Calculating a radius… 3 points known
davidmeiland
| Posted in General Discussion on
I have some arches to frame. They are all specified in terms of 3 points… the overall width of the opening (window opening or hallway width), and the amount that the arch rises in that width. If I understand correctly, this means I have 3 point of an arc, and it’s a straightforward calculation for someone that actually knows how.
Can anyone give me the Construction Master keystrokes for this? Or, is there a website with calculators that would solve this?
Replies
Dave,
I don't know what CM you have but I have the CMP Trig Plus 111. I'll give you an example using 6' as the width and 1' as the height.
6 [Feet] [Run]
1 [Feet] [Rise]
Press {Conv] [Diag] Returns - 5' (Radius)
On my calculator right above the "diag" button is the word "radius" in yellow so when you hit [Conv] [Diag] is gives you the radius.
Looks like I didn't spend enough... got the CM IV. It has a 'Conv' button in blue, and there's a 'Circ' button with 'Arc' written above it in blue. However, it does nothing when I enter rise and run and then press (Conv)Circ, and if I just press Circ it returns 'Error 4'. I'm going to dig around and see if I can find the book that came with it.
David,You can use this formula for now.Width = 72"Height = 12"Formula - [W² ÷ 8 x H] + [H ÷ 2] = (Radius) 72 x 72 = 51848 x 12 = 965184/96 = 5412/2 = 654 + 6 = 60" (Radius)Joe Carola
Edited 6/27/2005 1:31 pm ET by Framer
Awesome... I'm sending you a virtual case of beer!
David,Just curious if you found the book that came with your CMIV and what the answer is because I've never used a CMIV before.Joe Carola
I didn't find it... it's here somewhere... but where?
There's a phone number on the inside of the flap of the Armadillo case that came with it. I called that and the lady said that someone would give me the keystrokes I needed (if in fact my calc would do it) but I was on hold too long and had to go.
To solve this equation we need to refer back to our highschool geometry text:
*******
* |h *
* d | d *
*----------+----------*
* | / *
* r-h| /r *
* | / *
*------------+------------*
I've called the length of the chord 2d, to simplify the calculations,
so half the chord is d. The distance from the chord to the arc (your
"chord height") is h, and the unknown radius of the circle is r. Then
the distance from the center to the chord is (r-h). Incidentally, the
technical term for h is the "sagitta" of the chord, and r-h is the
"apothem." You don't hear those words used much. "Sagitta" is Latin
for arrow. If you think of the arc as the bow and the chord as the
string, you can see why.
Look at the right triangle with sides d, r-h, and r. Using the
Pythagorean Theorem, we can say that:
d^2 + (r-h)^2 = r^2 ("^2" means squared)
which expands to:
d^2 + r^2 - 2rh + h^2 = r^2
and by subtracting r^2 from both sides we get:
d^2 - 2rh + h^2 = 0
Now we can add 2rh to both sides:
d^2 + h^2 = 2rh
and divide both sides by 2h to get:
d^2 + h^2
r = ---------
2h
So the radius is just the sum of the squares of the height and half the
length, divided by twice the height.
dude, you have way too much time on your hands to type something like that.
Thanks for taking the time to type something like that.
Does it have to be a circular arch?I would have thought that some kind of ellipse would be more common since If you were spanning a 6' wide window you wouldn't necessarily want the thing going up 6'.In fact, the original post gave 3 points of data to start with. I would assume that the center of the arch (the highest point) would be the centre of the window or whatever.Here's what I get.1. Assume the centre of the arch is the centre of the window. Set the width of the window = W and use the center of the window (horizontally) as the 0 point for the x=axis. Thus, the edges of the window would be at plus or minus W/2.2. Count the height of the arch H from the top of the window. 4X^2 Y^2
----- + ---- = 1
W^2 H^2Example.Width of window 3'. Desired height of arch 1'. 4X^2 Y^2
------ + ---- = 1
9 1 Simplify to, Y = Square Root [ 1 - 4X^2 / 9]If my house wasn't a 100 years old... there would go a really fine hobby.
Indeed, thanks for writing that.
So, I have [H^2 + (W/2)^2] / [2H]
Using that formula and a sample opening where W=63.5 and H=5.64, I get R=92.18.
I have several openings and just calc'd them all. Using Joe's formula and yours, I get slightly different results in some cases, identical in others. I'm not sure why, but they're very close, about .20 in approximately 90".
