Last updated: September 11, 2018
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. 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:
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 related 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:
A related formula can be used to calculate the height of the arch at any point - not just in the center. This formula can be used by those that want to build a dish by routing out a board to different depths for example. The first step is to calculate the sagitta s for the arc based on the radius r and the span l. Use the first calculator (above) to do this. Then you can plug the values for r and s into the following formula to calculate height h at any offset x from the center of the arc.
The formula is:
Here is one more calculator. This one calculates the arc length / arc circumference given the radius and length of the chord. This is useful in lutherie for things like calculating the length of the side of an instrument. Since the body shape is generally composed of a number of tangent circular arcs, you can calculate the length of a body side by calculating all of the arc lengths and then adding them together.
θ = 2 arcsin(l/r)
c = θr
See American Lutherie for a number of articles on construction of jigs and fixtures for building instruments with domed plates.