Im looking at the following diagram,
Here, I am interested in a relation between $a,s,c$, assuming that $r >> 1$ my first thought is to relate then using the pythagorean theorem, $$ c^2 = a^2 + s^2$$ Is making sure an approximation a good approach and is there an alternate way to relate my variables? Thank you in advance!
You can apply the cosine rule: $$ c^2=(a+r)^2+r^2-2r(a+r)\cos\theta, $$ that is: $$ c^2=a^2+2r(a+r)(1-\cos\theta). $$ As $\theta=s/r$, if $r\gg s$ we can approximate $\cos\theta\approx1-{1\over2}\theta^2=1-{1\over2}s^2/r^2$, which gives: $$ c^2=a^2+s^2\left(1+{a\over r}\right). $$ If $r\gg a$ the term $a/r$ is negligible and you obtain $c^2=a^2+s^2$.