Find the radius from a chord

[Unregistered]

By having only two measurments, first the length of the chord and the height of the curve, what is the formula for finding the radius of the circle.

[Robert Fogt]

I am not exactly sure what you mean.

The chord meaures the curve of a sector of a circle? The height is what? The height from the center of the circle or the height from the ends of the chord?

I have a book listing many formulas, but none to help knowing just the length of the arc and the height.

Take a look at the following diagram, and point out all the information you know and I can supply the correct formula.
http://www.onlineconversion.com/images/circle_diagram.png

My guess is you know the height m and the arc length XZ. Is this correct?

[Unregistered]

Yes that is correct. I know the length of the cord (c) and the height of the arc created by the cord (m). Is there a formula to find the radius with this information?

[Robert Fogt]

Just to make sure we are talking about the same thing, c is a straight line between points X and Z, called the cord, the curved line would be called arc length XZ.

Assuming you know c (not XZ) and m then the formula would be:

r = (m² + ¼c²)/2m

[Unregistered]

I have the circle and a chord through the circle then i know the hight but how do i fing the radius? Does it have something to do with triangles and the hypotenuse?

[Robert Fogt]

Using the image above, provide the symbols for what you know. I can look up the formula for what you need in my pocket ref. There is a section on circle segments, circle sectors, and circular zones.

[Unregistered]

An arch is in the form of a circle and has a span of 30 meters and height 10 m. What is the radius of arc.
May please give a solution for this

[Robert Fogt]

An arch is in the form of a circle and has a span of 30 meters and height 10 m. What is the radius of arc.
May please give a solution for this

You will also need to know the degree of the angle, which is Q in the above diagram.

The formula I have is:
XZ = Q * Pi/180 * r

You can solve the equation for r, and you know XZ, just need to know Q.

[Unregistered]

if you know only c, and m; what is the radius?

example, i'm standing on the out side of a semi-circular structure and i have no way of getting in to the interior to measure any angle. it is of such a nature that i can only measure a segment of the arc length of the semi-circular structure. i transfer that arc length to a piece of paper. i can now draw a chord of 'c' length and a can measure the sagitta (arc height from center point of chord or 'm'). what is the radius ?

what would be more useful is if i only know the arc length, what is the radius.

roadbump

[Robert Fogt]

If you know c and m, you can calculate the radius like so:

radius = (m² + 1/4c²)/2m

Knowing just the arc length XZ, you cant calculate the radius.

[Unregistered]

I am seeing your frustration. Please let me help you.
C= cord
H=height (sagitta)
solution #1: square one half of c
now divide by H
then add the value of H
finally divide by 2
DONE!!!

solution #2: Square C
now divide that by 8 times H
now Divide H by two and add
that to the former number.

solution #3: square H
multiply this by 4
Add to this, C squared
now divide all that by 8 times H

I sure hope this is helpful to find the radius of an arc knowing the cord length and the distance from the middle of the cord to the top of the arc

[naveen dhanerwal]

Yes, it's really works dear.



I am seeing your frustration. Please let me help you.
C= cord
H=height (sagitta)
solution #1: square one half of c
now divide by H
then add the value of H
finally divide by 2
DONE!!!

solution #2: Square C
now divide that by 8 times H
now Divide H by two and add
that to the former number.

solution #3: square H
multiply this by 4
Add to this, C squared
now divide all that by 8 times H

I sure hope this is helpful to find the radius of an arc knowing the cord length and the distance from the middle of the cord to the top of the arc

[Ken Cornine]

I just laid out formula #2 on the floor of my garage, and used a string to intersect my points. In my example C=27" & H=3". Using formula #2, the radius was 31-7/8" or 31.875. It crossed my points exactly. Thanks!

Ken

[Jeb Springfield]

Any simple formulas?

Practically speaking, I'm trying to find the length of an inside wall curve around an inside room corner. The over all corner angle (past the curved part) could be 90 degree, could be 45, or something else.

Yes, I could just measure it. But sometimes that's a bit difficult for one reason or another.

Thanks

[Unregistered]

Yes that is correct. I know the lenght of the chord (c) and the height of the arc created by the chord (m). Is there a formula to find the radius with this information?

Could you please let me know what the formula is by e mail.
[.. guest link removed ..]

[Mrs X]

Bumping for unregistered :)

[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.

[Unregistered]

Thank-you, this was really useful and I have not had any joy finding this formula elsewhere. HH.

