A simple paper and pencil method for calculating the area of an arbitrary plate (stringed musical instrument top or bottom) is presented. This is a reprint of an article that originally appeared in American Lutherie.
Last updated: Tuesday, May 08, 2012
[Originally published in American Lutherie #70, Summer 2002]
Copyright (C) 2002 by R.M. Mottola
There are a number of reasons one might want to calculate the area of the plate of a stringed instrument. Once calculated the area of a flat plate can be used to determine the volume of the instrument by simply multiplying the area by the depth. This value is useful in the design of electric guitars and basses to determine the weight of the body of the instrument before it is built. This info can aid in the design of an instrument that balances well when hanging from a strap or sitting on the leg. In the design of acoustic instruments the volume can be used to calculate the nominal Helmholtz resonance of the sound box, which may be useful in the tuning of the resonance characteristics of the instrument.
The technique specified here will work for any arbitrary shape and is both simple and relatively quick. It is the essential algorithm of a Computer Aided Design (CAD) script I use, and is based on a computer graphics rasterization technique. Modified and simplified for use with pencil and paper, it yields a good enough approximation of the area of a plate for the purposes outlined above.
The first step of the process is to copy the outline of the plate onto graph paper. Graph paper that is 2’ x 3’ is generally available but in a pinch smaller sheets can be taped together carefully so that the squares line up. Once the outline is traced the number of squares that fall either completely inside the outline or mostly inside the outline are added up and that number written down. This sum is then multiplied by the area of a single square on the graph paper to yield a very close approximation of the area of the plate. To calculate the area of a square, multiply the length of a side of the square by itself. For example the area of a 0.25” square is 0.0625” (0.25 * 0.25 = 0.0625).
The figure shows the outline of one of my “Mezzaluna” acoustic electric basses thus worked, with a calculated area of 147.875 square inches. The size of the squares is 0.25” x 0.25” which yields a value for the area accurate to within about 0.5%. Smaller squares will yield more accurate results but the effort is hardly worth it, at least for use in the volume calculations for the practical examples given at the top.