Question

What is the internal angle of a regular `17`-sided polygon?

Answer 1

`(15pi)/17` radians, or about `158^@ 49' 24`"

Explanation:

A heptadecagon (`17` sided polygon) can be dissected into `15` triangles whose internal angles sum to the sum of the internal angles of the heptadecagon.

The internal angles of a (plane) triangle sum to `pi` radians or `180^@`.

So the total sum of the internal angles of a heptadecagon is:

`15pi` radians `" "` or `" "15 * 180^@ = 2700^@`

So each interior angle in a regular heptadecagon is:

`(15pi)/17` radians

`2700^@/17 ~~ 158^@ 49' 24`"

Bonus

Note that `17` is a Fermat prime since `17=2^(2^2)-1` is of the form `2^(2^n)-1`.

As a result, the regular heptadecagon is one of the few prime sided figures constructable using an unmarked ruler and pair of compasses - that is using a classical construction.

Answer 2

`158.82°`

Explanation:

A useful way to work with the angles in polygons is using the exterior angles.

The sum of the exterior angles of any polygon is always `360°`

In a regular polygon with `n` sides, the size of each exterior angle, `beta` can be calculated from:

`beta = (360°)/n`

The interior angle `(theta)`of the polygon is the supplement of `beta`

`theta = 180°-beta`

So, for a polygon with 17 sides,

`theta = 180°- (360°)/17`

`theta = 180° -21.18°`

`theta = 158.82°`