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How to calculate the coordinates of the middle point of a given arc?

I am trying to calculate the green sides of this triangle:

right triangle in circle

I know/have:

  • the arc length,
  • the arch base,
  • the radius,
  • and the h (distance from the red dot to the center of the circle)
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migrated from stackoverflow.com Dec 15 '11 at 15:06

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marked as duplicate by Zev Chonoles Feb 17 '12 at 17:06


You only need to know the lengths or the exact position? Because two points (and the arclength) doesn't define the circle unambiguously. –  WebMonster Dec 15 '11 at 11:57

1 Answer 1

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Not exactly mathematical notation (which labels points and hence makes naming "lines" easier), but let me try to understand your question.

You have the arc - i.e the circular part of the circle and the cord (I believe is the term), i.e. the bit that separates the arc. (http://mathworld.wolfram.com/CircularSegment.html)

I.e. you have an accurate description of the segment of the circle if I am not mistaken from your post.

Now at this point, your segment can rotate around the circle without any issues at all. The radius as well as the height h will not change depending on where the segment will be located on the circle.

If you have the coordinates of one of the points or an angle however, you can start to draw triangles and work out the "unknown bits" step by step. But if my understanding of your question is correct, and you have only a few lengths, then the answer is no.

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Thanks, I got what you mean. But, what if I would know the coordinates of the left most black dot? Sorry I know this image sucks.. –  Jonas Dec 15 '11 at 12:04
In this case, you also have the coordinate of the centre pointof the circle (if not you can easily work it out as it touches the axis). This gives you a triangle with 3 known sides (chord 2* radius) plus a definite position in space. As a result you can work out the other coordinates based on that triangle and so on. Let me find a link on geometry & trig - because this is best explained in pictures. –  DetlevCM Dec 15 '11 at 12:18
Not that I like Wikipedia, but it is seems to be the only resource that has what I wanted for you: http://en.wikipedia.org/wiki/Triangle The rules with respect to three pieces of information defining a triangle. Hence, once you know at least one of the points you can start to work out the positions of the other points and hence the other distances. (You will possibly need to work out some angles along the way too) –  DetlevCM Dec 15 '11 at 12:25
Actually, the right term is chord (with an "h"). –  Michael Hardy Dec 15 '11 at 18:05
@Michael - indeed you are right. –  DetlevCM Dec 16 '11 at 11:35
