I have to admit upfront that while I did fine at high school Geometry, it probably remains one of the subjects where I thought, 'okay, when am I ever going to use this ?' And sort of blanked it out of my mind for direct reference.
Notably, this is not the only subject that has 'come back to bite me', or in undergrad studying first in Philosophy, I took a course on logic, where we learned about 'truth tables'-- And lo-and-behold, some 15 years later I find in FPGA's and system state logic, what do you have, but 'truth tables !'.
In any case, unfortunately my 'geometry' is maybe a little too far back and too fuzzy. But I am trying to work on a little side project at the time, and rather than 'theoretical' this actually applies to a 'real world' type example:
So say I have this series of arcs/chords, and I am trying to determine the radius of the circle they are composed of.
I know there are formula's out there such as this.
But that requires you to know 'h' or how far the center of the circle is.
Since these drawings/sketches come off a piece of machinery, let's just say it is 'not reasonably possible' for me to figure out the actual origin of the circle.
Yet, it seems there must be some other way I can determine the figures that I want, no ?
But... I'm a little confused as to how to go about it, or what formulae to use.
Sorry for being 'naïve' or just 'forgetting', but any assistance is greatly appreciated.
The formula you choose should depend on what information about the arc is easiest to measure. Two variables that seem convenient are the length of the intersecting chord (w) and the what i'll call the height of the arc (w)
this is your arc
This is the given arc shown in the circle we are looking for.
$$\begin{align}
\frac{w}{2}\cdot \frac{w}{2} = h \cdot (2r-h) && \text{by intersecting chords theorem}\\
\frac{w^2}{4} = 2rh -h^2 \\
\frac{w^2}{8h} + \frac{h}{2} = r\\
\end{align} $$
And there you have it.
Link to intersecting chords theorem. https://en.wikipedia.org/wiki/Intersecting_chords_theorem