Radius calculation given Arc Length and Chord length

[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

[JohnS]

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.

[JohnS]

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

[usnavy1955]

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!