radius from arc and chord length

Ever wanted to calculate the radis of a circle just knowing the length of an arc & it’s chord. Well after a lot of searching I found just that. The great step of faith was realising that you need to use a Taylor expasion for the trig function. Here’s a very kewl Java Applet that shows the expansion and its number of terms in action.

Then I ended up with the Cubic Formula provided by Graeme McRae of Math Help, which is really the same formula as given by Eric Schechter and number 43 as Wolfram decribes how to calculate not only the real root, but the two other roots as well.

update: Erica found me a really nice discussion on solving quartics and cubics in computer graphics. Just the sort of thing that I need :-)

update 2: A lot of people have commented asking for more circle formula. The 1728 website has a brilliant collection of formulas and calculators for almost anything you can think of. Please check out their website, their circle calculator is here.

update 3: It seems that I made a fatal error in the substitution of the trigonometric identity, resulting in an expansion that is only true when r = 1 (the equation should read: C² = 2 r² (1 – cos(θ))). If you read up at Dr. Math, you’ll find that there’s an iterative solution to this using Newton’s Method, or alternatively there is also an infinite series.

You might ask what’s the point ? Well I want to draw a circle knowing only it’s end-points and it’s curve index (how bendy it is). I already know how to find the center given the radius and end points :-)

Where C is chord length and A is arc length (from C to C*pi/2):

A = θ r
C = 2 r sin(θ/2)

or

C² = 2 r (1 – cos(θ))
So substitute in (1) and we get:

C² = 2 r (1 – cos(A / r))

Or assuming cos(θ) = 1 – θ^2/2! + θ^4/4! – θ^6/6! + θ^8/8!
Then letting x = r², we solve:
ax³+bx²+cx+d=0

where:
a = A² – C²
b = -2A^4/4!
c = 2A^6/6!
d = -2A^8/8!

Let:
f=c/a-b²/(3a²)

g=d/a+2b³/(27a³)-bc/(3a²)

x=cuberoot(-g/2 + sqrt(g²/4+f³/27)) – cuberoot(g/2 + sqrt(g²/4+f³/27))-b/(3a)

r = sqrt(x)

now plug in r to part #2 to find the center of the circle!

36 comments on “radius from arc and chord length

  1. got a good one for ya.

    A bridge crosses a river that is 1 mile (5280 ft) wide (cord). The length of the bridge is 1 mile and 1 foot (5281 ft) (arc). What is the distance between the water and the bridge at the center of both (sagitta).

    note: The arc formed by the bridge is part of a circle. The river represents a cord of the circle. We are looking for the sagitta. You can only use Trigonometric fundamentals, no calculus allowed.

  2. Come now, that’s too easy. You have two knowns: A and C, which given the first two equations will find us r and θ. The distance you’re looking for is r – (length from center to arc). That length is basic trig, it’s the root of (r squared subtract half the width of the river).

  3. It’s an excellent approximation but keep in mind that the power series expansion of cos(x) is an infinite series and anything less is just estimation. Unless you can solve infinitely powered polynomials, this idea won’t help you with exacts. As for appoximations however, it might be easier and more direct to solve C = 2r sin(A/2r) numerically e.g. Newton’s Method.

  4. For this application you only need to be able to solve for values of x between 0 and 360. I did a quick mockup in excel with the graph of the taylor approximation for each power and found that you only need to go to the 4th power before it’s close enough. I ended up using the quartic equation to solve this, which was even hairier than the cubic – but I found some source that already did it all for me.

  5. Nevertheless, it is still an approximation, try cos(1.5) in your calculator and then try 1-1.5^2/2!+1.5^4/4!-1.5^6/6!+1.5^8/8!. I gaurantee that the answers won’t be identical (remember to use radians). On the other hand, using succesive applications of Newton’s method will give you any desired accuracy.

  6. Stacy: That’s actually really easy to find – substitute radius and chord length into C = 2 r sin(θ/2). This gives you angle θ. Then substitute radius and angle θ into A = θ r to get arc length.

  7. I’m a steel fabricator, not a Mathematician, quite regularly i have to fabricate flanges, base plates etc. These items will usually have an X number of holes/chords on a PCD. Every item will have different dimensions and number of holes/chords. To the mark the location of the holes prior to drilling i use lengths of chord
    How do I find the chord length, when given the diameter and number of chords?
    I know simple formula for the following- 3 Chords C=Dx0.866; 4 Chords C=Dx0.7071; 5 Chords C=Dx0.5878; 6 Chords C=Dx0.5; 8 Chords C=Dx0.3827. What can I use after 8 chords?
    I hope I have written this in too “English Terms” for you to be able to assist me. Cheers

  8. You’ll need a calulator to work it out, but it’s: C=D.sin(pi/n) (assume you’re working in radians)

  9. i often need to make stone arches over windows before window are delivered. if all i know is the width of the window (chord) and the perpendicular height up from midpoint of this chord. can i calculate radius? angle of sector? arc length? as you can tell, i’m not a math major.

