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# Polygons

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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.

Dec 18, 2023

#1
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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.

Dec 18, 2023
#2
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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 180o.

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

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

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

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

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Dec 18, 2023