How do you determine the number of sides on a regular polygon given the measures of each of the interior angles?

Suppose you have a polygon with n sides, the measure of each interior angle is given by the equation [180(n-2)]/n. We can prove this expression with a regular triangle. We know that the triangle has 3 sides. Plugging this into our expression gives us: [180(1)]/3=60 degrees per angle, which is the standard value of an angle for a triangle.

Now, your situation is the opposite of what we have shown above. You have the measure of each angle, but not the number of sides. We can find an equation for the number of sides of a polygon by manipulating the expression mentioned above algebraically. Assume that the measure of each interior angle is given by the variable x. :

Original equation: [180(n-2)]/n=x.

Multiply each side by n: 180(n-2)=xn

Expand the left side: 180n-360=xn

Move the terms around: 180n-xn=360

Simplify: n(180-x)=360

Final expression: n=360/(180-x)

To better illustrate this, I will provide an example. Let's say you have a regular hexagon. You are given that each interior angle is 120.

Plugging this into the expression for number of sides: n=360/(180-120)

We have n=6, which is indeed the number of sides a hexagon has.