What the heck is a sagitta and why would you want to calculate one? Here’s the deal. It turns out that there are a number of lutherie applications that make use of spherical domes or cylindrical sections. The plates of modern so-called flattop guitars are generally domed, and the plates of some other instruments often describe a cylindrical section. Such instruments are built on dished and trough-shaped forms (work boards) which force the thin plate into the final shape. To build such work boards, or to figure out the depth of the sides needed to mate with shaped plates, or even to make radius sanding blocks for shaping fingerboards, one needs to know the relationship between the radius of a circular arc, the length of the chord connecting its two ends, and the deflection of the highest point or that arc from the center of the chord. This latter quantity is called the sagitta, or sag for short. A Javascript calculator is provided for those that don't want to do the math.

The diagram below will help to visualize the quantities involved and their relationship to each other. The circular arc is in red and is of radius r. The chord (span) connecting the ends of the arc is divided in half, and that is labeled l in the diagram. Finally the sagitta, the displacement or deflection of the highest point of the arc from the mid point of the chord is labeled s. I’ll get to the formula for calculating the sagitta in a bit, but first let me answer the question of why you’d want to calculate it for lutherie applications. If you want to draw an arc for some design application it is a simple matter to use a compass to do so. But things can get a little tricky when the radius of the arc is big. For example, the domed plates of typical flattop guitars have radii that fall in the range of 12’ to 30’. Practical approaches to drawing such large radius arcs include use of the long compass (described nicely by math professor Jon Sevy on his Lutherie Resources website). Circular arcs can also be approximated by bending a spline (thin strip of wood or metal) around three small pins or nails. To do the latter, you’d need to know where to place the nails for a given radius of arc, and this is where the sagitta calculation comes in. Given the radius of the arc you want to draw and the length of the chord connecting the ends of that arc (corresponding to, say, the width of the dished work board you want to make) the length of the sagitta can be calculated. Once done, the end points and displacement point for the arc can be laid out on a board, nails inserted at those points, a spline bent around the nails, and the curve of the spline penciled onto the board.

The formula for calculating the sag is:

where:

s = the sagitta (sag) or displacement;
r = the radius of the arc;
l = ½ the length of the chord (span) connecting the two ends of the arc;

The formula can be used with any units, but make sure they are all the same, i.e. all in inches, all in cm, etc.

A similar formula can be used to derive the radius of an arc from span and displacement measurements. This can be used to, say, figure out the radius of an unmarked dished workboard. lay a ruler across the dished surface and then drop another ruler from the center of the first ruler down to the surface of the dish. The length of the first ruler is the span and the distance from the first ruler to the surface of the dish is the sagitta or displacement. The formula is:

where:

r = the radius of the arc;
l = ½ the length of the chord (span) connecting the two ends of the arc;
s = the sagitta (sag) or displacement;

See American Lutherie for a number of articles on construction of jigs and fixtures for building instruments with domed plates. Don’t want to do the math yourself? Check out Jon Sevy’s Lutherie Resources website again for a nice applet and spreadsheets that will do it for you.