[Unregistered]

After using your equation to find my arc I found a slight error.
Maybe this will suffice.


radius = (m² + 1/4c)²
-----------
2m

[Unregistered]

By having only two measurments, first the length of the chord and the height of the curve, what is the formula for finding the radius of the circle

[Unregistered]

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

[JohnS]

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)

[kasten_rich]

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?

[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???

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]

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.

[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?

[JohnS]

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.

[Unregistered]

Thanks. Done!

[Unregistered]

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

[amanda dubuisson]

[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.

[JohnS]

[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.

The chord is the straight line across the end points of the sector. The longest possible chord is a diameter (twice radius).

[Unregistered]

The formula was very helpful to me. It help me calculate a very large radius on a small part.

[Unregistered]

what is /2m if m = 20 in your equation

[JohnS]

what is /2m if m = 20 in your equation

Divide by 40. Perhaps it would be clearer as /(2*m)

[Unregistered]

I know the radius of a circle and the angle formating at the center by the cord,How can I find the lenght of the cord?

Exp.-If radius=10mtrs. and the angle is 120 degree then the lenght of the cord is?

[JohnS]

I know the radius of a circle and the angle formating at the center by the cord,How can I find the lenght of the cord?

Exp.-If radius=10mtrs. and the angle is 120 degree then the lenght of the cord is?

If the angle is theta, the chord is
2*r*sin(theta/2) = 2*10*sin(60°) = 17.32 m

[Nisar1238]

By having only two measurements, first the length of the chord and the height of the curve, what is the formula for finding the radius of the circle

[JohnS]

By having only two measurements, first the length of the chord and the height of the curve, what is the formula for finding the radius of the circle

see posts #2 and #4 for diagram and formula.

[6178]

I have the radius and the length of the arc segment how do
I find the length of the chord

[JohnS]

I have the radius and the length of the arc segment how do
I find the length of the chord

The central angle, theta, is s/r radians or (s/r)*(180/pi) degrees

The chord is 2*r*sin(theta/2)

The height is r*(1 - cos(theta/2))

[Unregistered]

hi, i have a problem.. looking back at the first page of this post , the circle image,

how do i find the radius of the circle? just by using the height which is m and the curved line arc length XZ

[Unregistered]

hi looking back to the first page circle image, how do i find the radius just by using
m and the curved line arc length XZ?

[JohnS]

hi looking back to the first page circle image, how do i find the radius just by using
m and the curved line arc length XZ?

See post #27 which sets up the equation.
But you have to solve a transcendental equation by iterative methods.

[Unregistered]

R= radius in meters
C= chord in meters
m= middle height in meters (heightfrom chord to top of arc)

R=C^/8m + m/2

C=10
m=.25

R= 10^/8(.25) + .25/2
R= 100/2 + 0.125
R= 50 + 0.125
R= 50.125 m

good luck.

[Unregistered]

Hi,thanks for this useful info.

One point though, does anyone not read the entire thread before posting????

The same question asked AND answered several times!

The answers are all there folks just read it.....

[Unregistered]

Yes that is correct. I know the length of the cord (c) and the height of the arc created by the cord (m). Is there a formula to find the radius with this information?

What is the formula?

[JohnS]

What is the formula?

See post #44 or numerous other posts in the thread.

[Unregistered]

Hi all!
I'm missing something that I have been looking at too long so I would appreciate some help
What if I know the Chord Length and the Radius or Diameter (Diameter of Corse being 2 * r)
I want to break it down into a usable example so here are some numbers to use
r = .625
c = .375
m = ?

Thanks!

[JohnS]

Hi all!
I'm missing something that I have been looking at too long so I would appreciate some help
What if I know the Chord Length and the Radius or Diameter (Diameter of Corse being 2 * r)
I want to break it down into a usable example so here are some numbers to use
r = .625
c = .375
m = ?

Thanks!

From post #4
r = (m² + ¼c²)/2m

This results in the quadratic equation m² - 2*m*r + 0.25*c² =0, which has two solutions

m = r ± r*sqrt(1 - 0.25*c²/r²)

Normally, you want the solution with the minus sign. The chord divides a circle into two arcs, the first, less than 180°, has the m with the minus sign. The other piece of the circle, greater than 180°, the plus sign.

Substituing, the answer is 0.0288

[Unregistered]

The answer using the above Numbers should be 0.0288

[JohnS]

The answer using the above Numbers should be 0.0288

Agreed. I must have made an error keying the numbers in. Thanks

[Unregistered]

Thanks! that formula worked like a charm for automating my radius calculations along a railroad track. It gave pretty much the same results as least squares circel fitting.

