What is the Sagitta?
The sagitta (Latin for "arrow") is the height of a circular arc measured from the midpoint of its chord to the midpoint of the arc itself. It describes how far an arc bows away from its chord. The term appears in optics, archery, structural engineering, woodworking, and road and rail design where curved shapes are everywhere.
How to Use This Calculator
Enter the radius r of the circle and the chord length c (the straight-line distance between the two endpoints of the arc). The calculator returns the sagitta, the arc length, and the central angle subtended by the chord. Make sure the chord is not longer than the diameter (\(c \le 2r\)); otherwise the geometry is impossible.
The Formula Explained
The sagitta follows directly from the Pythagorean theorem. The half-chord (\(c/2\)), the apothem (distance from center to chord), and the radius form a right triangle. The apothem equals \(\sqrt{r^{2} - (c/2)^{2}}\), so the sagitta is the radius minus the apothem:
$$s = r - \sqrt{r^{2} - \left(\dfrac{c}{2}\right)^{2}}$$
The central angle is \(\theta = 2\cdot\arcsin(c / 2r)\), and the arc length is \(L = r\cdot\theta\) (with \(\theta\) in radians).
Worked Example
Suppose \(r = 5\) and \(c = 8\). Then \(c/2 = 4\), and \(r^{2} - (c/2)^{2} = 25 - 16 = 9\), so \(\sqrt{9} = 3\). The sagitta is \(5 - 3 = 2\). The central angle is \(2\cdot\arcsin(4/5) \approx 1.8546 \text{ rad} \approx 106.26°\), and the arc length is \(5 \times 1.8546 \approx 9.273\).
FAQ
What if I know the sagitta and chord but not the radius? You can rearrange: \(r = (s^{2} + (c/2)^{2}) / (2s)\).
Why is the chord limited to 2r? The longest possible chord in a circle is the diameter, which equals \(2r\). A longer value has no real geometric solution.
Is the sagitta the same as the segment height? Yes — sagitta, arc height, and the height of a circular segment all refer to the same measurement.
