Question

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

Answer 1

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`.