What the heck is a sagitta (also called the versine) 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. Javascript calculators are provided for those that don't want to do the math.

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:

where:

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:

where:

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:

where:

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

where:

See American Lutherie for a number of articles on construction of jigs and fixtures for building instruments with domed plates.