Leather Didgeridoo

Step 4 Calculations

Now that you have the bore design, you need to do some calculations to determine the shape of leather to cut out. The idea is that at each "section" in the design, the leather needs to be wide enough that when you bend it into a circle, it is the diameter that you want.

At first, you might think that you just need to calculate the circumference of a circle with the given diameter, but that will actually give you a smaller bore than you want, due to the thickness of the leather.

In general, if you have a piece of leather with a certain width and thickness and you form it into a circle, the inner diameter of the circle will be:
InnerDiameter = (width/Pi) - thickness

And the outer diameter will be:
OuterDiameter = (width/Pi) + thickness

The difference between the two is twice the thickness of the leather, which makes sense, because you go through 2 thicknesses of leather going from the outer edge of one side to the outer edge of the other side.

You need to calculate the width of leather than will give you the bore (inner) diameter that you want. From above, we know that
InnerDiameter = (width/Pi) - thickness

And we need to solve for the width.
width = (innerDiameter + thickness) * Pi

Now you need to find the average thickness of the leather. You can measure it at various places around the edge and guestimate the average thickness. It will typically be around 4-6mm.

I happen to have a custom-made tool that can measure the thickness of the leather anywhere in the middle of the leather, so I was able to get a more accurate thickness measurement at each position (shown below)

0m: 5.2mm
.628m: 5mm
1.256m: 4.6mm
1.78m: 3.9mm

Now you just need to plug in the thickness and the diameter for each section to get the width of leather at that position.

0m: (30mm + 5.2mm) * 3.14159 = 111mm
.628m: (40mm + 5mm) * 3.14159 = 141mm
1.256m: (60mm + 4.6mm) * 3.14159 = 203mm
1.78m: (100mm + 3.9mm) * 3.14159 = 326mm

And finally, you need to calculate the curve at the wide end. If this was straight instead of curved, the bell of the didgeridoo wouldn't be flat on the bottom, but it would stick down where the seam is.

The curve that is needed is a circular curve, so you need to calculate the radius of the circle. Later, you will use a string that is the length of the radius in order to draw the curve.

The derivation of the calculation is a bit involved, so I'll skip it and just give the calculation itself.

CurveRadius = (((LargeRadius - SmallRadius)^{2 + Height}2)^{(1/2)) * LargeRadius / (LargeRadius - SmallRadius)}

See the images for this step for a better formatted version of the equation.

In this equation, LargeRadius is the radius of the bore at the bell end, plus 1/2 the thickness of the leather.

SmallRadius is the radius of the bore at the section just before the bell, plus 1/2 the thickness of the leather.

Height is the distance between the bell and the section just before the bell.

For the didge that I am building:
LargeRadius = 50mm + (3.9mm/2) = 51.95mm
SmallRadius = 30mm + (4.6mm/2) = 32.3mm
Height = 1780mm - 1256mm = 524mm

And then plugging these into the CurveRadius equation:

CurveRadius = (((51.95 - 32.3)^{2 + 524}2)^{(1/2)) * 51.95 / (51.95 - 32.3) = 1386mm}