Question 624495: The ratio of each interior angles to each exterior angle of a regular polygon is 4:1. Find the number of sides of the polygon
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! I = interior angle
E = exterior angle
the sum of I and E is equal to 180 (they are supplementary).
the ratio of I to E is 4:1
this means that I = 4E
since I + E = 180 and I = 4E, you get:
4E + E = 180 which simplifies to:
5E = 180 which results in:
E = 36 degrees
since the sum of the exterior angles of a polygon is equal to 360 degrees, you divide 360 by 36 to get the number of sides of the polygon.
360/36 = 10 which means the polygon has 10 sides which makes it a decagon.
working back from the knowledge that the polygon is a decagon, we can use the formula for the interior angle of a polygon to find the interior angle which should be equal to 180 - 36 = 144 degrees.
the formula for the interior angles of a polygon is:
I = 180(n-2)/n
since the decagon has 10 sides, this formula becomes:
I = 180(8)/10 which is equal to 1440/10 which is equal to 144 degrees.
Since I = 144, E = 180 - 144 = 36 degrees.
Decagon is confirmed.
One last confirmation:
ratio of I to E is 4:1
144:36 is the same as 4:1 because 4*36 = 144.
We're good all around.
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