The Measure of Each Interior Angle of a Regular Polygon is Eleven Times That of an Exterior Angle: Solving for the Number of Sides

Understanding the relationship between the interior and exterior angles of a regular polygon is crucial for solving various geometric problems. This article delves into a specific problem: given that the measure of each interior angle is eleven times that of the exterior angle, we will determine the number of sides of the polygon. We will explore multiple methods and provide a thorough analysis to solve this problem.

Method 1: Using Basic Geometry and Algebra

To begin with, we introduce some basic definitions:

Let n represent the measure of an exterior angle. The measure of each interior angle is then 11n, as given in the problem statement. The sum of the interior and exterior angles is 180 degrees.

Based on these, we can write:

$$22n 180 Rightarrow n frac{180}{22} approx 8.18$$

This seemingly incorrect approach stems from an overlooked relationship. Instead, we should express the interior angle as:

$$11n n 180 Rightarrow 12n 180 Rightarrow n 15$$

Now, using the property that the sum of all exterior angles in any polygon is 360 degrees:

$$frac{360}{n} 15 Rightarrow n 24$$

Hence, the polygon has 24 sides.

Method 2: Applying Polygon Angle Relationships

Revisiting the problem with a more systematic approach, we use the relationship between the exterior angle and the number of sides:

The measure of an exterior angle of a regular polygon is given by: $$frac{360}{n}$$ The measure of an interior angle is related to the exterior angle by: $$180 - frac{360}{n}$$

According to the problem, the interior angle is eleven times the exterior angle:

$$180 - frac{360}{n} 11 left(frac{360}{n}right) Rightarrow 180 frac{360}{n} (12) Rightarrow n 24$$

Thus, the polygon has 24 sides.

Additional Example: Solving with Interior Angle of Seven Times the Exterior

Another interesting case is when the interior angle is seven times the exterior angle:

Let the exterior angle be e and the interior angle be 7e. The sum of the interior and exterior angles is 180 degrees: $$7e e 180 Rightarrow 8e 180 Rightarrow e frac{180}{8} 22.5$$

The number of sides is then:

$$frac{360}{22.5} 16$$

Hence, the polygon has 16 sides.

Conclusion

Understanding the relationship between interior and exterior angles allows us to solve a variety of geometric problems. By using basic algebra and the properties of polygons, we can accurately determine the number of sides of a polygon given specific angle relationships. In this article, we have provided step-by-step solutions to two such problems, demonstrating the power of mathematical reasoning in geometry.