 # Chain Length Calculation?

2989 Views 15 Replies 8 Participants Last post by  AZ081000
Could someone direct me to where I can find how to calculate chain length?
I have search, read, found one calculator, but no idea how the length is generated.
I assume the distance between the drive sprocket center and rear axle is needed, along with sprocket diameters for a given chain size, along with the given chain link dimension between pins.
Am I headed in the right direction?

Appreciate some assistance.

Thanks
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What are you trying to do chain-wise? Change sprocket diameters? >
When I change chains I buy the common length (120 links) which is almost always cheaper than ordering one that's been sized in advance. Rather than measuring, I reset the tension adjusters and run the chain around both sprockets, check for a minimum of slack and then mark where to break it to fit the master link.
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Here's a rabbit hole for ya:
I've been using this site for years. Super helpful.
I found the sprocket calculator, tried it, just unsure if I can believe it.
The Gearing commander site; I am going to need to spend more time on there......

And yeah, in the beginning process of changing sprockets. Have read a lot of postings about using a combination of F/R that used this length of chain, then read where someone else used same F/R but a different length chain. So I want to understand the numbers.

So far I have:
the 500 chains have a 5/8" pitch, so is it accurate to do: a 41 rear sprocket has a circumference of 41 * 5/8 = 25.625", divide by pi to get diameter of 8.156"..which seems to be about right when I did a rough measure of the 41T rear sprocket on my bike.
Using the circumference of 25.625", divide by 2 as only half the sprocket is covered by the chain...yields a distance of 12.8125"; divide this by 5/8" and I get 20.5 links are on the sprocket.
But say I want to go to a 45T rear sprocket; doing the math again, I get there will be 22.5 links on the sprocket...so I need to increase the chain length by two links to 114.

I tried the sprocket calculator, it came back with 117 links, so we disagree by a lot!

The Gearing Commander came back with 112 links no matter what size F/R sprockets I entered....?????
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Don't forget that it's more than half of the rear sprocket, and less than half of the front sprocket, that's covered by the chain. Plus the two chain runs from front to back don't run parallel. This is due to the sprocket sizes not being equal.

If you go two teeth (one link) up on the rear sprocket, that's a little over half of one link on the rear sprocket, plus a certain fraction of a link to allow for the non-parallel runs, minus a tiny fraction on the front sprocket as the chain now covers a slightly smaller circumference of the front. Not an easy calculation. And since the chains don't run in parallel, you also need to know the exact distance from the front sprocket center to the rear sprocket center to do it correctly.

I did a very crude drawing, leading to a precise but entirely theoretical calculation. Can't post the drawing right now so let me know if you're interested. But here's the calculation:

Let:
d be the radius of the front sprocket (in whatever unit you prefer, inches is ok)
D be the radius of the rear sprocket (same)
Both d and D can be calculated given the tooth count and the pitch of the chain.
x be the distance between the center of front and rear sprockets (same) - start out with the distance when the rear is in the middle of the adjustment range.

If you draw this out and shift the hypotenusa, you end up with a triangle with a 90 degree corner with y and (D-d) as the bases and x the hypotenusa.

Let y be the length of the top and bottom run. y^2 + (D-d)^2 = x^2 so you can calculate the length y of the top and bottom run: y = sqrt( x^2 - (D-d)^2 )

Let a be the angle (in degrees) that the top or bottom rung makes compared to the line drawn through the center of the sprockets. a can be calculated as sin( a ) = ( D-d ) / x. With this angle a you can calculate how much of the chain wraps around front and rear sprocket: The chain wraps around the front over ( 180 - 2a ) degrees, and around the rear by ( 180 + 2a ) degrees. So the chain length for the front wrap is 2*pi*d * ( 180 - 2a ) / 360, for the rear wrap is 2*pi*D* ( 180 + 2a ) / 360.

Add these four numbers up and you have the theoretical length of the chain. You need to factor in a bit of chain slack and you need to round it (down) to the nearest whole number of links. On the other hand, since the chain wrapping around the sprocket is not going to form a perfect circle, you do get a bit of extra from that already. Then calculate again to make sure that the number you get for x indeed falls within the adjustment range of the rear sprocket, and keep in mind that a chain "stretches" over time.
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Agree!! Yep a bit more than half on the rear sprocket due to the chain angle at the bottom take-up point...which looked to be about a link+ more than half. Did not look at the front; I know it is original and needs to be changed....which I will do a complete set this Winter.
.And a funny here...I was counting links on the rear sprocket and noticed the part number on the sprocket had a -42. Previous owner told me he replaced the rear, but thought he went stock 41....apparently not.
Anyways, I know to round-up from my numbers to allow for the bottom take-up angle, slack, etc.

