Here's the math:
Radius = R, center height = h, full chord length = C.
Pythagoras: (R - h)^2 + (C/2)^2 = R^2
Simplify: R^2 - 2Rh + h^2 + C^2 / 4= R^2
2Rh = h^2 + C^2 / 4
R = (h^2)/(2h) + (C^2 / 4)/(2h)
* R = h/2 + C^2/8h *
This can be combined into one fraction, but why bother ...just stick
the numbers in the above.
So with h = 3" and C = 36", you'd have
R = 3/2 + (36 x 36)/(8 x 3) = 55 1/2"
Also easily checked with a CAD program.
While I agree that, with the problem as originally posted, a perfect
arc will probably not fit into most work done by a mason, I'll have to
disagree with your statement above.
With the with design and manufacturing techniques used today there are
a lot of instances where the mathematical solution will yield very
good results when trying to work with existing architectural elements.
I was part of the earlier discussion about finding a radius for a
window while knowing just the cord length and height of the arc. I
can testify that not only did the formulas work great but the
resulting jam extensions and trim pieces were very accurate. The time
savings involved in not making a template (or using trial and error to
find the radius) makes it well worth keeping a few formulas in the
notebook along with the calculator.
I'm a machinist. Your idea of fit and mine obviously differ. At work, I have
two calculators. One at each end of the shop and they are used extensively.
At home, to fit something like what was under discussion, I would not
blindly rely on the numbers, even though I know they are correct,
Variations are very common. In the case of your windows, you have a lot
better chance of getting a good fit by formula when fitting to factory made
windows which are jig built and machine made. With something that was built
in place, I wouldn't chance it. A test piece would be a good idea and don't
be surprised if hand fitting is needed.
You are correct. There has to be an element of both; practical and
theory for the best advantage. Been there, done that in the steel
business. In particular I assisted my brother [structural steel] by
doing calculations on an arch for the recovery after the 911 disaster.
His practical knowledge far exceeded [he's now passed on] mine, but he
said I saved him many hours of work. On another occasion he couldn't
get the CAD dawings to give him what he expected. I saw the flaw
through calculation that the had been given two sets of unmatching
detail on two different drawings. Both are necessary for a better
job: practical experience and base theory. He knew there had to be a
mistake, and I found it. It's not enough to use a sledgehammer to get
the motor started. You have to know where to hit it.
Get you a strip of hardwood say 1/8" thick by 5 or so feet. Tie a string
to one end. Now loop it around the other end and form a bow. Put it in
the arc and let it spring fit to your opening. Tie off the sting so the
bow will not spring back when removed. Mark the center of the bow from
the center of the arc. Do the same with the ends. All you do then is lay
it on your wood and trace.
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