Find the radius from a chord

What if you know the arc-length, and the chord? Is there anyway to calculate the radius?

[JohnS]

Originally Posted by Unregistered

What if you know the arc-length, and the chord? Is there anyway to calculate the radius?

Yes, but it gets ugly:
c must be less than s, and c/s must be accurately measured, even so it works poorly for small central angles, appreciably less than 1 radian (about 58 degrees).

s = R*theta (note: central angle, theta, must be in radians)
c = 2*R*sin(theta/2)

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

You have to solve this transcendental equation for theta, then R = s/theta

Four approaches:
1) Plot the function, enter with c/s, look up theta
2) Prepare table of values and do reverse interpolation
3) Expand the first few terms of Taylor series and solve
(three terms (quadratic to solve) work pretty well to 1 radian)
4) Use Newton's method to iterate a solution. (Use #3 for initial guess, quite unstable near theta=0, because derivative is 0)

Here's the formula that I have found that works. This is from the Machinist's Handbook.

R= C squared + 4 h squared / 8 h.

C is the length of the chord; h is the height. As an example:

C= 53.125, h = 3.25, so C squared = 2282.3 + 4 X 3.25 squared or 42.25. 2282.3 + 42.25 = 2864.55. 8 X 3.25 = 26. 2864.55/26 = 101.175. So R = 101.175.

Does that help?

hate to be the devils advocate here but what happens in the case that the height H (or m) in some formulas is zero, i.e. a point on the arc, shouldn't the formula still result an answer???

Originally Posted by Unregistered

I stumbled on this through a google search. I was also looking for the same formula which I couldnt find in any of my study books. It's in the machinist handbook but I dont have one here at home.

Here's the Formula R = ((C/2)² + H²)/2H

So in your case with a chord of 30m and a height of 10m

R = ((30/2)² + 10²)/2*10
R = ((225+100)/20
R = 325/20
R = 16.25m

I know I'm a little late on the thread but should help others.

[JohnS]

Originally Posted by Unregistered

hate to be the devils advocate here but what happens in the case that the height H (or m) in some formulas is zero, i.e. a point on the arc, shouldn't the formula still result an answer???

No. H isn't a point on the arc. It is the point of maximum height, at the center. If it is zero the "arc" is a straight line and the radius infinite.

You'd have to set up a different equation for an arbitrary point on the arc.

so i need to find the radius having only the height from the chord to the arch and the arch length.. I do NOT know C or R or an angle? how can this be done?

[JohnS]

Originally Posted by Unregistered

so i need to find the radius having only the height from the chord to the arch and the arch length.. I do NOT know C or R or an angle? how can this be done?

Much like #22, you can set up and solve an ugly transcendental equation

s = R*theta (theta must be in radians)
h = R*(1 - cos(theta/2))

2h/s = (1-cos(theta/2))/(theta/2)
Solve above, for theta/2, then theta, then R = s/theta

Expanding the above in Taylor series, first term gives initial guess of theta approx. equal 8*h/s. If this is a common problem, you could assume theta, calculate h/s, and build a lookup table. You could then interpolate between points.

Thanks. Done!

I have a problem ...tricky for me simple to some
please help I am building a radius bulkhead from a drawing. I am able to scale the distance(chord) and the height (furthest distance from the straight wall)

length of wall is 396 inches
the furthest point of the radius away from the wall is 3O inches.

I need to have a supplier crimp and bend my track but I need to provide the radius.(which is not provided on the drawing) Thanks to lazy Mr Architect
I would call the architect for this but this is a long weekend and I need to finish... please help

[QUOTE=amanda ]I am not exactly sure what you mean.

The chord meaures the curve of a sector of a circle? so what can u explain to me like in a question for this Circle A has a radius of 3 m. What is the length of the longest chord in circle A? can you help me on that and to show me how you did it.