Final answer:
A regular polygon where the exterior angle is one-third of its interior angle has 8 sides, which means the polygon is an octagon. The interior angle is calculated to be 135 degrees, and the exterior angle is 45 degrees.
Step-by-step explanation:
The question deals with the relationship between the exterior and interior angles of a regular polygon. It states that the exterior angle of the polygon is one-third of its interior angle. To find the number of sides of the polygon, we need to use the relationship between exterior and interior angles. For any polygon, the sum of an exterior angle and its corresponding interior angle is 180 degrees because they form a linear pair.
Let's denote the interior angle as i and the exterior angle as e. Given that e = i/3, and knowing that i + e = 180, we can set up the equation i + i/3 = 180. By solving this, we find that i = 135 degrees and e = 45 degrees. Since the sum of exterior angles of any polygon is 360 degrees, we divide 360 by the measure of one exterior angle to find the number of sides: 360 / 45 = 8. Thus, the polygon has 8 sides, which makes it an octagon.