# How do you find radius of curve?



## SE18 (Feb 21, 2008)

Dumb stupid beginner question.

I'm redoing my trackwork to ensure minimum radius of 36 inches or diameter of 72.

I have some Lionel track in that curve I could cart outside but just wondering if there's an easier way.

I'm handlaying track. Thanks

Dave


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## chuck n (Jan 2, 2008)

If you just want to know if the radius is 36 inches or longer, I would put a stake in the ground somewhere near the middle of the curve, halfway between the rails. I would then tie a piece of string to that stake and then move toward the center of the imaginary circle that you have with your track. Measure 36 inches along that string and put in another stake. That is your radius. You can now take the string off the first stake, marking where 36 inches is located and swing the string along the track. If the radius is smaller than 36 inches the mark will soon pass over the outside rail. If it is longer than 36 inches the mark will pass over the inside rail. if it is 36 inches it will stay between the rails as you swing it through to arc.


By changing the length of the string until the mark stays in the center between the rails you can then determine the radius of your curve. 


When laying track you can reverse the process. Put a stake in the ground near the center of the curve and swing a 36 inch string through the curve. Put a stake or something to mark the curve every foot or so. You can then lay your track along the curve that you have just surveyed. 


Chuck


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## SE18 (Feb 21, 2008)

Ok, thanks


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## ThinkerT (Jan 2, 2008)

I have some Lionel track in that curve I could cart outside but just wondering if there's an easier way. 

Hmm... through some bizzare fluke, I seem to have ended up with quite a collection of lionel track, so.... 

1) The larger radius of lionel track is actually effective 32 inch radius (64 inch diameter measured outside to outside) unless they made some wider stuff I am not familiar with. 

2) The 64 inch diameter lionel track is actually a slightly tighter curve than is Aristocraft/LGB track.


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## kormsen (Oct 27, 2009)

similar, as Chuck explained. 
take any two points in the curve. put from each of these two points a line (or rod, or board...) towards the inner side of the curve. 
your lines must be each at a "right angle" (90°) to the track. 

where the two lines cross each other is your middle of the circle, your radius-point. (both lines should have the same length from the point to the rails)


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## John J (Dec 29, 2007)

I have used the above methods to determin radius of a cuve I have also use it to lay track curves. On thing to remember. Make sure you string does not streach easily. 

I have had some string when held taught streached and messed up my curve.


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## hcampbell (Jan 2, 2008)

I just cut a cardboard template with the curve for the inner rail. Then it's easy to tell if the track radius is too tight. 
No problems with stretchy strings, trees other other outdoor type obstacles. 

Harvey C.


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## Totalwrecker (Feb 26, 2009)

First off Diameter or radius is measured from track centerline. 
At 36" r you're lucky because yardsticks are that length, simply screw two small screw eyes into the ends of the stick. Use a brad at one end for the pivot and a pencil in the other eye to describe your arc. 
With a table saw or band saw you can cut your own yardsticks of any length. In those I just drilled holes for my lengths. I usually cut 3/16ths off an edge of a board. 

John


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## chuck n (Jan 2, 2008)

If you are worried about the string stretching you can always use a piece of wire. I doubt that a piece of string would stretch very much over a length of 3 feet.

Chuck


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## Dwight Ennis (Jan 2, 2008)

If trying to determine the radius of a curved piece of sectional track, it can be done mathematically.









If the red curve represents the inner rail, you can measure the length between the rail ends AB in a straight line (called the "chord" btw). At the exact midpoint of chord AB, measure the offset to the inside of the rail C.

The formula is:

r = (C² + ¼AB²)/2C 

where r = radius, AB = chord, and C = offset

The actual radius of the curve will be r + half the track gauge.


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## Del Tapparo (Jan 4, 2008)

I was going to post the chord method, but didn't have time to dig up the article I wrote on it years ago.

This method is extremely accurate, and the best part is you don't have to find the center of the circle, which is likely in the middle of your pond or something. In using this on a few tight spots on my railroad, I was shocked to find out that my minimum radius of 4 feet, was actually only 2.5 or 3 feet in a couple of spots. No wonder that is a problem spot!

The formula is a little cumbersome, so I made up a spread sheet with all of the likely measurements and calculations on it. Then out on the railroad, it was just a simple table look up.


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## Semper Vaporo (Jan 2, 2008)

Take a 1-ft ruler and at the exact center and perpendicular to it attach a short mechanics rule that has markings in hundredths.

