I would like to calculate the arc length of a circle segment, i.e. I know the start coordinates

Willow Pratt

Willow Pratt

Answered question


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?

Answer & Explanation



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 θ, 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 = ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2

According to the law of cosines, cos ( θ ) = r 2 + r 2 d 2 2 r r = 1 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,
θ = arccos ( 1 d 2 2 r 2 )
Lastly, the length is calculated -
L e n g t h = r θ
Where θ is expressed in radians.

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