Find the number of sides of a regular polygon if one interior angle is 120 degrees.

A.) 6

B.) 5

C.) 4

D.) 3

Question

Find the number of sides of a regular polygon if one interior angle is 120 degrees.
 
A.) 6

B.) 5

C.) 4

D.) 3

mathstudent55 · Answer

Do you know the formula that tells you the sum of the measures of the interior angles of a polygon?

anonymous · Answer

No

mathstudent55 · Answer

Here is the formula:

$S = (n - 2)180$

where S = sum of the measures of the interior angles, and n = number of sides.

mathstudent55 · Answer

Do you know what "regular" means when you say regular polygon?

anonymous · Answer

When all angles are equal?

mathstudent55 · Answer

It means a polygon that has all sides of the same length and all interior angles of the same measure.

mathstudent55 · Answer

Correct. All angles are congruent, and all sides are congruent.

anonymous · Answer

So would it be 3?

mathstudent55 · Answer

Now we can solve this problem.
Since you are dealing with a regular polygon, all angles have the same measure.
Let's say the sum of the measures of the interior angles is S.
Also, let's say your polygon has n sides.
Then each angle would have the measure S/n
Ok so far?

mathstudent55 · Answer

No, it's not 3.

mathstudent55 · Answer

Just follow what I am explaining . We'll find the answer soon.

anonymous · Answer

Okay!

mathstudent55 · Answer

If the sum of the measures of the interior angles is S and the polygon has n sides, then each interior angle measures S/n.

mathstudent55 · Answer

We have a formula for the sum of the measures of the interior angles.

$S = (n - 2)180$

If we divide it by n, the number of sides, we will have the measure of one angle.

$\dfrac{S}{n} = \dfrac{(n - 2)180}{n} $   is the measure of one angle.

We are told one angle measures 120 deg, so we set our expression equal to 120 and ewe solve for n.

$\dfrac{(n - 2)180}{n} = 120$

anonymous · Answer

I think I got it now! thank you so much!

mathstudent55 · Answer

Now we have to solve that equation for n.
We multiply both sides by n:

$(n - 2)180 = 120n$

Distribute 180 on the left side:

$180n - 360 = 120n$

Subtract 120n from both sides, and add 360 to both sides:

$60n = 360$

Divide both sides by 60:

$n = 6$

Since n is the number of sides of the polygon, we can answer:
The polygon has 6 sides.

mathstudent55 · Answer

You're welcome.