My Math Forum Finding Chord length given arc length and arc height?

 Trigonometry Trigonometry Math Forum

 May 2nd, 2019, 07:26 PM #11 Newbie   Joined: May 2019 From: Australia Posts: 7 Thanks: 0 What I'm trying to ascertain is this. A lot depends on getting this calculation right, so I thought I would ask someone who knows. Let's make some assumptions to avoid further discussion over the accuracy of my figures. Let's assume that all of my figures, Arc length, Arc Height, etc are 100% accurate. Given that I only have this information Arc Length and Arc Height, I cannot see how one could realistically calculate the Chord Length. But for some reason the spreadsheet I showed you does this (so I thought it must be an estimate, or it makes some assumption). I believe this because a circle with a smaller radius with an Arc Height of 10cm for example, and a circle with a larger radius with an Arc Height of 10cm, given the same Arc Length for both examples, would give me a different Chord Length! Am I right in assuming this, or am I getting confused? I'm not trying to trick anyone here. I'm simply implying that there's something that I probably don't know about Chord Math. So I'm asking yourself if my assumptions above are right or am I failing to consider something? Thanks for your attention to this. I'm probably not being clear so I have tried to clarify things above. Last edited by skipjack; May 2nd, 2019 at 08:29 PM.
 May 2nd, 2019, 08:39 PM #12 Global Moderator   Joined: Dec 2006 Posts: 20,623 Thanks: 2076 There may well be shorter chord lengths that initially look possible, but I suspect they turn out to correspond to an arc that is more than half of the circle that it's part of. The spreadsheet is probably finding the largest possible chord length, as I suggested earlier.
May 2nd, 2019, 11:15 PM   #13
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Quote:
 Originally Posted by skipjack There may well be shorter chord lengths that initially look possible, but I suspect they turn out to correspond to an arc that is more than half of the circle that it's part of. The spreadsheet is probably finding the largest possible chord length, as I suggested earlier.
Thanks for your answer! I've pretty much come to a parallel conclusion but needed some reasoning. I think your suggestion is probably more accurate.

The geometry in making a fancy tent is harder than I thought.

May 3rd, 2019, 02:26 AM   #14
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 Originally Posted by dominover What I'm trying to ascertain is this. A lot depends on getting this calculation right, so I thought I would ask someone who knows. Let's make some assumptions to avoid further discussion over the accuracy of my figures. Let's assume that all of my figures, Arc length, Arc Height, etc are 100% accurate. Given that I only have this information Arc Length and Arc Height, I cannot see how one could realistically calculate the Chord Length. But for some reason the spreadsheet I showed you does this (so I thought it must be an estimate, or it makes some assumption). I believe this because a circle with a smaller radius with an Arc Height of 10cm for example, and a circle with a larger radius with an Arc Height of 10cm, given the same Arc Length for both examples, would give me a different Chord Length! Am I right in assuming this, or am I getting confused? I'm not trying to trick anyone here. I'm simply implying that there's something that I probably don't know about Chord Math. So I'm asking yourself if my assumptions above are right or am I failing to consider something? Thanks for your attention to this. I'm probably not being clear so I have tried to clarify things above.

Let us be quite clear that solution is possible, though the best way depends upon the numbers.

If you look at my sketch you will see there are 5 basic quantities.
(Yes, the spreadsheet adds some more, but these are just two or more of the basic ones added up.)

There are also three basic (independent) equations connecting these 5 quantities.

Thus, supplying any two of these quantities leaves 3 to be determined (by the 3 equations).

Note I have used the half chord and half arc as it make things simpler.

In construction, to the accuracy of available construction theodolites, it is normal to make the approximation that for full chord or arc lengths less than R/20 the approximation sin(θ) = θ may be made.
Attached Images
 arctochord.jpg (23.4 KB, 5 views)

Last edited by skipjack; May 3rd, 2019 at 05:53 AM. Reason: add sketch

 May 3rd, 2019, 06:37 AM #15 Global Moderator   Joined: Dec 2006 Posts: 20,623 Thanks: 2076 For the figures that dominover supplied, your "arc length less than R/20" condition isn't satisfied.
May 3rd, 2019, 08:57 AM   #16
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Quote:
 Originally Posted by skipjack For the figures that dominover supplied, your "arc length less than R/20" condition isn't satisfied.
Which is why I did not post the simplified formulae, resulting from the approximation.
Nevertheless the approximation is often very useful and I doubt the ability of the OP to measure the dimensions quoted to the accuracy quoted.

May 10th, 2019, 04:09 AM   #17
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 Originally Posted by dominover I'm not sure this is possible
In case anyone doubts whether this is possible, here is an extract from a junior school geometry textbook.

Note theorem 32

There is one circle and only one circle that passes through three given points not in the same straight line.

So it is clear that taking two of these points as the endpoints of the chord and one other on the arc fully defines the one and only circular curve available.

The rest of the extract shows constructions to find this circle and centre.
Attached Images
 chord3.jpg (97.3 KB, 0 views)

Last edited by skipjack; May 11th, 2019 at 10:59 PM.

May 11th, 2019, 01:39 PM   #18
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Quote:
 Originally Posted by dominover I'm not sure this is possible but I'm in the process of building a tent and am trying to get two curved surfaces to meet.
Given the opening post, I wondered whether the OP was trying to make a tent roof something like this?

Of course the roof is not a plane figure, but cone-shaped.
Attached Images
 tent1.jpg (99.3 KB, 0 views) tent2.jpg (100.2 KB, 0 views)

Last edited by skipjack; May 11th, 2019 at 11:01 PM.

 Tags arc, chord, finding, height, length

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