Calculating the arc length of a circle segment

I would like to calculate the arc length of a circle segment, i.e. I know the start coordinates (x/y) of the circle segment, the end coordinates (x/y) and the x and y distances from the starting point to the center point of the circle segment.

I know that I can calculate the circumference with 2 * radius * PI. Consequently, I would have to calculate the radius and the angle of the circle segment via Pythagorean theorem and sin and cos. My question: Is there a simple formula where I just have to put in start-coordinates, end-coordinates and the circle origin point coordinates?

Thanks.

enne

• I have never seen the phrase "bow length" used before. What does it mean? – Gerry Myerson Jun 11 '14 at 12:34
• I mean the circle length, I replaced the term "bow" with "circle". – enne87 Jun 11 '14 at 12:40
• Your question is still not very clear. If you have translated this, please provide the original text also. – M. Vinay Jun 11 '14 at 12:41
• @GerryMyerson, I think OP wants to find the length of the arc between the two points given. – Vikram Jun 11 '14 at 12:42
• @GerryMyerson Arc length. Could be a word-for-word translation from the German Bogenlänge. – Daniel Fischer Jun 11 '14 at 12:42 Use $(x-h)^2+(y-k)^2=r^2$...(eq I), where $(h,k)$ is the center and $r$ is the radius. Your third point can be calculated as $x_3=x_1+$(horizontal displacement) and $y_3=y_1+$(vertical displacement), you have mentioned that you know x and y distance.

the x and y distances from the starting point to the center point of the circle segment.

Now, in eq (I), put $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$ to get three equations, solve these three equations simultaneously to get $(h,k)$. Next, use $(h,k)$ and any of the three points we know to get $r$ with the help of eq (I).

Now shift the circle so that it's origin coincides with $(0,0)$, this will help you find the angle $\theta$ easily. Now, find $\theta$ and the length of the arc=$r*\theta$

EDIT:Consider a circle with 3 points on it.

$x_1\equiv(2+2\sqrt{3},0) \hspace{12 pt}x_2\equiv(6,2)\hspace{12 pt}x_3\equiv(2+2\sqrt{3},4)$

If you use eq (I) you will get following 3 eqns:

$29.85+h^2-10.91h+k^2=r^2$...(eqn 1)

$45.85+h^2-10.91h+k^2-8k=r^2$...(eqn 2)

$40+h^2-12h+k^2-4k=r^2$...(eqn 3)

subtract eqn 3 from eqn 1 to get one eqn

subtract eqn 3 from eqn 2 to get second eqn

solve these last two eqn and you will get $(h,k)\equiv(2,2)$, the center of the circle, substitute $(h,k)$ in eqn 3 (or 1 or 2), you will get $r=4$

No need to do transform the center of the circle to the origin, we know the radius, so find what angle points $x_1$ and$x_3$ makes with the horizontal line using trigonometry

($\sin \theta=2/4=1/2 \therefore \theta=\pi/6$)

Length of the arc=$r*\theta=4*\large\frac{\pi}{6}=\large \frac{2\pi}{3}$

• I don't know why you are give 3 points on the circle, only two are enough to find the center,(2 unknowns, 2 equations) – Vikram Jun 11 '14 at 13:36
• replace the word "origin" in second last line with "center" – Vikram Jun 11 '14 at 13:47
• Sorry, probably I'm too stupid, but can you please give me an example how this works? – enne87 Jun 12 '14 at 10:03
• Hey, cool thing, thanks :D – enne87 Jun 12 '14 at 15:32
• @enne87, you’re welcome – Vikram Jun 12 '14 at 16:01

You can derive a simple formula using the law of cosines. In fact, while all the planar geometry is helpful for visualization, there's really no need for most of it. You have 3 points: your arc start and stop points, which I'll call $A$ and $B$, and your circle center, $C$. The angle for the arc you're wanting to measure, I'll call it $\theta$, is the angle of the triangle $ABC$ at point $C$. Because $C$ is the center of the circle that $A$ and $B$ are on, the triangle sides $AC$ and $BC$ are equal to your circle's radius, $r$. We'll call the length of $AB$, the remaining side, $d$ (see picture).

(By the way, if $A=(x_1,y_1)$ and $B=(x_2,y_2)$, then $d=\sqrt {(x_1-x_2)^2 + (y_1-y_2)^2}$.) According to the law of cosines, $\cos (\theta )={{r^2+r^2-d^2}\over {2rr}}=1- {{d^2}\over {2r^2}}$.

So all you need is the distance between the end points of your arc and the radius of the circle to compute the angle,

$\theta = \arccos (1- {{d^2}\over {2r^2}})$

Lastly, the length is calculated -

$Length = r\theta$

Where $\theta$ is expressed in radians.

• Hi Zimul8r, thanks for your help. I just tried your approach with a circle segment where θ = 90° and r = 4. This means this is an exact quarter circle segment. So d = sqrt(4²+4²) = 5.6568; Consequently, cos(θ) = (1-5.6568²) / 32 = -0.9687. arccos(-0.9687) = 14.373rad ?? What am I doing wrong here? – enne87 Jun 12 '14 at 9:40
• Ah sorry, now it works.Thanks for the great approach. – enne87 Jun 12 '14 at 15:32