Simple - Go here: Circle Calculator Mathematically, there is no closed formula. This problem is very similar to the Squaring the Circle problem.
The cut & fit method of design works but is tedious & difficult to digitize accurately. Before CAD it used to be the quickest way to go, but CAD makes everything so crisp & accurate & fast you do the entire model on the computer before picking up a scissors to do it for real. I solve my "rod & rope" problem doing iterations with a macro in Excel - I just didn't like the inelegance of not having a simple f(x). The iteration produces an answer much closer than the finest resolution of the printer, so I'm not stymied if it turns out there's no f(x), as Tom suspects. It's more the principle of the thing that bugs me - that there're apparently some mathematical relationships that cannot be defined as some f(x) - and I thought perhaps there IS an f(x) for this problem but getting to it is beyond my meagre ten cents worth of knowledge of algebra & trig. So maybe I'm bats after all, chasing an answer I can get along without, but, like a musical melody in your head that you can't name and can't get out of your head, or an itch you can't reach, some things just won't leave a poor mind alone.
I ground through the math again. You had the proper approach, but your problem is that it ends up being a function of sin(x)/x, which is also known as sinc(x). Like many transcendental functions, there is no direct solution. You can either use numerical methods to get an approximation, or use other measurements for which an exact solution is possible. For example, if you use the chord length and the perpendicular distance from the chord to the arc, the formula is trivial. Tom
Thanks, Tom. The critical word here is "transcendental", which I looked up in Google. There ARE relationships with no f(x), something I should have learned in at least one of the many math classes in my long forgotten past, but never got until just now.
Interestingly, the circle calculator does not give what we've been calling the string length above as a permitted or provided parameter. Maybe that's because it's transcendental.
Most transcendental functions are pretty nasty. Even all of the common trig functions such as sine and cosine are bad, but because we use them all of the time, we have nice calculators programmed to do the infinite series calculations necessary to get approximate values. You could do the same thing with the sinc function, in which case such a calculator, if it existed, would make your problem easy to work. Note that it would still be a numerical solution, and not a function. Exactly. The circle calculator exploits an interesting geometric theorem called the Intersecting Chord Theorem. The Intersecting Chord Theorem states that whenever two chords of a circle (AB & CD) intersect at point E, then AE * EB = CE * ED. This little beauty of geometry simplifies the math for the circle calculator, but it doesn't get around the fundamental problem with the arc-length/chord-length/Radius calculation. There isn't an easy way around it. Tom