There are a bunch of these to frame tomorrow, several archtop windows and a barrel vault hallway. I can knock it out in one day since now I won't spend a bunch of time playing around and doing trial and error drawings on the floor. Throwing out the calculator would be about the same as losing money on the job.
I believe I have a perfect stick of VG fir lattice in the barn for a 10-foot trammel arm.
Y'all have a good night!
I would like to know how you use a trammel without lines to put it on. There is no trial and error in my method. While you were figuring out to use the calulator I would be nailing them up.
I did'nt come up with 85degrees....it's the 'example' of a radius in the little booklet that comes w/the CM IV.
That's all there is.
Did I mention I don't give a rat's azz how you do it? You seem intent on being smug and unhelpful... so shove it!
I'm sorry you feel that way. I thought I was trying to help until you made the rude remarks in 16.
If you'd read the thread a little more carefully... the title and what I've posted... you'd see that it's specifically a question about CALCULATING the value in question, and that I already wrote twice that I can solve for it using layout and am interested in the math. Your comments are that I should throw out the calculator and that you'd be nailing 'em while I'm still calculating. You could be the world's greatest carpenter, but that's not the information I was looking for.
I use a CAD package extensively in my work, and one of the many things my package (Cadkey for Windows) will do is draw an arc or circle given three points.
We're up on the top now of the frame we are building (see my Blogger site for progress) and have built everything from daily pages I generate from my full-scale 3D CAD model. Trusses arrive shortly after July 4.
The only mistakes we've made is nailing a few studs and joists on the wrong side of the line.
You've seen all the replies so far that show you the formula, which is derived using trig and analytic geometry.
All these guys that say it is quicker to lay it out by using the floor and tape measures are full of crap. Twenty seconds with any calculator with a square root function will do it, and you don't even have to bend over.Gene Davis, Davis Housewrights, Inc., Lake Placid, NY
You should read through before posting some arrogant bull ship like that. Just toot you own horn I'll build next to you any day. And you can choose the game.
How long did it take to put the studs and joists on the RIGHT side of the line?
All day long.Gene Davis, Davis Housewrights, Inc., Lake Placid, NY
Here is a website with the formula for deriving a radius from a chord and height. Remember that the height of the cord is measured or called out from where the jamb breaks from plumb into the curve to the center of the arch:
http://mathforum.org/library/drmath/view/55037.html
dang... can't we all just get along?!
Easy, quick. David asked for "how to calculate" and all you guys want to tell him is how to do it with strings and nails.
I finished up some rough stairs today, a two-landing winder, worked solo, and I'll bet you could not have beat my time. Didn't use a calculator either.
When you still were wearing Pampers, I was running union surveyor gangs, laying out line and grade for a work force that included about 80 carpenters, building steel mills, and greenfield pulp and paper mills.
I see you are in the D.C. area. Check out Somerset House, 5600 Wisconsin Ave., a 22-sty luxury condo in Chevy Chase. I ran the curtainwall part of that job. Might have seen you there.
Let's move this to the Tavern where it belongs.
Gene Davis, Davis Housewrights, Inc., Lake Placid, NY
Edited 6/29/2005 8:18 pm ET by Ima Wannabe
Careful with the pamper line old man. It was cloth back then But OK. Plus I wasn't telling anybody to do anything if you read my posts.
Edited 6/29/2005 8:27 pm ET by quicksilver
Even though I'm the guy who said you should lay out stairs with a computer program, I'd probably solve this problem by drawing it on the floor and doing the two perpendiculars, simply because that's an easy scheme to remember and envision, and any error introduced is apt to be minor.To use a calculator I'd have to dig out the formula or reinvent it on the fly -- would never be able to remember it unless I used it regularly. So for me it would be faster and more reliable to draw it out.
The simplest approach is by construction. You draw lines through each pair of adjacent points, then erect a perpendictular at the exact midpoint of each line. Where the perpendictulars intersect is the center of the circle (and, obviously, the distance from there to any of the points is the radius.).
Thanks for the replies.
DanH, I can definitely do this stuff using 'descriptive geometry' but I'm tired of being a 97-lb math weaking and my Dad would definitely be embarrassed. When he was alive I called him more than once from jobsites with trig questions and he always had the answer.
Old School? draw a small case "t" full scale on floor. Leg of t is centerline, Arms of t is width you want, portion of leg above arms is height you want.
Two tape measures. Hold end of one tape on end of left arm, hold other tape on top of leg, pull both tapes down leg, where same measurement on both tapes have same measurement crossing is your radius. get a stick put two nails in that distance. set your cut piece on top of arms and scribe mark.
Throw the caculator away.