  10. i have an enclosed area consisting of a arc with radius R and a chord Length. I am analysing the stability of a slope using the bishop method and would like to create a excel sheet to avoid tedious hand calculations. The enclosed area is to be divided up into equal slices of width B. to do this perpendicular lines are to be drawn from the chord to meet the arc. These lines start at 0 height at one end of the segment and increase in height and then decrease to zero at the other end of the segment. My question for you is how do i calculate the length of each of these lines which divided the segment into slices?

  11. Dear Sir/Madam
    I am a student of Year 8 and have discovered a small formula to find the height of a segment of a circle. It is really a very nice blog. So I felt like asking you a question. Could you possibly tell me al the formulas which have been already discovered on the above subject ?

  12. I still do not understand, how you solved jeffs problem on May 21st, 2005 at 2:02 am. I cannot find r or theta.

  13. got one for u…..if you have a chord length of 80″, and the height from the centre of the chord to the arch is 11″, how can i figure out the arc length?

  14. Our kitchen light plastic panels are curved when installed to form an arc. I can measure from the end the height of the arc, the distance between the two ends and of course the panel measurements when flat. The 2 corner panels facing each other at 90 degrees match when installed. My problem is how to measure the radius of the arc when flat in order to mark the curve on the two corner panels for cutting so that they will match each other when fitted in a curved state. I cannot match the old curves due to breakage.

  15. @Boos: Check out the 1728 calculator, it says your arc would be around 84″

    @fred: Tough one, the best I can think of is to temporarily pre-curve them on the ground before installing, and mark the curve with a piece of string or something that will “fit” to the curve dynamically. Then you can mark the panels with a pen, and cut before installation.

  16. Hi folks. I notice my circle calculator is really popular here. I recently updated it because
    1) It seems the significant figure option wasn’t working but it is now.
    2) I decided to make it solve for the condition when you know just the chord length and arc length.
    Instead of solving this by Newton’s Method or the Dr Math solution, I decided to use the trial and error method whereby, the central angle is assumed to be 4 radians and then through a series of iterations, the angle becomes more precisely defined.
    If nothing else, please check it out.
    Thanks.

  17. Hi i’m a sheet metal worker and i need to know the chord lengh of a cone i’m making at work. The cone is 800 base 550 top and a height of 600mm please could you show me an easy way to work out how to find the chord . ps better with my hands than brian thanks .

  18. Hi,
    I have a challenge I am designing a new 1200mm Heavy duty open end Spanner and going off a 36mm open ender I need the radius of the three different arcs used in the shape of the open end head of the spanner, how can I find the Radius fron the given arcs?

    Regards
    Jose

  19. If I am laying out a segmented head, say a 30 foot diameter hemisphere with 15 segments. How do I solve for the right angle on 2 degree increments on my LDC. Lenght down center. Solving for the right angle every 2 degrees will give me the proper measurements to lay the segment out flat for cutting the blank before pressing. I am not strong on math. I need a simple formula. Thank you for your help.

  20. @Jose and tankguy,

    Unfortunately I just don’t have the time to help everyone that posts on my blog, I suggest that you make a scale model of what you’re trying to achieve so that you can work out the measurements before you start on full scale production. Additionally I suggest that you try some of the resources linked to on this page, as they will probably help out too.

  21. Hello,

    I dont understand what – is supposed to stand for in the equation C² = 2 r² (1 – cos(θ))) or anywhere else on the site. Is it maybe a type that i dont have? or an axiom that I dont know?

    thans in advance

    m

  22. i need to calculate the arc height from the centre of chord of length 100cm and arc lenght 101 cm. so help me to solve in step by step process to get the final answer.

  23. @arnav: Have a look at the circle calculator on the 1728 website, it has an option to calculate this.

  24. Hi Nurm,

    I am trying to follow your solution to the radius given chord and arc angle.
    I am using objective c since I want to use a computer to display the answers.
    So far having programmed the formula, I end up with an answer which is wrong
    Using 20 for chord length and 25 for arc angle, I end up with 14.564 as the radius.

    Can you help me decide what’s gone wrong
    Each step produces results like this

    A= 225
    B =4.94066 e-324
    C = 1.80108 e-226
    D = 2.07401 e-183
    F = 0.004438
    G = -9544371.768889
    X = 212.118924
    R = 14.564303

    I don’t really understand your formula and think I have programmed the steps properly, but hope by studying the outputs of each equation you can point me to the mistake in my calculation

    Many thanks

  25. Hi Gary, As I noted in my post, I made a fatal error in my calculations. Check out Dr. Math’s FAQ on Circle Segments, it has the correct equations.

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