Sag Calculator - Arc depth from radius and chord

Sag calculator finds the sagitta, the perpendicular distance from the midpoint of a chord to the arc it spans on a circle. Carpenters and fabricators usually call this the rise or arc depth, and it is the single number you need to lay out a curved member when you already know the radius you want the curve to follow and the clear width it must cover.

The sag calculator takes two inputs you can measure directly on the job: the radius of curvature of the arc and the chord length, which is the straight-line width between the two ends of the arc. From those it returns the sagitta, the full diameter of the circle, and the central angle the arc subtends.

Knowing the sagitta lets you scribe, cut, and test-fit curved parts without guessing. When you frame a doorway arch from a known radius, the sagitta is the height the arch reaches above the spring line at its centre. If you are working the other way around, the arch calculator turns a known span and rise into the radius you should feed into this sagitta tool.

The same number shows up outside the workshop too. Pipe fitters use the sagitta to set the rise of a long chord between two supports, and surveyors use it when laying out a vertical curve. In each case the geometry is identical: a known radius and a known width fix the depth of the arc between them, which is exactly what this tool returns. For long runs the chord is often only a small slice of the full circle, so a tiny change in radius produces a surprisingly large shift in the installed sag, which is why the rise matters as much in drainage and grade work as it does in finish carpentry.

The sag calculator uses the circular-segment relationship s = R - sqrt(R^2 - (d/2)^2), where R is the radius of curvature and d is the chord length. The term under the square root is the distance from the circle centre to the midpoint of the chord; subtracting that from R leaves exactly the arc depth.

When the chord equals the diameter the square root becomes zero, so the sagitta equals R: that is a half circle, the deepest arc possible for a given radius. As the chord shrinks toward zero the sagitta falls toward zero, which matches a nearly flat, tiny sliver of the circle.

Because the formula depends on R squared, a small change in radius moves the sagitta a lot when the arc is shallow, and only a little when the arc is deep. That same behaviour is what makes segmental arch layouts forgiving near the crown and sensitive near the springing points, and it is worth checking a shallow arch against the beam deflection calculator so the curve does not flatten under load once it carries weight.

As published by Wolfram MathWorld, the sagitta of a circular segment is the height from the midpoint of the chord to the arc and follows s = R - sqrt(R^2 - (c/2)^2).

Worked example: with R = 48 in and d = 72 in, s = 48 - sqrt(48^2 - 36^2) = 48 - sqrt(2304 - 1296) = 48 - 32 = 16 in. The arch rises 16 in above its spring line at the centre.

The tool also reports the central angle of the arc. For that same 48 in radius and 72 in chord the half-angle is asin(36 / 48) = 48.6 degrees, so the arc spans about 97 degrees of the circle. That angle tells you how much of a full ring the segment represents, which is handy when you repeat the curve as one unit of a larger assembly. If you are matching several identical pieces, the central angle is the quick check that each piece is cut from the same arc rather than a slightly different one.

Four quantities describe a circular arc completely, and this tool works with three of them. Once you know any two of radius, chord, sagitta, and central angle, the other two are fixed, so you can solve for whichever value your layout is missing. This interdependence is why the sagitta is so useful on site: you rarely measure it directly, but you can always measure the radius of an existing curve and the width it must span, and the calculator gives you the rise you need to cut. The same relationship drives a full ring or enclosure, which is where the round pen calculator takes a target radius and panel count and works the arc geometry from the other direction.

According to Omni Calculator, the sagitta is the practical layout value you need when working curved surfaces from a known radius and width.

1. 1Find the radius of curvature R, either from the design drawing or by swinging a string to match the existing curve.
2. 2Measure the chord d, the straight clear width between the two ends of the arc at the spring line.
3. 3Enter R and d into the sag calculator and read the sagitta s, the arc depth at the midpoint.
4. 4Use s as the rise when you scribe, cut, and test-fit the curved member before final assembly.

A calculated sagitta removes trial-and-error from curved work. You cut the first piece to the right rise instead of bending and re-bending stock until it looks right, and every repeated segment matches because the same radius and chord always produce the same sagitta. Keep the units consistent between the radius and the chord as well, since the formula has no way to flag a radius entered in inches next to a chord entered in feet, and a mixed-unit input quietly returns a wrong rise.

A calculated sagitta saves material. You cut the first piece to the right rise instead of bending and re-bending stock until it looks right, and you avoid scrapping a board that was ripped to the wrong depth because an eyeballed curve ran deep at the centre.

It keeps repeated curved units consistent. When you build several identical arch segments, the same radius and chord always produce the same sagitta, so every piece matches without hand-fitting each one against its neighbour.

It catches impossible requests early. If the width you need is larger than the diameter of the radius you picked, the tool flags it before you waste material on a curve that cannot close. When the curve is one repeat of a segmented ring, size each stave and glue-up from the same radius and arc geometry rather than trusting the arch alone.

It also documents the layout. Write the radius, chord, and resulting sagitta on the back of the template and anyone on the crew can re-cut the part weeks later without re-deriving the curve. That record is especially useful on site-built arches where no shop drawing exists, and it lets a second crew finish a run you started without re-measuring the spring line.

The sagitta is exact for a true circle, but real jobs introduce a few effects that shift the finished rise away from the number on screen. The points below cover the ones that matter most when you are cutting material to fit.

According to Wikipedia, the circular segment is bounded by a chord and the corresponding arc, and the sagitta grows sharply as the chord length approaches the diameter.