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May 1st, 2019, 06:04 AM  #1 
Newbie Joined: May 2019 From: Australia Posts: 7 Thanks: 0  Finding Chord length given arc length and arc height?
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. How can I find the length of a Chord of a circle if I have the Arc length and Arc height? I'm not sure this is possible without the radius but maybe there is something I'm failing to consider. This spreadsheet seems to be able to do it but I'm not sure how it comes up with the answer it does. For starters, the answer comes out wrong if I enter the Arc length (S) and Arc Height (h), it gives me a chord length, but it's wrong. http://mathforum.org/dr.math/gifs/ChordMath.xls 
May 1st, 2019, 07:21 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
What are the values of S, h and the chord length given by the spreadsheet? What is the approximate value you expected for the chord length?

May 1st, 2019, 08:05 PM  #3  
Newbie Joined: May 2019 From: Australia Posts: 7 Thanks: 0  Quote:
Arc Length: 164.9336  my input Arc height: 8.46  my input Chord length: 163.7707 as given by the spreadsheet. I can be specific about the the expected chord length because the only way to find it is to join two points at either end of my material 1.96 meters apart and work out the chord length based on the chord calculated at the other end of the material, all using the principles in this spreadsheet (which may be wrong). The ends of each chord should meet at a length of 196.7797cm (that won't mean anything to you) . But it doesn't. I'm just interested to know if it is possible to calculate the chord length if I have the arc length and arc height. I suspect not, but I'm not a mathematician either. However, the spreadsheet seems to indicate that it is possible ? Another way to explain this is that I have an arc which is a semi circle. I want to join a piece of material to that arc (which should have a joining edge the same length as the arc length of that semi circle arc I just mentioned). But in fact, I will slope the material back about 7 degrees, so this will distort the straight (joining edge) of my material. So the joining edge will still be the same length as the arc length, but the chord will change length because the top edge will be lower (due to the slope of the material). I have attached a picture to illustrate. Last edited by skipjack; May 1st, 2019 at 08:22 PM.  
May 1st, 2019, 09:04 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
The chord length given by the spreadsheet is correct. S = sin$^{1}$(4hc/(4h² + c²))(4h² + c²)/(4h) The above formula can't be rearranged to give c in terms of S and h, but the spreadsheet found a plausible value for c by some method of approximating the (largest) solution. 
May 1st, 2019, 10:32 PM  #5  
Newbie Joined: May 2019 From: Australia Posts: 7 Thanks: 0  Quote:
 
May 2nd, 2019, 04:25 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
Is your figure for the arc height precisely accurate or an approximation?

May 2nd, 2019, 04:54 AM  #7 
Newbie Joined: May 2019 From: Australia Posts: 7 Thanks: 0  
May 2nd, 2019, 08:46 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
How do you know that the arc length is exactly 164.9336 (cm, presumably)? The spreadsheet produced an approximation, but one that appears accurate to seven significant figures. Won't that suffice for your purposes?

May 2nd, 2019, 03:46 PM  #9 
Newbie Joined: May 2019 From: Australia Posts: 7 Thanks: 0 
The arc length is calculated by finding half the circumference of a circle and using that surface area to create a new arc with a wider chord. The height of that arc and the circumference is accurate. You've mentioned above that in this spreadsheet they have some how found c using some kind of approximation. All I am asking is, is this accurate? Is this something you could rely on to the mm or is it just an approximation? 
May 2nd, 2019, 04:54 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
Half the circumference of a circle of what radius? If the approximate value of c found in your example is accurate to 7 significant figures (which is a lot better than to the nearest mm), it's likely to be similarly accurate for other examples. As I don't know how the spreadsheet does the calculation, I can't guarantee that it's always accurate to 7 significant figures.


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arc, chord, finding, height, length 
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