Math help, need a formula for some data on an arc

[Debater Slayer]

Presumably R-h = R Cos φ . So h = R (1-Cos φ ). Or have I made a mistake somewhere?

(Blasted software refuses to display theta symbol, which is why I've resorted to phi instead.)

And then if we call the length along the circumference from the apex c, φ in radians will be c/R, won't it?

So finally we have h =R(1- Cos c/R), unless I've screwed up somewhere...........which is far from unlikely .

Yes, your explanation is very clear, and it appears correct to me (so if it's wrong anywhere, we're both wrong ). Personally, if I wanted to calculate θ, I would probably just use arccosine of (R-h)/R.

The trickiest part for me to wrap my head around was the change in the point from which one measured the height, since I wasn't entirely sure exactly which values aside from height would change in the process.

 

[exchemist]

hmmm..

imagine a set of right triangles where h is always the same (164)
now imagine we construct the various triangles by using various short side lengths along the O line (starting at O and extending left or right). The length of these short sides will be the same as the length along the chord line, right?

So if we take an example from near the middle of BM, we'll have a right triangle with h = 164 and the short side being, perhaps, 18. pythagagoras takes us home from there, right?

Yes, having had another look this morning I don't see a more elegant way of doing it. You have a formula in which if d is the distance inward from the end of the arc, i.e. your 6in increments, and h is the height at that point above the chord, you have (36-d)² + (164-h)² = 164². So 164-h = √(164²-(36-d)²), i.e.

h=164- √(164²- (36-d)²).

It looks a bit clumsy but that seems to be the formula.