# Standard sprocket

Anybody know the effective radius of the 10T sprocket? Like for the chain centerline.

Using a forumula I found at http://taylormhc.com/wp-content/uploads/2016/11/Sprocket-Engineering-Data.pdf,

Pitch Diameter = Pitch / SIN(180/Nt)

where Pitch = 1/4" and Nt = 10, yields 0.809" or half that for the radius.

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Thank you

This is very interesting find. At least in one spot, the software code assumed that the sprocket diameter equals Pitch*Teeth/Piâ€¦ not Pitch/Sin(180/Teeth)). In metrics, the software calculates the standard sprocketâ€™s radius as 10.10635 mm but this other formula calculates it as 10.27452 mmâ€¦

I donâ€™t know what the effect of this difference is. I feel confident that a standard sprocket with 10 teeth will feed out 63.5 mm of chain (6.35 mm pitch * 10 teeth = 63.5 mm) per revolution, regardless of what the actual diameter is. However, the triangular calibration and the chain wrap around sprocket calculations use Pitch*Teeth/Pi to calculate the diameter of the sprocket. I still havenâ€™t figured out the math of the chain over sprocket calculation, but I wonder if it shouldnâ€™t be using the pitch diameter formula, Pitch/Sin(180/teeth), instead.

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The software is doing math in radians, not degrees which is why there is the
difference.

David Lang

I wonder if there is a difference here between circular and point-to-point distances. Imagine you wrap a piece of chain all the way around the sprockets (donâ€™t panic, weâ€™re not going to run the motor with it wrapped, even in our imaginations!) Now form a decagon (a 10-sided polygon), by drawing lines connecting each of the pins (the red shape in the figures). The circumference of that polygon would be Pitch*Teeth = 63.5 mm. Divide by pi (as in the formula from the code) and you have a diameter of a circle that has that same circumference. That circle is smaller than the green circumscribed circle in figure B, and larger than the blue inscribed circle. I havenâ€™t worked out the math, but I suspect the 10.275 mm is the circumference of the green circle.

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So, the true diameter is the pitch diameter and is equal to pitch/sin(180/teeth). If thatâ€™s correct, I think I understandâ€¦ great job explaining @jwolter. If not, you did a terrible job and you should be ashamed!

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Yeah, itâ€™s one of those two.

So does it sound right that based upon the orientation of the sprocket, the radius is between a minimum value of:

gearPitch x numberOfTeeth / (2*pi)

and the maximum value of:

gearPitch / sin ( pi / numberOfTeeth)

Perhapsâ€¦ perhapsâ€¦ the radius of the sprocket thatâ€™s used to calculate chain wrap should be the average of those two unless we want to do some real fancy math and change the radius based upon the orientation of the sprocketâ€¦ all to gain very marginal incerase in accuracy. I favor the former.

The reference above to the equation by jolter has a complete set of completed tables with exact values for all chains and all numbers of teeth, about page 20 or so. The table also says 0.809 for a #25 chain and 10T.