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

Share

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

60 degrees.

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.