section_path: Properties/Sagitta
section_name: Sagitta
paragraph_text: Sagitta
  
  The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.
  
  Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: r = y 2 8 x + x 2. {\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}
  
  Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (2 r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2 r − x) x = (y / 2). Solving for r, we find the required result.