Radius help

[Mike O.]

I have a question for all of the math guys here. Thanks in advance for your help.

A few months ago we were trimming a house with custom moldings. We had several arched top windows that needed trim to match. These windows had a top section that was arched and the arch was really just a part of a true circle but less than a half circle. In other words if you knew the radius you could make the trim piece to match. Now, I've done a lot of these and I normally mark a plumb line in the center of the window and use a stick long enough to move up and down the line and change the length until my stick follows the window. This gives me the radius and I'm good to go. In this case, since the moldings were to match, the millwork sent out the sales guy and he got out his tape and measured the length across the arch (where the arch hit the vertical sides) then measured the distance from that line up plumb in the center of the window which would be the highest point of the arch. He wrote down the measurements and left. My partner and I both had this look on our faces that said "I'll believe it when I see it!". A few days later (very much to our surprise) the trim pieces arrived and were correct.

My question is how the hell did he do that? Does anyone know what formula might be used to find the radius when you only have the two measurements mentioned above?

I have two arched windows to trim in the house we're currently working on. Window A arch length 57 1/2" rise 10 7/8" Window B arch length 53 1/2" rise 9 1/2"

My preliminary stick method on window A is 43 3/8" radius and surprisingly window B is about the same.

If anyone could post a formula for me I would appreciate it very much. If not, I'll just keep poking a stick at it.:-)

Thanks.

Mike O.

[Swingman]

IIRC:

r = (c² / 8h) + (h / 2)

Where c = the chord (your "Length" above) and h= height (your "rise" above)

My brain is too tired to attempt to prove it at the moment ... try it and see if it comes close to your expectations. if it is wrong, rest assured it will be well pointed out, ad infinitum. :(

[Guess who]

Pythagoras [and a little algebra]. End result:

Distance across chord = "2c", height from chord to circle [called "segment height" or "saggita"], = "h" R = (h^2 + c^2)/(2h)

[Morris Dovey]

| r = (c2 / 8h) + (h / 2) | | Where c = the chord (your "Length" above) and h= height (your | "rise" above) | | My brain is too tired to attempt to prove it at the moment ... try | it and see if it comes close to your expectations. if it is wrong, | rest assured it will be well pointed out, ad infinitum. :(

Check the link below. Works for both coves and arches.

-- Morris Dovey DeSoto Solar DeSoto, Iowa USA

[Guess who]

It's right. I used "c" for semi-chord; you use it for the entire chord length. I use one fraction, you separate it into two. Otherwise it's the same end result.

Radius = "R", semi-chord = "c". By difference the other side of a right triangle from chord to center = "R - h".

By Pythagoras... R^2 = (R-h)^2 + c^2

Expand and simplify for "R" in terms of "c" and "h".

[Leon]

If you have the width of the opening and the distance from the top of the arc down to where the arc stops you can use a CAD program to determine the radius. I suspect that is what he did.

[Duane Bozarth]

Probably not... :)

See other posts for what he "probably" did.

[DJ Delorie]

That, and a form to do the math for you...

[Leon]

I do them that way.

[CW]

I have to agree with Leon. He probably used a cad program, though it certainly doesn't have to be done that way. Most anyplace that build anything these days uses cad. Work like this generally go through the planner, or programmer if done on a CNC, and that would be the cad man. If he had it, no reason he wouldn't use it.

[bridger]

how he did it, I bet is he got a measurement close enough to pick the arch out of the seven or whatever number they make. in other words, it wasn't an exact measurement....

[Guess who]

Yes, it can be done that way:

1. Draw the chord.
2. Draw the height from chord center.
3. Join the height top point to the chord ends.
4. Draw the perpendicular bisectors of those two smaller chords.

Where they meet will be the circle center. Dimension the radius from there to a chord end point.

[Jack Fearnley]

I believe the formula is as follows: Let a = half the arch length Let b = rise

then radius = (a*a + b*b)/(2*b)

where * means multiply.

In your two examples above I get 43.44" and 42.21" respectively.

It took Pythgoras and a bit of manipulation.

Best regards, Jack Fearnley

[Chris Melanson]

Hope this helps a: 43 7/16 b: 42 13/16 CAD works great for things like this if you need any other help just ask.

Chris Melanson BLH Millwork LTD.

[Mike O.]

It worked great! My stick was only off by 1/16! I also went to DJ's site and plugged in the numbers and everything matched.

Thank you very much and thanks to everyone else who answered!

BTW, now I just feel like a dumb ass! ;-)

Mike O.

[Mike O.]

My very scientific stick was only off by 1/16". :-)

Thank you!

Mike O.

[Leon]

With AutoCAD the radius dimension tool will show the radius. Simply draw a line the width of the window. From the midpoint draw a perpendicular line the height of the arc. Then draw an arc through all 3 points and use the radius dimension tool to show the radius of the arc.

[Jeff Gorman]

"Mike O." wrote

It might be useful to know that he first measured the chord of the arc and then the sagitta.

Jeff G

[Lowell Holmes]

What in blazes is the sagitta?

I suppose you fellows have never heard of a Smoleys book of tables. :-)

[Guess who]

Sigh. Yes, you can do that too. :-) [using an older version of DeltaCad]