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sagitta — Definition, Formula & Examples

The sagitta is the distance measured along a perpendicular from the midpoint of a chord to the arc it subtends. It represents how much the arc 'bulges' away from the chord.

Given a chord of length ll in a circle of radius rr, the sagitta ss is the length of the line segment joining the midpoint of the chord to the midpoint of the intercepted arc, measured along the perpendicular bisector of the chord. Equivalently, s=r−r2−(l2)2s = r - \sqrt{r^2 - \left(\tfrac{l}{2}\right)^2}.

Key Formula

s=r−r2−(l2)2s = r - \sqrt{r^2 - \left(\frac{l}{2}\right)^2}
Where:
  • ss = Sagitta (arc-to-chord distance at midpoint)
  • rr = Radius of the circle
  • ll = Length of the chord

How It Works

Draw a chord across a circle and then draw the perpendicular bisector of that chord; it passes through the center of the circle and intersects the arc at two points. The sagitta is the segment of this perpendicular bisector that lies between the chord and the nearer arc. You can derive the formula by noting that the perpendicular distance from the center to the chord is d=r2−(l/2)2d = \sqrt{r^2 - (l/2)^2}, and the sagitta is simply s=r−ds = r - d. If you know the central angle θ\theta instead, the sagitta equals r(1−cos⁡θ2)r(1 - \cos\tfrac{\theta}{2}).

Worked Example

Problem: A circular arch has a radius of 10 m and a chord (span) of 12 m. Find the sagitta of the arch.
Identify half-chord length: Divide the chord length by 2.
l2=122=6 m\frac{l}{2} = \frac{12}{2} = 6 \text{ m}
Find the apothem (center-to-chord distance): Use the Pythagorean relationship with the radius.
d=r2−62=100−36=64=8 md = \sqrt{r^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ m}
Compute the sagitta: Subtract the apothem from the radius.
s=r−d=10−8=2 ms = r - d = 10 - 8 = 2 \text{ m}
Answer: The sagitta is 2 m.

Why It Matters

The sagitta appears in optics when describing lens curvature, in civil engineering for designing arched bridges and tunnels, and in surveying for computing curve offsets. It also provides an efficient way to find a circle's radius when you can measure the chord and the arc height but cannot access the center directly: r=s2+l28sr = \frac{s}{2} + \frac{l^2}{8s}.

Common Mistakes

Mistake: Using the full chord length instead of the half-chord in the formula.
Correction: The Pythagorean step requires l/2l/2, not ll. Always halve the chord length before squaring and subtracting from r2r^2.

Related Terms