Finding # of Sides in Polygon Given Measure of Interior Angle

[dustie]

Homework Statement

the measure of an interior angle of a regular polygon is given. need to find the number of sides in the polygon. i cannot find the formula to be able to do this.

Homework Equations

The Attempt at a Solution

 

[Chaos2009]

On any polygon, the measure of the exterior angles always adds up to 360 degrees and they are supplementary to the interior angles. Because the interior angles in a regular polygon are going to have the same measure, the exterior angles will as well, so the exterior angles will have the measure 360/n where n is the number of sides. See if you can use that to get started.

 

[HallsofIvy]

Another way to do this is to draw a line from one vertex to every other vertex. The sides of polygon alredy connect that vertex to the vertex on either side so you draw n-3 "diagonals" and that divides the polygon into n-2 triangles. Since every triangle has angle sum 180 degrees, the n-1 triangles and so the total angles in the polygon have angle sum 180(n-2). Since there are n interior angles, what is the measure of each angle in a regular n-gon? Set that equal to the angle you are given and solve for n.

 

[Chaos2009]

I believe it is n-3 "diagonals" and n-2 triangles. A square (n=4) has 1 diagonal (n-3) and 2 triangles (n-2).

 

[HallsofIvy]

Right. Thanks. I wrote too fast. I will edit what I wrote.