Anyone know what the name of the formula is and/or a scientific paper i can refer to it?

[JohnS]

Thanks! that formula worked like a charm for automating my radius calculations along a railroad track. It gave pretty much the same results as least squares circel fitting.

Anyone know what the name of the formula is and/or a scientific paper i can refer to it?

I have never seen a name given to it. It is included in most math reference books. In the CRC Standard Mathematical Tables, it is covered in the Geometry section, under sector and segment of a circle. (Page numbers vary in different editions)

[Unregistered]

m^2 + (1/2c)^2 / 2m = r

[Unregistered]

Just to make sure we are talking about the same thing, c is a straight line between points X and Z, called the cord, the curved line would be called arc length XZ.

Assuming you know c (not XZ) and m then the formula would be:

r = (m² + ¼c²)/2m

brilliant, thankyou , i needed this info too. spot and many many thanks. Andy

[Unregistered]

The formula your looking for is....
(x/2)squared + Y squared/ 2Y = R

I cant figure out how to write this on the computer but its (X over 2) squared + Y squared and all of that is over 2Y = Radius

Hope that helps.
Its in Circular Work in Carpentry and Joinery on the 9th page

[Unregistered]

where that formula come from? will you please explain me how to get to that formula r=(m2 +1/4c2)/2m

[Unregistered]

Find the chord length when we know the diameter of circle

[Dale]

I know the arch length and the radius..need to know the cord length

[JohnS]

I know the arch length and the radius..need to know the cord length

Switch calculator to radian mode. The central angle is
theta = arc length/ radius
Chord = 2*R*sin(theta/2)

This only works in radian mode.

[Unregistered]

Thanks Robert. Your suggestion solved my problem.

[Unregistered]

I have the Delta Angle and the Cord Length

I need the Radius

[JohnS]

I have the Delta Angle and the Cord Length

I need the Radius

You have to use half the chord length and half the central angle

R = 0.5*c/sin(0.5*theta)

[maha]

please i would like to know how i can measure the radius knowing the length of the chord xz and the lenght of the arc??

[maha]

any one answer plzzzzzzzz

[JohnS]

please i would like to know how i can measure the radius knowing the length of the chord xz and the lenght of the arc??

see post #22

[NABUNWA GERALD]

formula to find the length when given only the chord

[Unregistered]

I just had a great time solving this one. It's a fun one!
If the chord length is 2b, and the sagitta/height is s and the apothem (radius minus the height) is a and the radius is c, then pythagoras gives

(1) a^2+b^2=c^2
and since the radius is the sum of apothem and the sagitta:
(2) c=a+s

then solve pythagoras for b^2 and sub (2) into it
(3) b^2=(a+s)^2-a^2
(4) b^2=a^2+2as+s^2-a^2
(5) b^2=2as+s^2

solve for a:
(6) a=(b^2-s^2)/(2s)

sub (6) into (2)
(7) c=(b^2-s^2)/2s +s
(8) c= (b^2-s^2+2s^2)/2s
(9) c= (b^2+s^2)/(2s)

as long as you remember that b is HALF the chord length.

[Krystal]

it is given that ab is the diameter of a circle and o is its centre. ab bisects the chord cd at e such that ce=ed= 8 cm and eb= 4cm. find the radius.

[alexuslara]

how about calculating the area of a segment when the only information available are the length of the cord and the hight of the segment? (in your figure it would be "C" and "M"). Thanks, Gabriel

[JohnS]

how about calculating the area of a segment when the only information available are the length of the cord and the hight of the segment? (in your figure it would be "C" and "M"). Thanks, Gabriel

You can't express it easily in those variables. Introduce two variables
R = 0.5*M + 0.125*C²/M
Theta = 2*arcsin(0.5*C/R) Be sure to be in radian mode
then Area = 0.5*R²*Theta

I suppose you can substitute the equation for R in both the theta and area equations, but it would be a mess. I'd calculate the intermediate variables.

[Unregistered]

using the posted diagram, (m squared plus 1/2c squared) divided by 2m

[Unregistered]

Ok i am reading but not getting it, maybe someone can help me out understand something. Our business we install frames for theater screens and some have a curve, our guru at getting the cord for us or radius whichever it is has left and no longer works with us. Now when he used to give u the measurements it would be simple, for example the frame is 50ft wide, he would say the ends need to be 10 feet from the backwall and the center of the frame needs to be 7feet from the back wall giving us that nice curve, so my question how the heck does he know those measurements, i think we had one job where they told us either cord was 2.5 which i have no flippin clue what that meant untill one of their engineers told us how to set it. what numbers would i need to be able to determine how much we bring out the ends from the center.?

