How many sides does a regular polygon with an exterior angle measure of 40 degrees have and how do you do it?

(The exterior angles of a regular polygon total 360 degrees. 360 divided by 40 = 9 sides. For a more difficult way to come up with the same information, read on.)

Okay, this is probably not the most efficient or simple way to do the problem, but I'll show you the way I did it.

First you determine the measure of the interior angles that correspond with the exterior angles. The exterior and interior angles form a straight line. Since a straight line is always 180 degrees:

40+y=180

where y= the measure of any interior angle (they're all the same)

40+y-40=180-40

y=140

so, the interior angles are 140 degrees

Then I wrote a formula to find the total measure of the interior angles of a regular polygon given n sides.

A regular triangle has 180 total degrees, a square has 360, a pentagon has 540, etc.

A formula that works for all of these is:

x=180(n-2)

where n is the number of sides and

where x is the total measure of the interior angles

To get the measure of each angle you simply divide by n, so we now have:

x/n=180*(n-2)/n

I'll replace the x/n with y for simplicity

(y=the measure of each interior angle)

y=180*(n-2)/n

Now simplify the equation to simplest terms

y=180*(n-2)/n

y=180*((n/n)-(2/n))

y=180*(1-(2/n))

y=180-(360/n) (Distributive property)

We have already figured out y from earlier in the problem, so just plug it in and solve for n.

140=180-(360/n)

-40=-360/n

40=360/n

n(40)=(360/n)n

40n=360

n=9