SAG Calculator
Our SAG calculator can help you determine the sag of a surface based on the radius of curvature and diameter.
Sag (sagitta)3.17542 mm0.12502 in
Radius of curvature100 mm3.93701 in
Chord50 mm1.9685 in
Arc length50.53605 mmalong the curve rather than across it
Half angle14.47751°subtended at the centre
Full angle subtended28.95502°
Curvature (1 ÷ R)10 per m10 dioptres if this were a lens surface
Shallow approximation c² ÷ (8R)3.125 mmaccurate when the sag is small — here it is off by 0.050416 mm
Sag as a fraction of the radius3.1754%
The formula
s = R − √(R² − (c ÷ 2)²)
The height of an arc above its chord
Sagitta is Latin for arrow: the chord is the bow and the sag is the arrow’s height at the middle. It comes up wherever a curve has to be measured across a gap — the depth of a lens surface, the crown of a road, how far a straightedge sits off a curved workpiece.
For shallow curves the approximation c² ÷ 8R is much easier and very nearly right, which is why it is the form usually quoted. It only starts to drift once the sag becomes a noticeable fraction of the radius.