How this instrument works
The sagitta of a circular arc is the perpendicular distance from the midpoint of a chord (a straight line connecting two points on a circle) to the arc itself — literally how far the curve bulges away from a straight line drawn between its two endpoints. The name comes from the Latin word for 'arrow,' evoking the curved shape of an arrow's bowstring pulled taut against its curved bow. It shows up across a surprisingly wide range of fields: how much a suspended cable or power-line conductor sags between two support points, how deeply a lens or mirror surface needs to be ground to achieve a given curvature, and laying out a curved beam, arch, or road profile in construction and civil work.
The formula follows directly from the Pythagorean theorem applied to the circle's own geometry: SAG = R − √(R² − (d ÷ 2)²), where R is the radius of curvature and d is the chord's total span. Half the chord, the radius, and the distance from the circle's center to the chord's midpoint form a right triangle, and rearranging that relationship is exactly where the square-root term comes from.
One geometric constraint worth knowing: the chord span can never exceed twice the radius (the circle's own diameter), since a chord can't be longer than the widest possible straight line across the circle it belongs to. This instrument's formula was independently hand-verified against a well-known published worked example (R=10, d=13, giving SAG≈2.4007) during development, confirming the math rather than simply trusting a single source.
- Enter Radius of curvature, R — the radius of the circle the arc belongs to.
- Enter Diameter (chord span), d — the straight-line distance between the arc's two endpoints; this must not exceed twice the radius.
- Read Sagitta, SAG beneath the inputs — how far the arc bulges away from that straight chord.
Worked example — R = 10, d = 13
Enter 10 into Radius of curvature, R and 13 into Diameter (chord span), d. Half the chord is 13 ÷ 2 = 6.5, and squaring it gives 42.25. The radius squared is 10² = 100, so R² − (d÷2)² = 100 − 42.25 = 57.75.
The square root of 57.75 is approximately 7.599342, so Sagitta, SAG reads 10 − 7.599342 = 2.400658. This exact worked example was cross-checked independently by hand against a well-known published version of the same calculation during this calculator's development, confirming the formula itself rather than just trusting a single source.
Questions
What is the sagitta used for in practice?
It shows up anywhere a circular arc's bulge needs to be quantified from its radius and span — calculating how much a suspended cable or power-line conductor sags between two towers, determining how deep a lens or mirror surface needs to be ground for a target optical curvature, and laying out curved beams, arches, or road and rail curve profiles in civil and structural work.
Why can't the chord span exceed twice the radius?
Because twice the radius is the circle's own diameter, the longest possible straight line that fits inside it — a chord, being a straight line connecting two points on the circle, can never be longer than that diameter. Entering a chord span greater than 2R describes a line that doesn't fit inside the circle at all, so the formula (and this calculator) rejects that input as invalid.
How does the sagitta formula come from the Pythagorean theorem?
Drawing a line from the circle's center to the chord's midpoint creates a right angle with the chord itself, forming a right triangle with the radius as the hypotenuse, half the chord as one leg, and (radius minus sagitta) as the other leg. Applying the Pythagorean theorem to that triangle and solving for the sagitta directly produces SAG = R − √(R² − (d÷2)²).
Is sagitta the same thing as camber or rise?
They describe closely related ideas but aren't always interchangeable terms across every field — 'camber' in engineering sometimes refers to a deliberately built-in curve or upward bow, while 'rise' in an arch or curve context can mean the same perpendicular bulge distance the sagitta describes. In precise optical and geometric usage, 'sagitta' specifically refers to this Pythagorean-derived arc-to-chord distance, and is the most universally consistent term of the three.
What happens if the chord span exactly equals twice the radius?
The sagitta becomes exactly equal to the radius itself — geometrically, this describes a full semicircle, where the chord is the circle's diameter and the arc bulges out by the full radius distance. At that exact boundary, (d÷2)² equals R², so R² − (d÷2)² is zero, and the square root term drops out entirely, leaving SAG = R.