Edited 6/27/2005 7:31 pm ET by doodabug
Very cool. I CAN visualize that. I have a similar method I use, but I don't agree with throwing the calculator away, but I do think that a carpy should not be so attached to one that we have to layout tomorrow because I forgot the construction master. The best calculator carpy I ever knew told me to keep away from those. Said they round down to quickly. I don't know I've kept away. I use a simple one on my Palm Pilot now. About the only thing I use calc for is stairs. geometry is first in my book
Old School? draw a small case "t" full scale on floor. Leg of t is centerline, Arms of t is width you want, portion of leg above arms is height you want.
Two tape measures. Hold end of one tape on end of left arm, hold other tape on top of leg, pull both tapes down leg, where same measurement on both tapes have same measurement crossing is your radius. get a stick put two nails in that distance. set your cut piece on top of arms and scribe mark.
Throw the caculator away.
I know your method works because I used to do the same thing before I even new about a calculator, but why would you say "Through the Calculator away"?
My example was 6' width and 1' height.
It takes all but 5 seconds to get the radius with the calculator which comes out to 5' Radius. I tack the piece of plywood on the floor that I will be cutting and mark my width, center line and height. I then square the center line down to 4' and that's where I mark the radius from.
You can draw you "t" as you said but forget the two tapes because it takes a lot longer than 5 seconds to do all that. Make your "t" and just mark 5' and your done.
There's nothing wrong with learning something new on a calculator from us young punks...........;-)
Joe Carola
Edited 6/28/2005 6:21 pm ET by Framer
I'm glad to hear somebody actually knows how to use one. I don't mind learning knew things. If I had one I would learn how to use it. You still have to draw the lines and mark the arc. Five seconds your way, two minutes mine.
I'm glad to hear somebody actually knows how to use one. I don't mind learning knew things. If I had one I would learn how to use it. You still have to draw the lines and mark the arc. Five seconds your way, two minutes mine.
Yes you do still have to draw the lines and mark the arc but the way I do it using the plywood it's done already, the plywood is scribed. Your marks are on the deck, you still have to set up your plywood to mark everything.
Joe Carola
you can get a copy of the book at http://www.calculated.com/ under product support
FWIW i use dodabugs method
Heres the example given in the CM IV manual;
Arc lengths.
find the arc length of a 85degree portion of a circle with a 5 foot diameter.5[feet][circ]
85[conv][circ]
[feet]This help you any?
How did you come up with the 85 degrees?
It just occured to me what was meant about specifying 3 points on the circle. Too much math, not enough building experience...I just figured out what everyone else already new about where the center of the circle is supposed to be :)and with that astounding flash of the obvious I verified that...Radius = W^2 / (8 x H) + (H/2) is correct.If you wanted an elliptical arch what I posted earlier is correct.If my house wasn't a 100 years old... there would go a really fine hobby.
Hey guys, speaking of the CM IV, but using a pencil first
If you think in Pythagorean terms, then using a^2 + b^2 = c^2,
Assign the rise to “a”,
Assign the run (1/2 the width of the arch) to “b”
The simplest formula will be (a^2 + b^2) / (2a)
or [c^2 / (2a)= radius of arch].
Basically the same as clarkconst’s excellent post, #6.
I quote (abridged), “So the radius is just the sum of the squares (c^2) of the height (a^2) and (+) half the length (b^2) [or = (c^2)], divided by twice the height (2a).” [c^2 / (2a)]
(CM IV method) I used my CM IV like this for years to solve for this task. I use the CM Pro for onsite calcs these days. The same method works on it too.
Enter the rise (a) and run (b)of the arches triangle to find the hypotenuse (c) of the first triangle. Divide the hypotenuse by 2 and enter this result as the rise of the second triangle. When you hit the Diag key it will return the radius of your arch.
As an example to solve for a 3’ arched opening with an 8” rise:
8 [Inch] [Rise]
18 [Inch] [Run] (This is 1/2 the width of the arch and I usually do this simply calc in my head)
[Diag] (returns) 19-3/4 inch, [Pitch] (returns) 5-3/8 inch
Now return to the [Diag] 19-3/4 inch
[divide] 2 (returns) 9-7/8 inch
[Rise] 9-7/8, [Run] 22-1/8 inch, [Diag] 24-1/4 inch
The radius for this example arch is 24-1/4”
I hope this is helpful.
Lost me there... how'd you get the 22-1/8" run of the second triangle in your example?
David,
<!----><!---->
In my example the two (or three) triangles are proportionate.
