In a regular polygon the wxterior angle is one fifth of an interior angle. How many sides has the polygon?

1 Answer
David O.
Feb 22, 2018

Denoting interior angle by #theta#, the exterior angle by #phi#, and the number of sides in the polygon by #n#,

It is given #theta#/5 = #phi#,
that is,
#theta# = 5 #phi#

It is noted
#theta# + #phi# = #pi# (angles on a straight line)

Substituting
5 #phi# + #phi# = #pi#

that is
6 #phi# = #pi#
which implies
#phi# = #pi#/6

Noting
n #phi# = 2 #pi# (sum of exterior angles of a polygon)

this implies
n #pi# / 6 = 2 #pi#

which implies
n = 12

Summary
As the relative size of the internal and external angles is know, it is possible to express the sum of the internal angle and the external angle as a multiple of the external angle. The size of the external angle may then be calculated by noting that the sum of the internal and external angle is #pi#. This is then used to calculate the number of sides in the regular polygon by noting that the sum of all of the external angles is 2 #pi#.

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