The Measure of an Exterior Angle of a Regular Polygon

In this article, we will explore how to determine the number of sides of a regular polygon when given the measure of its exterior and interior angles. Let's delve into a specific example where the exterior angle is 2x and the interior angle is 4x.

Relationships Between Interior and Exterior Angles

First, let's establish the relationship between the interior and exterior angles of a regular polygon. The sum of an interior angle and its corresponding exterior angle is always 180 degrees:

Sum of Interior and Exterior Angles:

For any regular polygon, the sum of an interior angle and its corresponding exterior angle is:

( text{Interior angle} text{Exterior angle} 180^circ )

Given that the exterior angle is 2x and the interior angle is 4x, we can set up the following equation:

( 4x 2x 180^circ )

Combining like terms:

( 6x 180^circ )

Solving for x:

( x frac{180^circ}{6} 30^circ )

Calculating the Angles

Now that we have x, we can calculate the actual values of the exterior and interior angles:

Exterior Angle:

( 2x 2 times 30^circ 60^circ )

Interior Angle:

( 4x 4 times 30^circ 120^circ )

Number of Sides of the Polygon

A key relationship in regular polygons is that the measure of an exterior angle is given by:

Exterior Angle Formula:

( text{Exterior angle} frac{360^circ}{n} )

where n is the number of sides. Given that the exterior angle is 60 degrees, we can set up the equation:

( 60^circ frac{360^circ}{n} )

Solving for n:

( n frac{360^circ}{60^circ} 6 )

This indicates that the polygon has 6 sides, making it a hexagon.

Conclusion

A regular polygon with 6 sides is called a hexagon. Therefore, we have determined that the number of sides is 6 and the type of polygon is a hexagon.

Additional Examples

Let's explore a few more examples to solidify our understanding:

Example 1:

Suppose the exterior angle is x degrees and the interior angle is 4x. Since the sum of the interior and exterior angles is 180 degrees, we can set up the equation:

( x 4x 180^circ )

Solving for x:

( 5x 180^circ )

( x frac{180^circ}{5} 36^circ )

The exterior angle is 36 degrees, and the number of sides is:

( n frac{360^circ}{36^circ} 10 )

This indicates a regular polygon with 10 sides, known as a decagon.

Example 2:

If the exterior angle is x degrees and the interior angle is 4x, their total is 180 degrees. Therefore:

( 4x 180^circ - x )

( 5x 180^circ )

( x frac{180^circ}{5} 36^circ )

The number of sides n is:

( n frac{360^circ}{36^circ} 10 )

Thus, the polygon is a regular decagon.

Bonus

Solving for the number of sides when the sum of interior angles is given:

If the number of sides are n, the sum of the interior angles is:

( 2n - 4 text{ right angles} 2n - 360^circ )

Given each interior angle is 144 degrees, the equation becomes:

( 2n - 360^circ 144n )

( 2n - 144n 360^circ )

( -142n 360^circ )

( n frac{360^circ}{-142} 10 )

Therefore, the polygon is a decagon.