In a regular polygon the ratio of the measure of the exterior angle to the measure of the adjacent interior angle is 4 to 1 how many sides does the polygon have Please show your work?

Answer 1

This is not possible for a normal regular polygon. (A regular polygon has all equal angles and all equal sides. A normal polygon has no intersecting edges.)

The smallest regular polygon is an equilateral triangle (a three sided polygon), whose exterior angle measure is twice the measure of its interior angle. A four-sided polygon (a square) has equal interior and exterior angle measures of 90⁰. Starting from a five-sided polygon, the exterior angle measure is smaller than the interior angle measure.

Let's assume that the given information is true. So we need to verify it.

Let's say that the interior angle of the regular polygon has a measure of x degrees, and the measure of the exterior angle of that polygon is 4x degrees.

Since the sum of the interior and the exterior angles of the polygon is 180 degrees (a straight line), the interior angle is 36 degrees.

4x + x = 180

5x = 180

5x/5 = 180/5

x = 36

The sum of the angles of a polygon = 180⁰(n - 2), where n is the number of the sides of the polygon.

The measure of one of the angles of a polygon = 180⁰(n - 2)/n. Substituting the angle measure of 36⁰ into this formula, we have:

36⁰ = 180⁰(n - 2)/n (multiply by n to both sides)

36⁰n = 180⁰(n - 2)

36⁰n = 180⁰n - 360⁰ (add 360⁰ and subtract 36⁰n to both sides)

360⁰ = 144⁰n (divide by 144⁰ to both sides)

2.5 = n !!

That means that a such normal polygon does not exist.