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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.

 Sep 10, 2024
 #1
avatar+37107 
+2

Interior angle + exterior angle = 180 degrees 

  23 e     + e = 180 

    e = 7.5 degrees     so interior angle = 172.5 

 

Sum of interior angles of n-gon is  ( n-2)*180 

   each angle is then   (n-2)*180 / n    and this equals  172.5  degrees   solve for n 

         (n-2)*180 / n = 172.5 

           180n -360 = 172.5 n 

                n = 48 sides 

 

 

Maybe a bit simpler to realize exterior angles sum to 360 

                        n (7.5) = 360 

                             n = 48 sides 

 Sep 11, 2024
edited by ElectricPavlov  Sep 11, 2024
 #2
avatar+37107 
+2

User Bosco pointed out an error in my solution..... 

Here is a correction 

Interior angle + exterior angle = 180 degrees 

  2 e     + e = 180 

    e = 60 degrees     so interior angle = 120 degrees 

 

Sum of interior angles of n-gon is  ( n-2)*180 

   each angle is then   (n-2)*180 / n    and this equals  120  degrees   solve for n 

         (n-2)*180 / n = 120 

           180n -360 = 120 n 

                n =6 sides 

 

 

Maybe a bit simpler to realize exterior angles sum to 360 

                        n (60) = 360 

                             n = 6 sides 

 

THANX , Bosco !! 

ElectricPavlov  Sep 13, 2024

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