Question

i noticed something interesting

Answer 1

an equilateral triangles interior angles are all 60 degrees. and a squares angles are all 90. and a hexagons is 120. and an octagons is 135.

Answer 2

is their a relationship between number of sides and angles

Answer 3

yes.

you are looking at the interior angles.
for nice convex figures (and let the sides by equal)
the exterior angles (formed by how much you turn at each corner)
must add up to 360º (because we end up where we started... we turn a full 360º )
if the figure has n sides, and the the exterior angles add up to 360, then nX=360 and X= 360/n

the interior angle will be 180 - X (where X is the exterior angle)
but X = 360/n, so the interior angle will be 180 - 360/n
or
\[ \frac{180n}{n} - \frac{360}{n}= \frac{180n - 360}{n} \]
if we factor 180 out of the top, we can write it as
\[ \frac{180}{n} (n-2) \]

Let's test this. for n= 3 we get
\[ \frac{180}{3} (3-2) = 60 \]
which works!

also the sum of the interior angles (all n of them) will be n* interior angle, which is
180(n-2)

for n=3 we get 180(3-2) = 180