon.
Next to your angle is formed by a side
and an extension of an adjacent
So right here I've drawn
an exterior angle.
I could draw in two more by extending that
side and forming another exterior
angle, and I could extend this side
forming a third exterior angle.
But is there anything special about
the sum of an exterior angle?
To do that, let's look at a table.
And I have it separated into three parts.
The number of sides.
The measure of one exterior angle and the
sum of all of the exterior angles.
So we're going to start with regular polygons,
which means sides are the same
and the angles are the same.
So over here I'm going to draw an equilateral
triangle and I'm going to include
my exterior angles.
So we're going to assume that this
is an equilateral triangle.
If I look at the number of exterior angles,
that's going to be 3. So if
we go back here, number of sides is three.
We're going to ask ourselves what's
the measure of just one of these.
Well, if I look closely, this is a linear
pair, so it has to sum to 180 degrees.
We know in an equilateral triangle that
each degree measure of the angle is
Meaning that each of these exterior
angles is 120 degrees.
So I'm going to write in measure of
one exterior angle is 120 degrees.
So to find the sum, a shortcut
for adding is multiplication.
I'm going to multiply 3 times 120
and I'm going to get 360 degrees.
So let's see if it's different for a square.
So I'm going to draw in a regular quadrilateral,
also known as a square.
So, again, we're going to assume that we have
four congruent angles, four congruent sides.
And we know that this has to be 90 degrees,
which means its supplement would also be 90 degrees.
So every single one of these exterior angles
is going to be 90 degrees and we have four of them.
So the sum 4 times 90 is 360.
Looks like we're developing a pattern here.
I'm going to guess that for 5 I'm going
to multiply by something and I'm going
to get 360 degrees.
Let's check it out.
If I have a pentagon, and I draw in my
exterior angles here, again, this is
a regular polygon.
So all sides are congruent,
all angles are congruent.
We know that 108 degrees is the measure
of one angle in a regular polygon.
So its supplement is 72 degrees.
So the measure of one exterior angle is
going to be 72 degrees and sure enough
5 times 72 is 360 degrees.
So if we're going to generalize this for
any polygon with N sides, the sum of
the exterior angles will
always be 360 degrees. Always.
And I should include the dot, dot, dot
here if we want to find the measure of
just one of these if it's equiangular,
we're going to take the total sum
which is always 360 and divide
by the number of sides.
So a couple of key things here.
First one, if you want to find the measure
of one exterior angle in a regular
polygon, 360 divided by N. If you
want to find the sum of all of
the angles it's 360 degrees no matter
how many sides you have.