Willow Pratt

2022-07-02

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.

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.

tilsjaskak6

Beginner2022-07-03Added 14 answers

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, $\mathrm{cos}(\theta )=\frac{{r}^{2}+{r}^{2}-{d}^{2}}{2rr}=1-\frac{{d}^{2}}{2{r}^{2}}$.

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

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

Lastly, the length is calculated -

$Length=r\theta $

Where $\theta $ is expressed in radians.

(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, $\mathrm{cos}(\theta )=\frac{{r}^{2}+{r}^{2}-{d}^{2}}{2rr}=1-\frac{{d}^{2}}{2{r}^{2}}$.

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

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

Lastly, the length is calculated -

$Length=r\theta $

Where $\theta $ is expressed in radians.

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