Question

How many degrees will skaters turn if they go once around a regular hexagon? A regular octagon? A regular polygon with n sides? Explain

Answer 1

Let us start with the explanation:

A rule of polygons says that the sum of the exterior angles always equals 360 degrees.

First of all we need to know the Interior and Exterior angle formulas:

It says the sum of the measures of the interior angles of a polygon with
`n` sides is
`(n-2)* 180=(n-2) 180`.

Now If we count one exterior angle at each vertex, the sum of the measures of the exterior angles of a polygon is always 360°.

Let's talk about Hexagon first : It has got 6 sides so,
`n=6`.

Since the skater is going around the hexagon, we need to find the sum of the measure of exterior angles ,

`(n-2)180=(6-2)* 180=4* 180=4 * \pi= 720^ \circ=4 \pi`

Talking about Octagon, it has got 8 sides so,
`n=8`. So the sum of the measure of exterior angles is:

Plugging the value of 'n' we get:

`(8-2) * 180=6 * 180=1080 ^\circ=6 \pi`

Now, finding the sum of measure of exterior angles for a
`n` sided polygon. We get:

`(n-2)* 180= (n-2) \pi`

Therefore, when the skater goes around a hexagon he covers
`720^\circ` , when he goes around an octagon he covers
`1080^\circ`, and when he goes around a regular
`n` sided polygon , he covers
`(n-2)\pi=(n-2)* 180 ^\circ`.