SAG Calculator

Find a circular arc’s sagitta from its radius of curvature and diameter.

Eight metric and imperial length units, with every box solving backwards.

Updated October 24, 2026

CreatorFrank Zhao

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What a SAG calculation tells you

SAG, or sagitta, is the depth of a circular arc measured from the straight line across its two ends. This calculator relates that depth to two other lengths: the radius of the circle the arc belongs to, and the straight width of the opening, which is the chord diameter.

The three boxes are peers. Enter any two, and the third is solved. Most people start with a radius and a width to find a depth; someone checking a finished part often measures the depth and the width to recover the radius. The same numbers work in both directions, so the page is as useful for verifying a curve as it is for laying one out.

Box	What it measures	Often confused with
$R$ Radius of curvature	The radius of the circle that carries the arc.	The diameter, or the width of the opening.
$d$ Diameter	The straight chord from one end of the arc to the other.	The arc length measured along the curved surface.
$\mathrm{SAG}$ (sagitta)	The perpendicular depth from the middle of the chord to the arc.	A gap read off-centre, or the sag used in suspension tuning.

That last row matters. In suspension work, “sag” usually means a ride height or a travel figure that depends on spring rate, preload, rider weight and linkage. This page has none of those inputs. It computes the geometric depth of a circular arc, and the result is a length shown in the unit selected for that box.

Curved surfaces: check the depth of a concave or convex face against a straight edge and a measured width.
Arc layout: turn a chosen span and rise into the radius that a template, batten or cutter has to follow.
Reverse checks: recover a radius from a depth and a chord when the centre of the circle is not accessible.
Consistency checks: test whether a quoted radius and chord can describe the same circular arc at all.

A result is always a length

The calculator returns the missing length among radius, chord and depth. It does not return an angle, an arc length or a surface area. Those are different questions; this page only connects the three lengths.

Find the third measurement

The calculator works from any pair, so there is no required order. The only rule is that the two starting values describe the same arc.

Enter the two lengths you actually measured

A radius and a chord are the usual pair; a chord and a depth are the usual reverse pair. Each box can carry its own unit, so a width in millimetres and a radius in metres can sit side by side without hand conversion.

Leave the unknown box empty

As soon as two boxes hold values, the third is solved and marked as a result. If you later type a third value that disagrees with the pair already there, the number you just typed is kept and one of the earlier boxes is recalculated from it.

Read the result, then read the field message

A highlighted box is the solved length. A red box means a value cannot be used in the model; the message says which limit was crossed. A truly undefined result stays empty rather than showing a made-up number.

A worked example: a 400 mm chord with a 25 mm depth

Suppose you have a curved panel with a straight width of 400 mm and a 25 mm gap under a straight edge at the middle. The depth and the chord are the two measurements you have, so they are the two to enter. The radius that follows is:

R = \frac{d^{2}}{8\,\mathrm{SAG}} + \frac{\mathrm{SAG}}{2}

=

\frac{400^{2}}{8 \times 25} + \frac{25}{2}

=

\frac{160{,}000}{200} + 12.5

=

812.5\ \mathrm{mm}

The panel belongs to a circle with a radius of 812.5 mm. To check it the other way, enter 812.5 mm as the radius and 400 mm as the diameter: the depth comes back as 25 mm. That round trip is the quickest way to confirm that your two measurements and the calculator are using the same definition of the chord and the depth.

The geometry behind the three formulas

Half the chord, the radius minus the depth, and the radius itself form a right triangle. The radius is the hypotenuse, so the whole calculation follows from one relationship.

\left(R - \mathrm{SAG}\right)^{2} + \left(\frac{d}{2}\right)^{2} = R^{2}

Rearranging that relationship for the depth gives the forward formula. It is the form most people use when the radius and the opening are known:

\mathrm{SAG} = R - \sqrt{R^{2} - \left(\frac{d}{2}\right)^{2}}

The same relationship rearranges into the two reverse forms. The first recovers the radius from a measured chord and depth; the second recovers the chord from a radius and depth.

R = \frac{d^{2}}{8\,\mathrm{SAG}} + \frac{\mathrm{SAG}}{2}

d = 2\sqrt{\mathrm{SAG}\left(2R - \mathrm{SAG}\right)}

Why the valid range is narrower than the algebra suggests

A chord cannot be longer than the diameter of its circle, so the first limit is

\frac{d}{2} \le R

\Longleftrightarrow

d \le 2R

At exactly $d = 2R$ the chord is the diameter of the circle, the arc is a semicircle and $\mathrm{SAG} = R$ . That boundary is valid. Beyond it the expression under the square root turns negative, so the depth has no real value; the diameter box keeps the number you typed, turns red, and the depth stays empty with an explanation.