Place the 1-ft ruler so the ends press against the outer edge of the inside rail and measure the distance to the rail on the perpendicular part.

Then look up the value on the following chart, interpolating as necessary. (The chart takes into account the difference between the inner-rail radius and the track-center radius.)
Note the chart shows "C" (the measurment you took) is in Thousandths.

C (in.) --- R (ft.) --- D (ft.)
0.094 --- 16 -------- 32
0.101 --- 15 -------- 30
0.108 --- 14 -------- 28
0.116 --- 13 -------- 26
0.126 --- 12 -------- 24
0.137 --- 11 -------- 22
0.151 --- 10 -------- 20
0.168 --- 9 -------- 18
0.189 --- 8 -------- 16
0.217 --- 7 -------- 14
0.254 --- 6 -------- 12
0.305 --- 5 -------- 10
0.384 --- 4 -------- 8
0.516 --- 3 -------- 6
0.792 --- 2 -------- 4

But, note, the extreme accuracy you will need to determine the differences in large radii!

If your curves are too short to fit a 12-inch ruler in it, try a 6-inch ruler and use this chart (it is even worse for accuracy!)

C (in.) --- R (ft.) --- D (ft.)
0.024 --- 16 -------- 32
0.025 --- 15 -------- 30
0.027 --- 14 -------- 28
0.029 --- 13 -------- 26
0.031 --- 12 -------- 24
0.034 --- 11 -------- 22
0.038 --- 10 -------- 20
0.042 --- 9 -------- 18
0.047 --- 8 -------- 16
0.054 --- 7 -------- 14
0.063 --- 6 -------- 12
0.076 --- 5 -------- 10
0.096 --- 4 -------- 8
0.128 --- 3 -------- 6
0.196 --- 2 -------- 4


It would be easier using a 2-ft ruler:

C (in.) --- R (ft.) --- D (ft.)
0.377 --- 16 -------- 32
0.402 --- 15 -------- 30
0.431 --- 14 -------- 28
0.465 --- 13 -------- 26
0.504 --- 12 -------- 24
0.550 --- 11 -------- 22
0.606 --- 10 -------- 20
0.674 --- 9 -------- 18
0.760 --- 8 -------- 16
0.871 --- 7 -------- 14
1.020 --- 6 -------- 12
1.231 --- 5 -------- 10
1.554 --- 4 -------- 8
2.114 --- 3 -------- 6
3.359 --- 2 -------- 4


Or a 3-ft ruler:

C (in.) --- R (ft.) --- D (ft.)
0.850 --- 16 -------- 32
0.907 --- 15 -------- 30
0.972 --- 14 -------- 28
1.048 --- 13 -------- 26
1.136 --- 12 -------- 24
1.241 --- 11 -------- 22
1.368 --- 10 -------- 20
1.523 --- 9 -------- 18
1.719 --- 8 -------- 16
1.973 --- 7 -------- 14
2.316 --- 6 -------- 12
2.807 --- 5 -------- 10
3.574 --- 4 -------- 8
4.964 --- 3 -------- 6
8.614 --- 2 -------- 4


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## SE18 (Feb 21, 2008)

Thanks, all. You guys are creative geniuses. 

Been off the forum since posting this Friday. Used the string method. Actually used a 3' piece of dental floss. The thick variety that always gets stuck in my teeth, as I can only use thin kind and they don't label the thickness on the box. I attached it to a chisel and did that. Having to move it a lot. Plants kept getting in the way and it became tedius, so I used the string method to measure a curve segment, then using my rail bender, I bent a piece of 332 brass. I'm using that rail to do it, but am catching up on other methods that you folks have posted and may try each, although my math is challenged. 

Thanks and all the best 

Dave V


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## wigginsn (Jan 9, 2008)

For us lazy mathematicians, try this circle calculator.

It'll work out radius using the chord length method with less knashing of teeth. Of course a printed chart out on the railway is easier once you have the numbers...

*Circle Calculator *

Cheers
Neil


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## RimfireJim (Mar 25, 2009)

The chord offset method, described by Dwight, et al, gets my vote, too. That's the way curves are laid out in surveying, and why full-size railroad curves are stated in degrees rather than radius: degrees of curvature subtended by a 100' chord (there is another thread somewhere on this forum that goes into this in more depth). It can be very accurate, can work for any curve up to a half-circle, and doesn't require establishing a center somewhere in a lake.


