SOLUTION: How many sides has a regular polygon whose interior angle is 22½

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Question 1179357: How many sides has a regular polygon whose interior angle is 22½
Found 3 solutions by ankor@dixie-net.com, MathLover1, ikleyn:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
How many sides has a regular polygon whose interior angle is 22½
:
360%2F22.5 = 16 sides

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Since an interior angle is 22%261%2F2=22.5 degrees, its adjacent exterior angle is 180-22.5=+157.5 degrees.
Exterior angles of any polygon always add up to 360 degrees. With the polygon being regular, we can just divide 360+by 157.5 to get the number of sides n.
n=360%2F157.5=2.2857142857142856
so, there is no a regular polygon whose interior angle is 22%261%2F2°

double check this way:
Let n be the number of sides of a regular polygon whose interior angles are each 22.5°.
Then %28%28n-2%29%2Fn%29%2A180+=+22.5°.
%28n-2%29180=+22.5n
180n-360=+22.5n
180n-22.5n=+360
157.5n=+360
n=+360%2F157.5
n=+2.2857142857142856=>
so, there is no a regular polygon whose interior angle is 22%261%2F2°
Thus, 22%261%2F2° cannot be an interior angle of a regular polygon.



Answer by ikleyn(52804) About Me  (Show Source):
You can put this solution on YOUR website!
.
How many sides has a regular polygon whose interior angle is 22½
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Having two different answers that contradict each other,  may  PERPLEX  you.

So I came to make things clear to you.


                        Your problem in the post is posed  INCORRECTLY.

        There is no a regular polygon with the interior angle of 22 1/2 degrees.
        If the EXTERIOR angle of a regular polygon is 22 1/2 degrees,
        then the number of sides (same as the number of vertices) is 360%2F22.5 = 16.