[JohnS]

Ok i am reading but not getting it, maybe someone can help me out understand something. Our business we install frames for theater screens and some have a curve, our guru at getting the cord for us or radius whichever it is has left and no longer works with us. Now when he used to give u the measurements it would be simple, for example the frame is 50ft wide, he would say the ends need to be 10 feet from the backwall and the center of the frame needs to be 7feet from the back wall giving us that nice curve, so my question how the heck does he know those measurements, i think we had one job where they told us either cord was 2.5 which i have no flippin clue what that meant untill one of their engineers told us how to set it. what numbers would i need to be able to determine how much we bring out the ends from the center.?

The distance from the backwall at the center is arbitrary as far as the curve is concerned. The difference in distances from the backwall is the h in all the formulas given (10 ft -7 ft = 3 ft in your example). h has to be calculated.

You need to know the radius the theater owner wants (flat is a radius of infinity, which makes h zero, the ends and center are the same distance from the backwall). Then you need to know either the arc length (the width of the screen material, measured along the curve) or the chord (the width of the screen measured in a straight line, not around the curve).

With either of those and the radius, you can calculate h.

[rochawalter]

Robert, thanks for all the information you posted in here, it help me a lot.
Walter

[Unregistered]

If the arc and chord lengths are known, can one compute the radius of the circle?

[JohnS]

If the arc and chord lengths are known, can one compute the radius of the circle?

See post #22. You have to solve a transcendental equation. Ugly.

[jason campbell]

This is an old problem I had to solve when I was in college - It took me a cubic equation. I had it in a copybook but my wife cleaning the house throw it away thinking it was old rubbish. I can't remember how I solved it.
It is a semicircle in which there are 3 chords, measuring 3, 2 and 1 units. The 3 chords are one after the other, from one to the other extreme of the diameter.
Find the radius.
Many thanks!

[jasoncampbell123]

I'm not going to solve it, but one approach is take 0.0 as the center of the semicircle. The circle is given by x² + y² = R²

At one end of diameter, swing a unit circle, (x+R)² + y² = 1, the intersection of this represents the end of the 1 unit chord. Eliminate y and get x1 = (1-2R²)/(2*R). Plug this into R² equation to get y1, as a function of R.

At the other end of diameter, swing a 3-unit circle. The intersection is the end of the three unit chord. Similarly, determine x3 and y3 as function of R.

Require (x3 - x1)² + (y3-y1)² = 2². This will be some horrible function of R. Solve for R.

An alternate approach, which can only be solved nummerically is to note the sum of the central angles of the three chords is 180° (or pi radians)
2* ASIN(0.5/R) + 2* ASIN(1/R) +2* ASIN(1.5/R) = 180°
From the geometry of the problem, R has to be more than 1.9 and less than 3.
Tabulate for a few values of R, and interpolate between those that give a total closest to 180°. Iterate. 2.056545 =< R =< 2.056546 after a few trials.

[Moncery]

A circle by all three points can also be found in the construction with a compass and ruler. This also provides the position of the center point, and therefore the RADIUS. In the top of the page, the three orange dots can be used with this method. See a circle through three points to build.

[XIT]

The Diameter is 32.50 M

[Unregistered]

what is the radius of the circle of a square inside of it with a side of 5/8 inches

[JohnS]

what is the radius of the circle of a square inside of it with a side of 5/8 inches

The diagonal of the square must be a diameter of the circle
d = sqrt(0.625² + 0.625²)= 0.884"
Tghe radius is half that, 0.442"

[Unregistered]

Formula for calculating measure of arc angle using only chord length and length of portion of radius from chord to arc

[JohnS]

Formula for calculating measure of arc angle using only chord length and length of portion of radius from chord to arc

This is the problem discussed in posts #1-4, use formula in post #4.

[Unregistered]

i think what he means is how to find the radius point of an arch sometimes you can't have the radius point either because something is on the way or x this is how i do it with an app called construction master pro on my android phone example 16 feet opening 2-6 rise from chord you enter the rise first on my case i will enter 2-6 then press rise then enter only half of the total opening my case will be 8 foot then press run then diagonal then hit the convert key then the square rot key then divide by 2 then divided by rise = (14 foot 5/8) of an inch will be my radius point