SAG Calculator

How to use the SAG calculator

What this calculator does

This calculator finds the SAG (sagitta) of a circular arc: the greatest perpendicular distance between a straight chord and the arc. It uses the circle's Radius of curvature and the chord's Diameter (chord length). The result is exact for ideal circular geometry. It is not a cable-sag, spring-preload, suspension-tuning, or structural-strength model, because those problems require loads, tension, material properties, and boundary conditions that are not part of the circular-segment formula.

When to use it

Use this tool when laying out a shallow arch, checking the crown of a curved panel, estimating the rise of a circular roadway or rail profile, or converting a known circle radius and chord span into a measurable center depth. It is also useful for shop fabrication, woodworking templates, glazing layouts, and quality-control checks where an arc is assumed to be circular.

How to calculate

1. The calculator opens with a complete demonstration: a 10 m radius and a 13 m chord. The results and Excel export are immediately available.
2. Replace Radius of curvature with the radius of the parent circle. Select its unit beside the field.
3. Replace Diameter (chord length) with the straight-line distance joining the two arc endpoints. Choose its unit independently.
4. Select a Result unit. The calculator converts SAG, half-chord, arc length, and the linear values in the table. Segment area is shown in the squared result unit.
5. Read SAG (sagitta) as the maximum center depth. Review the secondary outputs when you also need the central angle, length along the arc, or circular-segment area.
6. Select Download Excel to save the current validated model as a real .xlsx workbook. Reset clears the demonstration and all calculated content; export stays unavailable until a complete valid state is entered again.

Input guide

Radius of curvature is required and accepts a positive decimal in mm, cm, m, in, or ft. A realistic example is 10 m. Increasing radius while holding the chord constant makes the arc flatter and reduces SAG. Do not enter the arc length or diameter of the parent circle here. The radius must be at least half the chord.

Diameter (chord length) is required and accepts a positive decimal in the selected unit. A realistic example is 13 m. Increasing the chord while holding radius fixed increases SAG and the central angle. The largest valid chord equals 2R, which produces a semicircle and SAG equal to R. The word “diameter” here refers to the straight span used by the source model; geometrically it functions as the chord length.

Result unit is required and changes presentation only, not the geometry. For example, 2.4007 m and 2400.7 mm describe the same SAG. Avoid comparing values without checking their units. The unit relationships follow the SI system and exact inch definition described in the NIST guidance on SI units of length.

Output guide

SAG (sagitta) is the primary result and is shown in the selected result unit. Zero would represent a limiting perfectly flat chord with zero span; larger values indicate a deeper circular arc. Half-chord is d/2 and is a direct identity. Central angle is the minor angle subtended at the circle center, displayed in degrees. Arc length is the distance along the minor arc, while Segment area is the area enclosed by that arc and its chord. All are geometry estimates based on ideal inputs, not fabrication tolerances.

The summary pills repeat the current radius, diameter, and dimensionless chord-to-radius ratio. The Geometry breakdown table lists each quantity, symbol, converted value, and practical meaning. A chord/radius ratio above 2 is impossible for a real circle and is rejected.

Worked example

For the startup values R = 10 m and d = 13 m, half the chord is 6.5 m. Applying s = R – √(R² – (d/2)²) gives 10 – √(100 – 42.25) = 10 – √57.75 = 2.4007 m. The same geometry produces a central angle of about 81.083°, an arc length of 14.1517 m, and a segment area of 23.6722 m². These values match the first-open display and workbook.

Learn more

For the underlying geometry, see Wolfram MathWorld's explanation of a circular segment and sagitta relationships. The key domain condition is d ≤ 2R; equality gives a semicircle, while smaller chords give shallower minor segments.