Create Arc by Segment Length

Hi everyone,
I am modeling a building and came across a challenge that I wasn’t able to figure out on myself.
I have a line that is 36 feet long and want an arc based on that line that has 10 segments that are 4 feet each (adding up to a 40 feet). How can I calculate the bulge of that arc. Is there a formula for that.
I hope this picture shows clearly what I am looking for.


Any help would be appreciated. I hope there is someone with better geometry skills than me. :sweat_smile:

The bulge on a 10 segment curve would be 7.513’ give or take a bit.

I’m sure there is a formula for that somewhere but my brain isn’t working optimal at the moment, sorry.
But here is an “approximation” image of what SketchUp tells me:


The total segments length isn’t what is ‘Entity Info’ is showing, that is the real arc length. You’ll get the sum of all 10 segments (which is less) by exploding the arc curve. That adds up to 40.000001’ in this model. I hope that’s close enough.

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I only did mine to 4 decimals which is why it is less accurate but still pretty close. And yes a bit of trig would give you the exact dimension but I’m guessing the use case has a bit of tolerance.

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I couldn’t find an exact answer online, but for the case of a circle, an arc length of 40 for a chord of 36, gives a circle of radius of 25.43. If you make a circle at that size and measure the angle from one end to the other end of the chord, it’s extremely close to 90 degrees.

Which means a 40 sided polygon would have the 10 sides in pretty much the exact spot you need them to be in. So, you can draw a 40 sided polygon of any size, tape measure the chord, type 36 Enter, and let the model resize. You now have ten segments that span 36 across, and each segment is 3.9945 long. You could resize that to be exactly 4, and would have a span of 36.0497.

Either option gives a tiny error, but it would be very easy to draw.

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i find keeping this graphic nearby helps me…


as does this:

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Those work for circles I think, do they work for polygons of unknown sides?

I got my starting figure from this calculator:

https://planetcalc.com/1421/

i honestly didn’t try to determine the size of a given number of segments based on the arg length. if i have time later today i’ll try it out. i just provided those two graphics as a means of helping people whose have geometry-memory impairment syndrome… :upside_down_face:

Thank you very much, that is close enough to what I was looking for.
It’s difficult to find something in geometry books etc. because they almost exclusively deal with circles and not with polygons.

I guess it’s too late and you might not be interested in using an extension but it’s possible to do this in SketchUp without any calculation.

TIG’s Arc By extension has an tool to create an arc from chord length, segment length, and number of segments. Leave ChordY blank.
Screenshot - 2_20_2021 , 11_27_31 AM

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That does seem better. His line segments come out at 3.999974, which is probably not the extension’s fault, and is some sort of internal rounding error. His ChordY comes out very close to what Wo3Dan showed.

edit: when working in feet the segments were 3.999919.

It’s not too late.
I will definitely check that out.
Thank you very much

yeah, it’s all floating point and i think SU is limited to 1000th of an inch for rendering the geometry… so rounding is definitely happening…

Interesting that the extension gives a bulge of 7.51’ and i posted 7.513’ 11 minutes after the question was asked, worked out using the dimension tool and cardinal point.

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Yeah, I alway’s pick the face area of circles because I am to lazy to do the math :wink:
Depending on the segment count, the accuracy usually is enough.

With nudging, a few steps provide the same kind of accuracy.
I would choose a polygon so you can snap dimensions on a segment without exploding the curve.
Two things are important, the concept of ‘cardinal points’ and the fact that you can change the movement distance.

(btw, I noticed the cursor did not change to the move-icon, probably a Mac thing)