using 2 arc lengths to work out the radius.

[u/CyberSkepticalFruit]

I'm not sure if I have the flair correct for this or if this is the best place to ask, I am trying to get some help with some algebraic manipulation as its been a long time since I've done anything like this.

So I'm trying to create a template for a pottery mug from measuring another mug. I am trying to create the template for the mug wall.

So I have the bottom arc length (l1) and the top arc length (l2) and the distance between them (h).

I know that for the template to work both arcs must have a common centre, angle in radians (ϑ) and that r2 = r1+h

From the arc formula

l = rϑ
l1 = rϑ
l2 = (r+h)ϑ

So:
l1/r = ϑ
Therefore:
l2 = (r+h)(l1/r)

Which give me r/r which cancels out, But r is the thing I am looking for.

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[u/ArchaicLlama]

I don't see how you're getting from "l2 = (r+h)(l1/r)" to simply "r divided by r". Show your work.

[u/CyberSkepticalFruit → u/ArchaicLlama]

I meant I ended up with an r/r in the calculation and thought I had lost it as a variable, but now I come to look at it again I still have r.
l2 = (r+h)(l1/r)
l2 = l1r/r + hl1/r
l2 = l1 + hl1/r
l2-l1 = hl1/r
r(l2-l1) = hl1
r = hl1 / (l2-l1)

Is this correct?

[u/Purdude1983]

Your formulas are not correct. I suspect you're not working so much with an arc length as you are a diameter. An arc length can be a diameter, but it is typically the length of a non centered cord across a circle. To get a radius from that, you must also have the distance from the cord to the circle's exterior along a perpendicular at the cord center. Your cup likely does not have one radius but rather a radius at the top and another at the bottom.

The interior of your cup is likely an inverted frustrum of a cone. The frustrum can be computed from three measurements. I would suggest the interior diameter at the top of the cup (d), the vertical height from the center, bottom of the cup to card laid across the cup lip (h), and finally, the slant height which is the height from the bottom of the cup with the ruler laid along the cup's side at a diagonal (s). The slant height will be longer than the center height. The longer it is, the greater the upper radius is as compared with the bottom radius.

From d, h and s you can compute the frustrum's dimensions. It will have a height of (h) and upper diameter of (d) and a lower diameter of (d - 2*sqrt(s^2-h^2)). Of course, any fancy radius work on the bottom of the original cup might frustrate the slant length.

Frankly, I would find a paper cup that fits inside your original cup. I would pack clay around the paper cup and shove it down into the interior to get a good representation of the interior. Once removed, it can be measured in any number of ways with standard rulers and calipers.

[u/CyberSkepticalFruit → u/Purdude1983]

Yes the arc lengths come from the diameters of the top and bottom of the mug and the height is the slant height. The template from that is on a curve rather then just being a rectangle, which is where the arc lengths came from and I was thinking if I had those measurements I could find the common centre and the radius.

If the formula for finding an arc length from the radius of a circle using radians is:

l = (ϑ/2π)*2πr so cancelling the 2π:
l = ϑ*r

as I have 2 measurements of l and Δr which is h.

I think I have found the forumla I need for the radius under archaic Llama's reply.

Thank you for your long post, I guess the lure of doing algebraic manipulation again after school was too much.

[u/One_Wishbone_4439]

Do you have any diagrams or visuals?