If I know the radius of a given circle, the length of the chord of the segment, and the height of the segment can I find the area of the segment?

  • $\begingroup$ If you know the radius and the length of the chord, you can calculate the central angle with the Law of Cosines. With that angle as well as the radius you can find the area of the sector. To find the area of the triangle is a little bit of trig: Area = 0.5*product of radii*sin(enclosed angle) The difference gives the area of the segment Sorry for my bad typing... $\endgroup$ – imranfat Feb 28 '15 at 0:34
  • $\begingroup$ Maybe can you provide us with an example? $\endgroup$ – imranfat Feb 28 '15 at 0:35
  • $\begingroup$ @imranfat: knowing the radius and chord length gives you one of two possible segments. $\endgroup$ – robjohn Feb 28 '15 at 0:47
  • $\begingroup$ @robjohn.Yes, I realized that once I made a drawing. I will honestly tell you though that when I wrote my comment that this fact did not immediately cross my mind. I asked the OP for an example so that we could work with that. $\endgroup$ – imranfat Feb 28 '15 at 1:49

A quick read of http://mathworld.wolfram.com/CircularSegment.html should help! The formula is; $$ A =R^2\cos^{-1}\left(\frac{R-h}{r}\right)-(R-h)\sqrt{2Rh-h^2} $$ or

$$ A =\frac{1}{2}\left(Rs-ar\right) $$ Where: 'R' is the radius of the circle, 'a' the chord length, 's' the arc length, 'h' the height of the arced portion(a.k.a Sagitta), and 'r' the height of the triangular portion.


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