How do you calculate the length of a belt around two pulleys? Lets say that the larger pulley has a radius of 20" and the smaller has a radius of 6" and distance between the center of the pulleys in 56".

Lets assume that belt has a thickness of 0.

I thought that a good way to estimate this would be to create a right triangle by moving the line between the center of each pulley up to the tip of the smaller pulley. This would create a right triangle with the sides being the centerline, the radius of the larger pulley minus the radius of the smaller pulley and the distance between the highest point on the two pulleys. Then I could calculate the distance between the pulleys using the pythagorean theorem, double it and add half of the circumference of the two pulleys.

However, this seems to assume that the belt cuts through the pulley, when in actuality the belt goes around more than half of the larger pulley. I suspect that the above method is a good estimation, but how could I calculate the length of the belt to a higher degree of accuracy?

(I’m sorry if the above solution is convoluted, its pretty difficult to describe these types of math problems without a picture )

Take 2 pieces of cardboard. Make one 6" dia. Make one 20" dia. Place them as desired, 56" apart from center.

Run the path of the belt with a piece of string, and then trim as needed.

Undo the string, and measure the total length of the string.

See… The answer will be a very accurate length, and you don’t need math.

edit: Ok, depending on what kind of belt this is that you are using, and what the pulleys or v-grooves are made from you may need a bit more than what you just measured, just to allow it to slide into the v-groove of the pulley’s.

Also, a tensioner for belts as well as with chains is always a good idea. Whether it is a separate roller, or making one of those pulley’s (6" or 20") that moves (like an alternator on your car) tension of the belt always needs to be adjusted over time.

You can also use Autocad to do what Elgin Suggested, Its faster, easier, and more accurate.

Also, If you are thinking of using this for timing belts, https://sdp-si.com/index.asp has a handy center distance calculator on the front page that is really easy to use.

Thank you all for helping me with this. Could you help me understand how the formula above works? I think that its the same process that I described in my first post. It seems to me that if you’re taking pi/2(D1 + D2), what you’re doing is taking half of the circumference of the two pulleys, which seems to me like you’re assuming that the belt connects to exactly half of each pulley. Please correct me if I’m wrong, but a belt will only connect to half of each pulley if the pulleys have the same diameter.

I guess for engineering purposes this estimation is more than sufficient, but I’m curious about the mathematics of it.

I was thinking that if you marked the point where the belt intersects each pulley and drew an angle, than the angle describing how much of the smaller pulley connects to the belt would be the same as the angle that describes how much of the larger pulley does NOT connect to the belt. If you could find this angle, than you could take its arc length and solve part of the problem. Than you would just have to find the length of the lines between the pulleys which you might be able to do with the pythagorean theorem. I’m just not sure how to get there. Anybody have any ideas?

Measure the distance between the two pulleys, write down and multiply it by 2. (56 * 2= 112 in).

Now, take one pulley (6 inches) then find the circumference of it (2pir).

2pir= 37.699 -> 38 in around.

Then divide that by 2 (the belt only contacts half the pulley) and you get 19 in. Repeat your steps for 2.

112in + 19in + 63in = 194 in will be needed. Of course, it’s totally theoretical.

That’s for when the belt touches half the pulley, so now you have to do some more findings. First you need to find the angle of the opening where the pulley doesn’t touch. Divide that by 360, then you’ll get a decimal, convert that to a fraction, then multiply that by the total circumference of the pulley. Subtract that number from the total circumference and that’s the length the belt touches the pulley.

I thought of it that way, but with a small change that will be more accurate because it compensates for the extra length between the two pulleys along the line that the belt travels.

I looked at the line that the belt travels between the two pulleys as the hypotonuse of a right triangle. The longest side is 56" (the centerline bewtween the pulleys) and the shorter side is 14" (large radius minus the smaller radius). Then you use the pythagrean therom a^2+b^2=c^2
I included my math so anyone can check it.

56^2+14^2=c^2
3136+196=c^2
3332=c^2
c=57.72…

So you would multiply that number by two and add it to half the circumference of each of the pulleys. Using the numbers in JosephM’s post…the distance would come out to roughly 197.5" of total belt length.