Introduction

Understanding the relationship between the interior and exterior angles of a regular polygon is a fundamental concept in geometry. This article explores the process of finding the number of sides of a regular polygon when the interior angle is given to be seven times the exterior angle.

Problem Statement

The interior angle of a regular polygon is seven times as large as the exterior angle. How many sides does the polygon have?

Solution

Let's denote the exterior angle of the polygon as ( x ) degrees.

The sum of an interior and its corresponding exterior angle is always 180 degrees: Interior angle 7x (given) 7x x 180 8x 180 x 22.5

Now, since the sum of all exterior angles of any polygon is always 360 degrees, we can use this knowledge to find the number of sides, ( n ), of our polygon:

360 n * 22.5

n 360 / 22.5

n 16

The polygon in question has 16 sides, hence it is a 16-gon.

Derivation and Explanation

Since the interior angle is 7 times the exterior angle, we can set up the following equation:

I 7E

For any regular polygon, the interior angle can be expressed as:

I 180 - E

Setting the two expressions for the interior angle equal, we get:

7E 180 - E

Solving for ( E ), we find:

8E 180

E 22.5

Now, using the fact that the sum of all exterior angles of a polygon is 360 degrees, and knowing that each exterior angle of a regular polygon is ( frac{360}{n} ), we can find ( n ) as follows:

360 n * 22.5

n 360 / 22.5

n 16

Conclusion

The polygon in question, with each interior angle being seven times the exterior angle, has 16 sides. This result is a clear application of the geometric properties of polygons, particularly the relationship between interior and exterior angles.

Further Exploration

Understanding the relationship between interior and exterior angles can be extended to solve other geometric problems. For instance, you can determine the number of sides of a polygon given different ratios or sums of angles.