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.

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?Thanks.

tilsjaskak6 · Accepted Answer

You can derive a simple formula using the law of cosines. In fact, while all the planar geometry is helpful for visualization, there&#039;s really no need for most of it. You have 3 points: your arc start and stop points, which I&#039;ll call A and B, and your circle center, C. The angle for the arc you&#039;re wanting to measure, I&#039;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&#039;s radius,   r. We&#039;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  θWhere   θ is expressed in radians.

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

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