This is a copy of an article from about 5 years ago posted to the USFG list that has way too many sums in it.

WARNING! CONTAINS MATHEMATICS
Hello Craig,

> For example.. say you have a piece of rough that is about 6mm
> deep, but is 8mm square.. your finished stone will not be
> larger than 6mm.. thats what I’m finding with my cuts but
> maybe it’s a coincidence or something..

I would suggest you measure a couple of stones that you’ve cut
and see what kind of ratio you have between your girdle width
and stone height. What you describe is absolutely horrible
recovery.

With a few sums complete predictability is possible. We will use
the cotangents of our main angles to determine the exact height
of the stone’s pavilion and crown expressed as a ratio of the
girdle width. This applies to brilliant cut stones with pavilion
main facets that touch both culet and girdle and crown mains
that touch both table and girdle.

If we have a square piece of rough we could try for a princess
style crown with a high crown angle and large table.
Our first example uses;
70 percent table, 40 degree crown 43 degree pavilion.
The height of the crown is 12.578194 percent of the girdle width.
The height of the pavilion is 46.625742 percent of the girdle
width.

Add room for the girdle height required and you will see that if
this design produces a stone with a finished 6 mm total depth
the girdle width needed exceeds availability by some 2 mm.

As we only have 8mm for our girdle width we only need 3.73mm for
the pavilion, 1.01mm for the crown and a bit for the girdle.
This makes our total depth 4.74mm plus .2-.4.  This gives us
plenty of room for an 8mm square stone. We can even change the
design to a true princess with it’s approx 10 percent extra
pavilion depth requirement. It fits.

Our next example has a more conventional crown angle and table
size;
55 percent Table, 35 degree crown 42 degree pavilion.

The height of the crown is 15.754669 percent of the girdle width.
The height of the pavilion is 45.020183 percent of the girdle
width.

Once again we are not wide enough for the 9.87mm girdle width
that this depth requires.

An 8mm width only needs 4.86mm plus .2-.4 for the girdle

Well it seems that we have more depth than width so maybe a trap
(emerald) cut would work;
55 percent Table, 20, 35, 40 degree crown 42, 49, 55 degree
pavilion.
The height of the crown is 15.754670 percent of the girdle width.
The height of the pavilion is 57.989058 percent of the girdle
width.

An 8mm width needs 5.89 mm plus the girdle which makes this cut a
perfect fit…….or does it?

If you have a block 8mm square and 6mm deep then the distance
between an edge of the top and it’s opposite edge on the bottom
is 10mm….hmm.

Imagine this line through the stone as the stones girdle plane.
Unfortunately we can’t get a 10 mm stone out of it but there is
room to get one over 9mm. I hope you realise that this stone is
over 3 TIMES as heavy as your 6mm stone!

To save you searching your trig books the formula for the crown
height is girdle width divided by 2 x cotangent crown angle
minus (crown triangle height before truncation x table percent) expressed as a percentage of the girdle width.
The pavilion height is girdle width divided by 2 x cotangent pavilion angle also expressed as a percentage of the girdle width.

As you can see from the above examples there is a very close
ratio, our popular example 2, a brilliant cut needs;
The Total depth to be close to 60 percent of the width
The pavilion depth to be close to 75 percent of the total depth.

The trap (emerald) cut needs;
The Total depth to be close to 75 percent of the width
The pavilion depth to be close to 80 percent of the total depth.