Why does a side of an angle.....?

Question

As the number of sides in a regular polygon increases, the angle of the side increase? Why?

Answer 1

Think of it as though the polygon is turning more and more into a circle as the number of sides increases. Ultimately it reaches 360º which is the maximum.

Does that make sense?

Answer 2

Let's consider a regular polygon as if it were an apple pie with N sides. Lets cut the pie into N pieces. Each piece will be a triangle. Therefore the sum of all the triangle angles is 180N.

Put them all back together into the pie pan. All the pie pieces touch at the centre of the pie pan and the total angle of the N pieces together at the centre adds up to 360 degrees ( a circle).

This means that the remainder of the degrees is shared equally at the N corners of the regular polygon.

So the angle at the corners is:

(180N-360)/N

N angle
== ======
3 60
4 90
5 108
6 120
7 128.5714286
8 135
9 140
10 144
11 147.2727273
12 150
13 152.3076923
100 176.4
200 178.2

looks like the limit will be 180

Answer 3

Consider toothpicks of all the same length. Two toothpicks form a two-sided "polygon" with zero degrees between them. Add a third toothpick, and room must be made: the original two must move away from one another. The angle that shrinks is the opposite angle, the one outside the polygon. It can never shrink to 180 degrees - that is what in calculus is called a "limit," since it can not go past that number.

Answer 4

A regular polygon with n sides has n interior angles of equal measure. They're each 180 - 360/n. As n gets bigger, 360/n gets smaller. If you're subtracting a smaller and smaller quantity from 180, then 180 - 360/n is going to get bigger as n gets bigger.

Answer 5

b/c as the side measures start to get shorter, the degrees and the angles have to expand to ajust with this.

Answer 6

it makes the shape bigger