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## Torby (Jan 2, 2008)

Cord method is a cinch, especially on a reverse polish calculator. And people wonder why I use backwards calculators


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## Dwight Ennis (Jan 2, 2008)

especially on a reverse polish calculator.Shoot, I haven't seen one of those in decades!!


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## Gary Armitstead (Jan 2, 2008)

The first HP calculator I had was "Reverse Polish Notation". I bought it in 1970 and it had everything. Still have it. Along with my slide rule!


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## wigginsn (Jan 9, 2008)

I 'misplaced' my HP32S calculator last year and was real upset when I found they no longer made them. It was a nice RPN scientific I've had for 10yrs or so. 

Definitely overjoyed to find it again a week or two ago at the bottom of a kit bag I rarely use.. Going to look after this puppy a whole lot more carefully from now on. 

Cheers 
Neil


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## NormG (Jan 2, 2008)

Hi All

FYI if you are in to RPN HP Calculators there is an HP App available for the iPhone (as well as other styles of RPN) I too still have my old HP, and several slide rules (dated ???) but it is very handy to have it on one device.

Thanks to all for the reminder and info on curves.

Norm Goodrich


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## Torby (Jan 2, 2008)

My HP 11C is 26 years old.

My dad had an HP 35 in 1970. Incredible piece of equipment that would be right at home on any engineer's desk today.

Alas, he doesn't have the 8 ft slide rule that used to hang in the lab. As a kid, I knew how to multiply and find trig functions with it. He taught electronics at the local junior college.


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## RimfireJim (Mar 25, 2009)

I use my HP 15C practically every day at work, and my HP41C at home. The 15C is great because it statistical functions, is programmable (I'm always crunching inches to millimeters or vice versa), and can slip into my shirt pocket when I need to go to the shop or production floor. I'd be lost without it. 
I switched from algebraic entry to RPN when I got my 41C in college back in '82, and choke whenever I have to use someone's TI. Not that one is better than the other, they're just different and you get used to what you use most. 
I should put my slide rules on my desk at work to give the young pups something to ask about, or ridicule  My favorite is my Pickens circular pocket rule - you never run off the end.


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## chuck n (Jan 2, 2008)

If there was a Nobel prize for marketing, HP would have won hands down. The basic HP-35 did only the four basic functions: add, subtract, multiply and divide. All for three hundred or so dollars. The next year they came out with a new calculator the HP-45 which had all the scientific functions: logs, square roots, squares and more. All for the same price as the HP-35. And so it went.

When I did my thesis in the mid 60s-- what I would have given for one of those babies. I had to do a lot of number crunching on a Frieden ca'chunk-ca'chunk adding machine and then when I needed the logs I had my slide rule. Sometimes I think that the current generation doesn't know how good it is!!! 


I have used both the RPN and the standard type of calculator. I think that it is a little easier to program RPN equations that the normally used calculators. But in the end they both work and it is what your are used to using.


Chuck


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## jfrank (Jan 2, 2008)

Posted By wigginsn on 15 Mar 2010 12:43 PM 
For us lazy mathematicians, try this circle calculator.

It'll work out radius using the chord length method with less knashing of teeth. Of course a printed chart out on the railway is easier once you have the numbers...

*Circle Calculator *

Cheers
Neil 



Thanks Neil, I tried that and it works great.


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## SE18 (Feb 21, 2008)

Most of these formulas work great if it is a perfect curve, but unfortunately none of my handlayed track curves are perfect curves. I've designed easements into the curves so the radius changes continually, I just don't want it to get below 36" radius. Isn't there calculus involved here


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## wigginsn (Jan 9, 2008)

Not unless you want there to be, depends on how accurate you want to go. 

I had the same Q after I laid some track with a dual rail bender. Have a close read of what Semper wrote. If I understand right I made something similar but a bit less detailed. 

Get a 2' ruler, at the half way point nail, glue, or stick something at 90 deg that has graduated marks. Either thou as suggested, alternatively I marked the defined diameters at the right(ish) spot. Now slide the ruler around the curve butting the ends against the sides of the track and the sticking out bit over one of the rails (inner or outer). As you move it you can read the radius off the graduated part. This is will give you the average dia over the 2' length. 

A picture is prob easier than those words but I'm at work so hard to do right now. 

Perhaps put a clear mark at 36" so you know when you're close. You may be surprised how tight the radius gets in the middle of the bends... 

Hope this helps 

Cheers 
Neil


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## wigginsn (Jan 9, 2008)

Something like this










Of course you can make it any size you want to suit your layout.

Cheers
Neil


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