8” over 18” = 9-7/8” over 22-1/8” = 5-3/8” on (over) 12 pitch (the pitch is the third triangle, or pitch setting that the CM IV will use to solve the radius/hypotenuse of the second triangle with.)
A triangle with a proportionate pitch of 5-3/8” in 12 with a rise of 9-7/8” will have a run of 22-1/8” and a hypotenuse of 24-1/4”.
Since they are proportionate, you can enter the rise of the second triangle with ½ the hypotenuse of the first triangle and the hypotenuse [Diag] of this second triangle will be the radius of the arch. Try doing it like in my example with your CM IV. You should get the same results. The run (22-1/8”) of the second triangle is really irrelevant and I probably should have not included it in the explanation except it helps show the equal proportions of the triangles.
David,
Take a look at this. It might help.
I've been using that method for quite a while and it has served me well, but instead of using a calulator I use geaometry. The only problem I could possibly suggest is that laying it out like this on the deck I find when I snap my two perpendicular bisectors the two chalk line thicknesses leave a little bit of the final call up to interpretation. So I would think for a purist a calculator would give a more exact radius. In the same sense that a carpenter calculating a rafter on the calculator will come up with a slightly different length than the carpenter stepping off with a framing square. This is basically minutia, because these methods, my geomaetricapproach, your formula oriented mathematical approach and doodabugs method (also geometric) are fine for carpentry. Once the radius is found it is accurately and repeatedly duplicating it that will make the work stand out . If the calc guy comes up with 97 3/4'' you come up with 98'' and doodabug comes up with 97 5/8'', in my experience all the jobs if well executed from this point on will be indistinguishable. Using the word purist above to describe the guy with the calc is really contrary to the way I feel in my gut. To me purists are geometry guys. To me purists prefer coping saws, still have a sharp 14'' smoothing plane. A purist is willing to put equal parts arse and brains behind his techniques and doesn't really care if someone comes up with some fancy new way because his ways are what gives him a sense of pride. I'm not talking about food on the table here, I hear a lot of guys say I don't have time for this or time for that. Carpenters that lose their discipline (I'm certainly not accusing any attendees of this site of this) because of monetary pressure can end up doing work that is really not up to their own standards. Some guys did better work in their twenties than their thirties because they no longer need this step or that step. Man if I had to be in such a hurry that it took the grace out of it, I'd be doing something else. A lot of approaches can make money. Its executing a fluid day that makes money. I think that to be the best carpenter in this world of many technological advancements one has to be a student of traditional carpentry, but also able to apply the advancements available to his trade. You got to be able to know that sometimes the quickest way to cut that odd angled piece of crown is to make a wooden miter box. as your standing at a table with a 12'' compound miter saw setting on it. Does that make sense to anyone?
Edited 6/29/2005 5:27 pm ET by quicksilver
(½ the length² + hieght²) /( height x2) = radius
above as an example
18² +8²/16=24.25
My calculator I got out of a cereal box is enough for this one.
I've recorded the formula in my palm pilot. I'll give it try. Do you have any other cool formulas while I have it out. Sorry to keep editing but is that 1/2 length of the chord?Edited 6/29/2005 5:31 pm ET by quicksilver
Edited 6/29/2005 5:33 pm ET by quicksilver
Yes ½ the chord.
Other formulas? Ya gotta give me a problem before the cobwebs will let them out.
Edited 6/29/2005 5:53 pm ET by JAYZOG
Something I've always wondered about is laying out a groin arch, I've done quite a few barrel vaults, and even a dome, but have never been confronted with a groin arch. I think I could lay it out with the same method I use to lay out an elipse. Which is plot the rise and run and divide the two lines into equal parts. The take the point on the run line closest to the intersection and connect it to the point on the rise line farthest from the intersection. Then do the point second closest on the run connected to the second farthest away on the rise and so on. The crrve created by the points where the lines converge will create a fair curve with any rise and run. But I don't know if this fair curve will match the true curve needed for a groin arch. So if you could help me learn a good way to describe a groin arch I could check my theory.
Just to make sure we are on the same page- I think of a groin arch is the curve formed by two intersecting barrel arches.
Is that what you are describing?
Yes. Been on jobs in the finish stage and seen them and I would like to know if I was confronted with one I could just get down to it.
I understand what you are saying. Also, I did two groin vaults in Chicago about ten years ago. If I can do it I am sure you can.
Thanks for the vote of confidence. I opened this forum yesterday and just smiled. Thought maybe they had ya on the ropes for a second but you came out allright.