Question

If each exterior angle or a regular polygon measures 15°, how many sides does the polygon have?

Answer 1

hmm, I get a no-so-regular polygon

Answer 2

Firstly, note that if the exterior angle is 15°, then one of the interior angles must be 180° - 15° = 165°. Now notice the following: A regular polygon with 4 sides, the sum of the interior is 360°, this polygon is a square. Now when we increase the number of sides by 1, we get a polygon and the sum of the interior angles increases by 180° so it's now 540°. If you make the number of sides 6, the sum of interior angles increases by 180° once again so it becomes 720°. Notice the pattern? The sum of the interior angles from this pattern is given by:\[\bf Sum \ of \ interior \ angles = 180+180(n-3)\]Where 'n' is the number of sides.

Now notice that we know that each interior angle of this regular polygon is 165°; 180° - exterior angle = interior angle = 165°. Now notice that the number of sides of a polygon and the number of interior angles is equal. e.g. a square has 4 sides and 4 interior angles, pentagon has 5 sides and 5 interior angles..etc. Hence if this regular polygon has 'n' sides, then it must also have 'n' interior angles. Since the sum of interior angles is the sum of all interior angles, and we know that each interior angle is 165°, then the sum of the interior angles will be the number of interior angles times 165° = 165° * n. Making this substitution in the aforementioned equation gives us:\[\bf 165n=180+180(n-3)\]Now simply expand the right side of the equation, rearrange, and solve for 'n', which is the number of sides of the polygon. Can you do that?

@dokolo54