The second limit is about which side of the circle the depth measures. This page treats SAG as the depth of the minor arc, the shallow cap you get from a straight edge laid across the opening. For that cap,

0 < \mathrm{SAG} \le R

The diameter formula can still be evaluated between $\mathrm{SAG} = R$ and $\mathrm{SAG} = 2R$ , but there it describes the major segment on the other side of the circle. The chord it returns would not reproduce the same depth through the forward formula, so the calculator treats a depth larger than the radius as a geometry error: the affected fields turn red, and the finite numbers stay visible instead of being accepted as the same minor arc.

Units are normalised before the arithmetic

Each box converts its own length unit into a common base before the formula runs, then shows the answer back in the unit you chose for that box. The conversion factors are exact definitions rather than rounded table values: an inch is 0.0254 m, a foot is 0.3048 m and a yard is 0.9144 m. Mixed units are therefore safe; the one thing to avoid is treating a diameter and a radius as interchangeable because their numbers happen to look similar.

What the reverse formulas assume

All three forms assume one circular arc of constant radius, a chord measured as a straight line between the ends, and a depth measured perpendicular to that chord at its midpoint. Change any one of those assumptions and the three numbers no longer describe the same curve, even though each box still accepts a value.

Two measurements at two scales

The formulas do not care about the size of the part. These two examples use a workshop scale and a building scale, and they approach the result from opposite directions.

A 100 mm chord on a 130 mm radius

A curved trim piece is quoted as having a 130 mm radius. You can measure the straight width of the opening as 100 mm, but the centre of the circle is behind the material, so the radius is difficult to check directly. The depth is the measurement you can take.

Enter the quoted radius of 130 mm and the measured chord of 100 mm. The calculator predicts the depth you should find by halving the chord and evaluating the right triangle:

\mathrm{SAG} = 130 - \sqrt{130^{2} - 50^{2}}

=

130 - \sqrt{16{,}900 - 2{,}500}

=

130 - 120

=

10\ \mathrm{mm}

A straight edge across the 100 mm opening should show a 10 mm gap at the centre. A noticeably different gap means the quoted radius, the measured chord or both need checking before the piece is accepted.

A 16 m span with a 4 m rise

An arch is set out from a 16 m straight span and a 4 m rise at midspan. The rise is the depth, so the unknown is the radius that a template or a bending jig has to follow.

Enter a diameter of 16 m and a SAG of 4 m. The reverse formula gives:

R = \frac{16^{2}}{8 \times 4} + \frac{4}{2}

=

\frac{256}{32} + 2

=

10\ \mathrm{m}

The arch follows a 10 m radius. If the rise on site is 4.2 m instead, the same 16 m span implies a radius of about 9.719 m: a slightly tighter curve. That is why the rise should be measured at the centre of the span and not inferred from a tape pulled along the curved face.

Getting the measurement right

The arithmetic is short; the measurement is where most disagreements enter. These are the checks that change the answer most often.

Measure the chord straight across, not along the curve

The diameter box wants the straight-line distance between the two ends of the arc. A tape laid along the surface measures the arc length, which is longer. On a shallow arc, using that length as the chord overstates the sagitta; on a deep enough arc it can push the values outside the valid range altogether.

Take the depth at the midpoint, square to the chord

The sagitta is the largest perpendicular distance from the chord to the arc, and it occurs at the middle of the chord. A depth read near one end is shorter; treating it as the sagitta implies a flatter arc than the real one.

Check whether the number is a radius or a width

A radius runs from the centre of the circle to the surface. A width runs across the opening from edge to edge. Entering half the opening as the radius, or the radius as the diameter, changes the result substantially even though both numbers are lengths.

Shallow arcs make small depth errors matter

A 100 mm chord with a measured depth of 1.25 mm implies a radius of about 1,000.6 mm. Read the same depth as 1.30 mm and the implied radius falls to about 962.2 mm. In a shallow arc, repeat the depth reading and use a consistent reference before trusting the last digit.

Where this calculation stops

The model is one circular arc of constant radius in the plane you measure. A cylindrical surface works when the chord and depth are taken in its circular cross-section. An ellipse, a parabola, a freeform surface or an aspheric lens does not have a single radius across the opening, so a circular fit describes only part of it.
The chord is taken as a straight line and the depth as a perpendicular measurement at its midpoint. A tape run along the surface, a depth read off-centre or a surface that is tilted relative to the reference edge does not match the model.
The result is a geometric length, not a manufacturing tolerance. Surface finish, edge chamfers, coating thickness, springback and measurement uncertainty are not included, and a red field is a warning about consistency rather than an automatic correction.
It is not an optical prescription. The radius of a lens surface is a starting point for optical design, but focal length, power, image distance and aberrations depend on more than this radius: the material, the other surface, the object position and the wavelength all matter, and none of them are inputs here.
It is not a structural or safety check. For load-bearing arches, glazing, pressure components or any safety-critical part, use the applicable standard and a qualified engineer rather than this page alone.

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