Solving for the Number of Sides of a Polygon with the Given Exterior and Interior Angles

In this article, we explore the mathematical problem of determining the number of sides of a polygon where the sum of the exterior angles is five times the sum of the interior angles. This problem requires an understanding of both interior and exterior angles of polygons and their relationships.

Understanding Polygon Angles

Polygons are closed two-dimensional shapes with straight sides. The basic properties of the angles in a polygon are crucial for solving more complex problems. There are two types of angles in a polygon: the interior angles and the exterior angles.

Interior Angles: These are the angles inside the polygon at the vertices. Exterior Angles: These are the angles formed outside the polygon at the vertices, when one side is extended.

Formulas for Polygon Angles

The sum of the interior angles of a polygon with n sides is given by:

Sum of Interior Angles (n - 2) * 180°

The sum of the exterior angles of a polygon with n sides is given by:

Sum of Exterior Angles n * 180°

The Problem: When the Sum of Exterior Angles is Five Times the Sum of Interior Angles

Let's delve into the problem itself. We need to find the number of sides (n) of a polygon where the sum of the exterior angles is five times the sum of the interior angles.

Rewriting the Problem

Mathematically, we can express this as:

n * 180° 5 * (n - 2) * 180°

By simplifying this equation:

n 5 * (n - 2)

n 5n - 10

10 4n

n 10 / 4 5 / 2 2.5 (This is incorrect; let's revisit the equation)

By correctly simplifying:

n 5 * (n - 2)

n 5n - 10

10 4n

n 10 / 4 2.5 (We need to recheck for correct simplification)

Correct simplification should be:

n - 2 2.5

n 6

Reinforcing with Multiple Solutions

Let's verify the solution by using multiple approaches:

Method 1: Expressing Each Angle ("x" is the exterior angle and "5x" is the interior angle): Total: 5x x 360° 12x 360° x 30° (each exterior angle) Number of sides 360° / 30° 12 Method 2: Using the given relationship between interior and exterior angles:

i e 180° (1)

i 5 * e (2)

Substituting (2) into (1):

5e e 180°

6e 180°

e 180° / 6 30° (each exterior angle)

360° / e 360° / 30° 12 (number of sides)

Method 3: Using the relationship given in the problem: Interior Angle: 5x Exterior Angle: x Total Angle: 180° x 5x 180° 6x 180° x 30° (each exterior angle) Number of sides 360° / 30° 12

Conclusion

We have demonstrated multiple methods to solve the problem, each confirming the same result that the polygon has 12 sides. These methods reinforce the understanding that the sum of the exterior angles is five times the sum of the interior angles.

Further Exploration

Delving further, we can explore the construction of a regular n-sided polygon. A regular polygon can be constructed with n isosceles triangles, each with an apex angle of 2π/n radians. The remaining angles are (1/2)π - (2π/n) radians, making each interior angle π - (2π/n) radians. Each exterior angle ψ is given by:

ψ π - (π - (2π/n)) (2π/n)

Given that each interior angle is five times each exterior angle, we have:

π - (2π/n) (10π/n) implies n 12

Thus, the polygon has 12 sides, confirming our earlier calculations.