Polygons

In a regular polygon, the sum of the interior angles is (n-2)*180 degrees, where n is the number of sides. Each interior angle of the given polygon measures 2x degrees, where x is the measure of each exterior angle.

 

Therefore, we have the equation:

(n-2)*180 = n * 2x

 

Simplifying this equation:

 

180n - 360 = 2nx

2x = 180 - 360/n

x = (180 - 360/n) / 2

 

Now, we know that the measure of each interior angle is 2x:

 

2x = 2 * (180 - 360/n) / 2

x = 180 - 360/n

 

We are given that the measure of each interior angle is twice the measure of each exterior angle, so:

 

2x = 2 * (180 - 360/n) = 360/n

 

Substituting the expression for x from before:

 

2 * (180 - 360/n) = 360/n

360 - 720/n = 360/n

720/n = 0

 

Since n cannot be zero, the only solution is if n = 720.

 

Therefore, the regular polygon has 720 sides.

 

In a certain regular polygon, the measure of each interior angle is $2$ times the measure of each exterior angle.  Find the number of sides in this regular polygon.     

 

The sum of the interior angle and the exterior angle at a vertex is 180^o.  

 

Since the value of the interior angle is twice that of the exterior angle, the exterior angle must equal 60^o.  

 

The sum of all the exterior angles of a regular polygon is 360^o irrespective of the number of vertexes.  

 

Since one exterior angle is 60^o there must be six of them to add up to the total of 360^o. 

 

Since there are the same number of sides that there are vertexes, the polygon has six sides.  

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