How To Calculate Length Of Screw Thread? I'm having a tough time searching for the answer to this as most of the results are in the realm of mechanical engineering and appear to be unrelated, I apologize if this is a really obvious question.
Say there is a circular arc in $2$ dimensions covering $90$ degrees at a radius of $21$. I know the length of the arc would be $\frac{21}{2}\pi$ or about $32.99$, but what if it were then stretched through the third dimension by some number $x$?  How do you calculate the screw thread length?

To define terms, let's say we have a helix with radius R, height H, and number of turn N (not necessarily a whole number). 
The length of  a small piece of the helix is
$$ds = \sqrt{dx^2+dy^2+dz^2}$$
Since the 2D projection of this curve is a circle, we have 
$$dx^2+dy^2 = R^2d\theta ^2$$
Where $\theta$ is the angle of this small segment. We need to relate $dz$ to $d\theta$. We know that the total angle is $2\pi N$ and the total height is H, and that there is a linear relationship between z and $\theta$, so 
$$dz = \frac{H}{2\pi N} d\theta$$
So we then have 
$$ds = \sqrt{\left(R^2+\left(\frac{H}{2\pi N}\right)^2\right)}d\theta$$
Ans the arc length is
$$s = \int_{0}^{\theta_{TOT}}\sqrt{\left(R^2+\left(\frac{H}{2\pi N}\right)^2\right)}d\theta = \sqrt{R^2+\left(\frac{H}{2\pi N}\right)^2}\theta_{TOT}$$
Where $\theta_{TOT}$ is the total angle of the helix (so if it winds around 4 times, it would be $8\pi$)
