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I'm in need of some gears, but having them made by a machinist is very expensive so I'm planning to have them laser cut. Loads are reasonably low for any grade of steel, so I'm hoping it should be fine.

Anyway, laser prices are given in amount per linear meter of cutting. In order to calculate how much it's going to cost me I need to calculate the perimeter of each gear and I just can not find how to do that...

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You need to know how the geometry of the tooth is calculated, which is also tied to the varies radius of the gear. The documents below can help you from the beginning to the end, especially the second article.

enter image description herehttp://www.robotpark.com/academy/LR/51031-SpurGears-Gear-Terms-ROBOTPARK.png

enter image description herehttps://grabcad.com/tutorials/how-to-design-a-custom-spur-gear-with-equations#:~:text=How%20to%20design%20a%20custom%20Spur%20Gear%20%28with,but%20remember%20to%20be%20realistic.%20More%20items...%20

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  • $\begingroup$ I don't think the question is about design of the gears. It's simply about calculating the perimeter length of a given gear. $\endgroup$
    – Transistor
    Sep 25, 2021 at 8:52
  • $\begingroup$ @Transistor If you don't know how to get your geometry, how do you draw? If you didn't know how to draw, how do you know the way to calculate the length. You don't know how to walk obviously you can't fly. $\endgroup$
    – r13
    Sep 25, 2021 at 14:18
  • $\begingroup$ See my answer and the OP's answer. The OP can do the drawings OK. $\endgroup$
    – Transistor
    Sep 25, 2021 at 16:32
  • $\begingroup$ @Transistor I don't think the OP really knows how to draw the gear as he has no clue about the geometrical parameters. Once he has a handle on it (how to draw), he should be able to calculate the exact perimeter length as you probably can do, though the steps are tedius. $\endgroup$
    – r13
    Sep 25, 2021 at 16:41
  • $\begingroup$ @Transistor Do not confuse the "geometric design" with "mechanics design". The former is the building block of any drafting, by hand or cad programs. Otherwise, we can approximate the length using 2*(pi)*(r) + 2h*(n), where "h" is the approximate tooth height with allowance for its slope or curvature, and "n" is the number of tooth, then call it done. I don't think you meant so though. $\endgroup$
    – r13
    Sep 25, 2021 at 16:58
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I would get the drawing and select the elements of one tooth pitch, get the length of each segment (which should be available from the CAD system) and add them up. Multiply the result by the number of teeth.

enter image description here

Figure 1. Gear-tooth path length measurement using OnShape.

  • Import your sketch into OnShape, select the line segments for one tooth and hit the measure button.

OnShape is free if you don't mind your creations being public.

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  • $\begingroup$ Yeah, I thought of this myself but I hoped there would be a formula for it that I just couldn't find. I made up one that approximates the perimeter and verified it using this method, I'll just post it as an answer. $\endgroup$
    – php_nub_qq
    Sep 25, 2021 at 13:28
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Turns out there is no single formula for finding the circumference of a gear.

So I came up with one that approximates it with about 5% error which is more than enough for my purpose.

$$P = 2\pi mz$$

Where m is module and z number of teeth.

How I came up with it: I drew a few gears and calculated their circumferences using the method described in Transistor's answer. For each I calculated the pitch circle's circumference and observed how it relates to the already known real circumference - turned out the double of it matched all gears.

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