Extending the range of calipers

[WmRMeyers]

I was surprised that no one asked how I arrived at the formula I used in the original post. It's not obvious from casual inspection how one can calculate the radius knowing only the chord and the sagitta.

So, I documented the derivation and it's shown in this diagram...



The starting point for the derivation uses something termed the "Intersecting Chords Theorem". Despite the fact that it's discussed in Euclid's Elements, many folks are unaware of this very useful geometry tool. Basically, it says that if two chords intersect inside a circle, the product of the two parts of each chord are equal. Again, this isn't terribly obvious so the proof is given here...



The proof uses the fact that two triangles are shown to be similar. Similar triangles are not the same as congruent triangles. In congruent triangles both the corresponding sides and angles match. In similar triangles, only the angles match. All 30-60-90 triangles are similar but they're not all congruent.

I have to take it on faith that SOMEBODY knows what they're doing. But it sure ain't me! Almost have to have told this story here before, but once again: Asked my Algebra teacher why I needed to learn this algebra crap in 10th grade. He said "So you can graduate from high school." I said "wanna bet?" Despite the fact that I could not do the math, he passed me, barely, in both algebra I & II. Did me no favors, either. 12th grade/senior year, had to take one more math credit, and they had one for practical math, including how to balance a checkbook. I managed to pass that one with decent grades. Though it took until the introduction of Microsoft Money for me to be able to keep my checking account balanced. They stopped producing of MS Money in 2007, but I still use it. https://answers.microsoft.com/en-us/...1-14763116568e Enlisted in the USAF in the Fall of 1973, after just barely graduating based on how stupid I was in the 9th grade, and working in several fields in the USAF I learned a bunch of uses for the algebra that my teacher didn't mention. I suppose it's because he didn't believe that I would find it useful. Could also be that he was tired of dealing with ID10T problems. For years I was mad at him about that. I wanted to be an astronaut, and if he'd told me I needed algebra (and a bunch of other math) to do that I might have applied myself earlier. At this point in my life I don't suppose I could reasonably blame him. I am pretty sure he died the day he found out I was teaching math, though.

Partially because I wanted to be a better teacher than he was, I told that story to my students, and often to their parents, quite a bit. Some folks just can't seem to learn higher maths, and I'm apparently one of them, but I don't know if early intervention might have helped me do better long term. Trying to learn such things as an adult is often more difficult than for the young. When I was stairstepping myself through the rock math classes at Rose State College, one of my classmates was an older lady (compared to my age back then, when I was in my 40's) who had taken each of her rock math courses three times, and was finishing up her second try at the Intro to Algebra class we were both in. She was a grandmother who'd dropped out of school in her teens. It took her three tries to get through each class, and learn all the basic info she needed to pass to the next level class. She knew that when she finished this class, she only had to take it one more time...

My oldest daughter got ASVAB scores sufficient to try the US Navy's Nuclear Power school. Wanted to do something Dad hadn't already done. She was the only student in her class of 38 who didn't have calculus and calculus-based physics in high school. The Navy expects a 70% fail rate in that class, and only 8 students graduated. She was the 8th. She said that the Khan Academy https://www.khanacademy.org/ got her through that class. She was the 8th graduate, and 8th in her class. She got a letter of commendation out of it. She was the only student so ill-prepared for the class to actually graduate in Navy history. Pretty sure she got her math brain from her mom, not her dad. And her stubborn from both of us.

Bill