Understanding Polygons and Their Interior and Exterior Angles

When dealing with polygons, understanding the relationship between their interior and exterior angles is crucial. In this article, we will delve into a specific scenario where the exterior angle of a polygon is one-third of its interior angle. We will explore how to determine the number of sides of this polygon and identify its name.

Overview of Interior and Exterior Angles

Interior and exterior angles of polygons are supplementary, meaning they add up to 180 degrees. This is because they form a linear pair on the same straight line. Given their relationship:

An interior angle and the adjacent exterior angle will always sum to 180 degrees. To solve for the interior and exterior angles, you need to set up an equation based on their relationship.

Problem: Exterior Angle is One-Third of its Interior Angle

Let's consider a regular polygon where the exterior angle is one-third of its interior angle. We aim to find the number of sides of this polygon and identify its name.

Plan

Using the fact that the interior and exterior angles are supplementary, we can derive the measures as follows:

Let the exterior angle be E and the interior angle be I. Since the exterior angle is one-third of the interior angle, mathematically this is expressed as E I/3. Solving for both angles, we start with the equation for the sum of an interior and an exterior angle:

Equation and Solution

Given that E I 180°:

Substituting E with I/3: I/3 I 180° Combining like terms: (I 3I)/3 180° 4I/3 180° Multiplying both sides by 3/4 to isolate I: 4I/3 * 3/4 180° * 3/4 I 180° * 3/4 135°

Now that we have the interior angle, we can find the exterior angle:

E 180° - I 180° - 135° 45°

The number of sides (n) can be found using the fact that the sum of all exterior angles in a polygon is 360°:

n * E 360° n * 45° 360° n 360° / 45° 8

Therefore, the polygon has 8 sides and is called an octagon.

Details and Additional Notes

To summarize, given the interior angle (θ), the exterior angle is 180° - θ. For a regular polygon, the relationship between the interior and exterior angles is:

Exterior Angle 180° - Interior Angle To find the number of sides (n) for a regular polygon, use the formula n 360° / Exterior Angle.

Applying this, we can also verify our solution:

Given the exterior angle (E) is 45°: Interior Angle (θ) 180° - 45° 135° And, n 360° / 45° 8, confirming an octagon.

As the interior angle must be less than 180 degrees and the exterior angle must be more than 0 degrees, it is indeed possible for the exterior angle to be one-third of the interior angle, leading us to an octagon.