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#1
02-11-2012, 09:08 PM
 superllanboy Junior Member Join Date: Feb 2012 Location: Shanghai Posts: 1 Rep Power: 0
Radius calculation given Arc Length and Chord length

I need help in finding a formula to calculate the radius (R) of a circle given the arc length (L) and the Chord Length (X)

Does anyone have any ideas how to solve this problem
#2
02-12-2012, 03:08 AM
 JohnS Double Ultimate Supreme Member Join Date: Dec 2007 Location: SE Michigan, USA Posts: 9,214 Rep Power: 18
Re: Radius calculation given Arc Length and Chord length

Quote:
 Originally Posted by superllanboy I need help in finding a formula to calculate the radius (R) of a circle given the arc length (L) and the Chord Length (X) Does anyone have any ideas how to solve this problem
Inconveniently. You can't solve for R directly, you first have to solve a transcendental equation for theta, the central angle of the arc. Working in radians

s = R*theta
c = 2*R*sin(theta/2)

Some manipulation leads to two equivalent forms in which R is eliminated
c/s = sin(theta/2) / (theta/2)
or sin(theta/2) - (c/s)*(theta/2) = 0

The first lets you set up a table for different theta of c/s and you can do reverse lookup and interpolation. The second can be used in Newton's method (along with the derivative) to rapidly converge on numerical value. Once you have theta/2, R = s/theta. Since Newton's method requires a decent starting guess, the table is still a good idea for the initial estimate (or a measurement off a drawing or something).

In theory, you could write theta = s/R and substitute in the chord equation. That does not lead to a general tabular method, and leaves a MUCH messier equation to solve by Newton's method, although it gives R directly.

Last edited by JohnS; 02-12-2012 at 05:43 AM.
#3
02-12-2012, 07:11 AM
 JohnS Double Ultimate Supreme Member Join Date: Dec 2007 Location: SE Michigan, USA Posts: 9,214 Rep Power: 18
Re: Radius calculation given Arc Length and Chord length

This calculator calculates ALL the attributes of an arc from any two given, including the the case of radius from chord and arc:
http://www.handymath.com/cgi-bin/arc18.cgi
#4
01-03-2013, 06:51 AM
 usnavy1955 Junior Member Join Date: Jan 2013 Posts: 1 Rep Power: 0
Re: Radius calculation given Arc Length and Chord length

It's impossible to compute the Radius with 100% accuracy. Came across this problem many moons ago. The process of getting close to the value is mentioned in the 3rd Edition of Mathematics at Work.

If you ever do find a method that will give you a result with 100% accuracy, PLEASE post it!
#5
10-31-2013, 10:36 AM
 Unregistered Guest Posts: n/a
Re: Radius calculation given Arc Length and Chord length

This is not possible......the reason for this is that the radius will vary with the angle subtended by the arc on the centre of the circle, i.e. as the angle subtended by the arc on the centre of the circle can have infinite number of values, the radius can also have infinite number of values.........hope this helps.....
#6
11-01-2013, 02:55 AM
 johannesmalan@gmail.com Guest Posts: n/a
Re: Radius calculation given Arc Length and Chord length

Y=Chord
L=Arc

Y^2 = (2*R^2)(1-cos((180L)/(Pi*R))

#7
11-01-2013, 04:45 AM
 johannesmalan@gmail.com Guest Posts: n/a
Re: Radius calculation given Arc Length and Chord length

Quote:
 Originally Posted by johannesmalan@gmail.com Y=Chord L=Arc R=Radius Y^2 = (2*R^2)(1-cos((180L)/(Pi*R))) Follow Newton-Raphson method and iterate to get an accurate R(radius).
For derivitave ,f'R use radians as angular measure.
#8
11-01-2013, 07:59 AM
 johannesmalan@gmail.com Guest Posts: n/a
Re: Radius calculation given Arc Length and Chord length

1)

Y=Chord
L=Arc

Y^2 = (2*R^2)(1-cos((180L)/(Pi*R))) .

You will probably need radians as as an angular measure,although written as :cos((degrees)) in the formula.

I wrote the equation ,but Prof. Villet(University of Johannesburg,Applied Mathematics) help me to apply the Newton-Raphson method .

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2)
Y=Chord
L=Arc
A=Theta(WANTED)

A^2+2((180*L)/(Pi*Y))^2(cosA-1)=0

A written as degrees in the formula.

Follow Newton-Raphson method and iterate to get an accurate A(theta).

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