I know the radius and length of an arc. how do I figure out the angle from center point to the ends of the arc? ( I'm trying to figure out the length of the arc if I offset/increase radius by $2$" and maintain the same angle from center point)
[Math] Radius & arc length = what angle from c.p.
geometry
Related Solutions
The length of the arc is proportional to the angle by proposition 6.33 in Euclid's Elements. We can choose units to make the constant of proportionality equal to $1$ for a circle of radius $1$. Those units are called radians.
[I give the Euclid reference since it uses the notion of angle that might be familiar to you. Nowadays we find it more economical to define the trignometric functions without any appeal to geometry (for eg. via the complex exponential), then define angles by inverting one of these functions, and then derive your result as an exercise in calculus.
Here is a simple hand-waving argument along those lines. Let OA and OB be two radii of a circle of radius $r$. Then by simple trignometry the length of the line segment AB is $2r\sin(\theta/2)$. For small enough $\theta$, $\sin(\theta/2) \approx \theta/2$ so the length of AB is approximately $r\theta$. In this case the length of AB is also close to the length of the arc AB hence that is also approximately $r\theta$. We can make the approximation as good as we like by making $\theta$ small enough. We can handle large arcs by approximating it by a large number of small segments of this sort.]
So you know the distance $d=CD$ between center and boundary. Then you can write
$$\cos\angle ECD = \tfrac dr \qquad \angle ECF = 2\angle ECD = 2\arccos\tfrac dr$$
Now the length of an arc is $r$ times its angle, so the outside arc is $r\angle ECF$ and the inside arc is
$$s = r(2\pi-\angle ECF)=2r(\pi-\arccos\tfrac dr)$$
You want that number to be equal to some given value, so you want to solve the above equation for $r$. Unfortunately, that equation is transcendental, so you can't expect a closed form solution to your problem. Your best bet is some form of iterative numeric approximation.
As you can see from that plot, you can expect that for many possible ratios of $\frac sd$, you get two distinct solutions for $r$.
The apex (with the vertical tangent) appears to be at
$$ s/d\approx 5.94338774142760424162091392488776998544210982523814509283191138267355981 \\ r/d\approx 1.06193134974748196175464922830803488867448733227482933642882008697882597 $$
Best Answer
In general, we have the following formula: $$ l=r\theta $$ where $\theta$ is the angle at the center of the circle (in radians) and $l$ is the arc length. In your case you just need to calculate $\frac lr$.