Finding a Chord Length of a Circle if I have Arc Length and Height?

Aug 2011
4
0
I am in the process of making a tent and need to get two curved surfaces to meet.
I need to find the length of a Chord of a circle given that I have the Arc length and Arc Height (that's all), no radius or anything else.

I suspect that I will need a radius to find this. Or am I missing a point here?

The below spreadsheet I found on the web proports to be able to do this but the answer is not correct. The edges of my material does not match up when I calculate the Chord length given that I've entered the Arc Length and the Arc Height.

Does anyone know how to do this?
Thanks

http://mathforum.org/dr.math/gifs/ChordMath.xls
 

ChipB

MHF Helper
Jun 2014
305
124
NJ
Call the known chord height 'H' and the known arc length 'A'. If we define R as the radius of the arc and theta as the half angle for the arc length, we have two equations in two unknowns (R and theta):

\(\displaystyle R\cos \theta + H = R\)

\(\displaystyle R \theta = A \)

Combine and rearrange to get:

\(\displaystyle A \cos \theta + 2 H \theta = A \)

I don't believe there is a closed-form solution for this, so you will have to solve for theta using a numerical technique. The radius can then be found using \(\displaystyle R = \frac A {\theta} \), and then the chord length will then be \(\displaystyle 2 R \tan \theta\).
 
Jun 2013
1,112
590
Lebanon
the spreadsheet you referenced seems to give the correct answer.

what values did you try?