# Circumference of a Circle of Track for Helix

Discussion in 'FAQs' started by TrainClown, May 5, 2003.

1. ### TrainClownMember

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I was perplexed by trying to design a helix. I tried drawing a circle and then bending a ruler around it with poor results. This affected my calculations for the grade.

I am pleased to say I have found the formula for finding the circumference of any circle, and I thought I would share this information.

The formula uses the mathematical principal of Pi.
Pi being 3.1459
So if you want to know the circumfrance of a 42" diameter circle, the formula would be:

3.1459 x 42" = 131.94678", or Pi x 42".

If you only know the radius of the circle then the formula would be:

2 x 3.1459 x 26" = 163.36268"

My wife says you can round off Pi to 3.14 and this won't change the outcome that much for our purposes.

I hope this helps.
2. ### billkActive Member

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Or for pi you can use 3-1/7, which is within about 0.04% of being right.
3. ### Mike RMember

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Pi was 3.1416 when I went to school, nevertheless, I am glad you have 'found' this way to calculate the circumference of a circle.
Trainclown....you are joking with us, I assume ???
If not, my apologies.
regards / Mike
4. ### TrainClownMember

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No Joke

I guess I'm showing my lack of education here, but it was a real problem for me and I did have to serch the web for a while before I found the formula. I couldn't even find the absolute definition of pi. The web page I got this from said 3.1459 was approximately right. I'm glad you could give a more exact number for it. Not everyone needs this formula. Not everyone knows this formula. But if I help one person, I will be glad I posted it.

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6. ### jkristiaMember

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Re: No Joke

As far as I know, there is not absolute definition. Last I read about it (several years ago now) some scientists still had computers crunching numbers for pi. They had calculated pi with a precision of several millions decimals, but they were still calculating.

This is what windows calculator uses for pi 3.1415926535897932384626433832795, so it's 3.1416

(If you open Windows calc, go to View and choose Scientific, then you have pi)

Jesper
7. ### TerryRMember

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There is no exact definition of pi. It's been calculated to millions of digits but it never ends. It can be proven that it never ends. Numbers like that have a name and I used to know what it is

The amount of precision here is irrelevant anyway - a loop of helix isn't quite a circle anyway - you need a bit extra track to make up for the height change. For calculating gradient you can probably get away with using pi as 3:

For a 40" diameter, a 3" rise, and ignoring the extra bit of track for the rise, the gradient is 2.387...% (IE 2.4% - one decimal place is more than enough accuracy).

The error is trivial and well within my construction tolerances

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They are called irrational numbers.
9. ### Mike RMember

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I thought Irrational Numbers were the ones used by federal, state,provincial, municipal & school board budget departments to calculate our taxes !!!

Anyway, I am sorry, Trainclown, I really thought you were kidding us, partly because of the nickname you chose.

BTW...there was a thread a few months ago on trains.com, about helix calculation. The proposal was a specific radius to fit a corner of a room, [I think it was 22"r].The calculation recommended was similar to Terry's here, however, there was an error in that calculation also...no offense Terry.

Anyway, the grade you get HAS TO INCLUDE THE THICKNESS of the roadbed you are using, otherwise the track cannot clear a train below.

So, here's my calculation.
Assume you want 3" clearance, you need to choose a roadbed thickness, say 1/4", 1/2", or 3/4".
The operative radius is always to the INSIDE rail, not the centerline, when calculating a helix clearance. So, if you have 20" radius, you really have 19.5" radius, x 2, is 39" diameter x 3.14 is about 123".

With 1/4" thick roadbed, the grade is approx. 2.6%, with 1/2" it's 2.8%, and with 3/4" it is 3%.

If your track can have a 24" radius [ actually becomes 47" diameter], you get 148" of length, and for the 3 roadbed thicknesses given, the grade is approx. 2.2%, 2.4%, and 2.5%

regards / Mike
10. ### TerryRMember

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Mike,

No offence taken. I just pulled 3" out of thin air as a somewhat reasonable example so I could check out how important the decimal places were in pi. I wasn't suggesting that the rise should actually be 3".

Thanks. Yes, that's what they were called when I went to school - sometime in the late Durassic
11. ### cidchaseActive Member

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As I'm sure y'all know, but I thought I should point out,
the EXACT definition of pi is the ratio of the circumference of a circle to it's diameter.

Bein' irrational does not mean that it's a moving target, just
means that it can't be written as a fraction. It's still a very specific value.

I have a book about the Search for Pi, almost none of which I
understand. Apparently the mathematicians have been trying to come up with an exact formula for hundreds of years. Think I'll go
read on it again, maybe it will come to me!!!!

FWIW
12. ### 60103Pooh Bah

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Pi is a very interesting number for mathematicians, and it shows up in all sorts of places. Most mathematicians don't worry about the actual series of numbers (why bother when you can say "pi" and be exact).
Take the operative phrase as "for all practical purposes" which means that you use 2 decimal places if you cut your bench work with a carpebter's saw, 3 decimal places for a hack saw, and 4 for a razor saw.
13. ### ezdaysOut AZ way

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I don't know what they'd do if they found the actual value of pi. Once you get to about 10 or 20 decimal places, there isn't anything in this universe that needs that kind of accuracy.

But, on the other hand, there's all those reseach dollars available for those that want to try, and I bet there are a bunch that want to use those bucks up.

Don
14. ### MasonJarIt's not rocket surgery

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I saw this on the Internet somewhere (don't remember now), labelled as the "Engineers' Creed", but it could easily be the model railroader's creed too:

- Measure with micrometer
- Mark with chalk
- Cut with axe
- Pound to fit
- Paint to match

Andrew
15. ### trainmasta13601Member

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i did not know pi was anything but 3.14
16. ### Mike RMember

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I don't know how that got out......it's Classified.......it's the repair instructions for a Canadian Sea King helicopter !
[]regards / Mike
17. ### TrainClownMember

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I just want to say

Mike R

It's ok Mike, no worries
The fact of the matter is, that I am a profesional clown and have been for 30 years (ouch!) I have been a train fan for longer than that and as I love trains as much as I love clowning, TrainClown seemed natural for me.

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