Would like to see your drawing somehow as I am not quite following the description.

As Fox stated, get longer than needed, then cut to fit.

Thanks for the replies.
Don't forget that it's more than half of the rear sprocket, and less than half of the front sprocket, that's covered by the chain. Plus the two chain runs from front to back don't run parallel. This is due to the sprocket sizes not being equal.

If you go two teeth (one link) up on the rear sprocket, that's a little over half of one link on the rear sprocket, plus a certain fraction of a link to allow for the non-parallel runs, minus a tiny fraction on the front sprocket as the chain now covers a slightly smaller circumference of the front. Not an easy calculation. And since the chains don't run in parallel, you also need to know the exact distance from the front sprocket center to the rear sprocket center to do it correctly.

I did a very crude drawing, leading to a precise but entirely theoretical calculation. Can't post the drawing right now so let me know if you're interested. But here's the calculation:

Let:
d be the radius of the front sprocket (in whatever unit you prefer, inches is ok)
D be the radius of the rear sprocket (same)
Both d and D can be calculated given the tooth count and the pitch of the chain.
x be the distance between the center of front and rear sprockets (same) - start out with the distance when the rear is in the middle of the adjustment range.

If you draw this out and shift the hypotenusa, you end up with a triangle with a 90 degree corner with y and (D-d) as the bases and x the hypotenusa.

Let y be the length of the top and bottom run. y^2 + (D-d)^2 = x^2 so you can calculate the length y of the top and bottom run: y = sqrt( x^2 - (D-d)^2 )

Let a be the angle (in degrees) that the top or bottom rung makes compared to the line drawn through the center of the sprockets. a can be calculated as sin( a ) = ( D-d ) / x. With this angle a you can calculate how much of the chain wraps around front and rear sprocket: The chain wraps around the front over ( 180 - 2a ) / 360 degrees, and around the rear by ( 180 + 2a ) / 360 degrees. So the chain length for the front wrap is 2*pi*d * ( 180 - 2a ) / 360, for the rear wrap is 2*pi*D* ( 180 + 2a ) / 360.

Add these four numbers up and you have the theoretical length of the chain. You need to factor in a bit of chain slack and you need to round it (down) to the nearest whole number of links. On the other hand, since the chain wrapping around the sprocket is not going to form a perfect circle, you do get a bit of extra from that already. Then calculate again to make sure that the number you get for x indeed falls within the adjustment range of the rear sprocket, and keep in mind that a chain "stretches" over time.
You win the prize. Actually, you ran away with the category for helpful answer of the year.
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Y'all doing algebra when all you need to do is place the unbroken chain onto your choice of sprockets to be 100% confident of the correct length Cool to know though
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Actually, you ran away with the category for helpful answer of the year.
Yeah, I admit I was bored tonight.
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I work with math for a living. I too just lay the chains side by side, add or subtract a link if I’m changing sprockets and cut it.
The Gearing Commander came back with 112 links no matter what size F/R sprockets I entered....?????
On the "Loaded bike" chain length table, click on the word "Links", in blue. It's explained there. You change the number of links and the sprocket teeth in the Custom column. That gives you a shaft-to-axle length. Trial and error to find the combo that fits your bike.
The reasoning behind doing the calculations is to make sure I order the correct length chain and not have to pay for extra chain lengths that I am not able to use.

Bentwee: I will take another look at that.
The reasoning behind doing the calculations is to make sure I order the correct length chain and not have to pay for extra chain lengths that I am not able to use.

Bentwee: I will take another look at that.
I get that part. The chain and sprocket sets I ordered always came with about 6 inches of extra chain. I could check one of the boxes if thats helpful. I recall 120 links.
The reasoning behind doing the calculations is to make sure I order the correct length chain and not have to pay for extra chain lengths that I am not able to use.

Bentwee: I will take another look at that.
I didn't know chain was priced by the link.
Not by the link, but lengths greater than 98 links or so costs more.
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