Diamond Math for the Trig Challenged

Ever wonder how to find the important pavilion and crown main angles from measurements of pavilion depth or table size and crown height? Here’s how to get those angles with the help of a couple of buttons on a $16 scientific calculator or the free Windows scientific calculator accessory on your computer.

You can use the following steps to check the consistency of angle and percentage results from devices like the Sarin, OGI, or your GIA proportion scope. (Note: You may be surprised at the results!) As always, the accuracy of the calculations depends upon the accuracy of the input measurements. This math assumes a reasonably symmetric diamond with reasonably centered table and culet.

First, find the tan and the cos buttons on your scientific calculator. On the calculator shown below, push the shift button and then the tan or cos to obtain the inverse tangent and cosign (tan^-1 and cos^-1 ). If you’re using the scientific calculator on a Windows-based computer—found under the “Start/Programs/Accessories/Calculator/View/Scientific” menu—check the “Inv” box before using the tan or cos buttons to obtain the tan^-1 and cos^-1 .

Pavilion angle. After measuring or obtaining the pavilion depth percentage and the culet percentage:

Double the pavilion depth percentage.
Subtract the culet percentage from 100.
Divide the result from step 1 by the result from step 2 and push the tan button to obtain the pavilion angle.

Example: What exact pavilion angle corresponds to the ideal depth percentage of 43.1% when there is no culet?

43.1 times 2 equals 86.2.
100 minus 0 equals 100.
86.2 divided by 100 equals .862 and tan^-1 (.862) equals 40.761 degrees.

So the exact pavilion depth percentage corresponding to Tolkowsky’s 40.75 is actually 43.083%, not the rounded off 43.1%. Close enough.

Crown angle. After measuring or obtaining the table percentage and the crown height percentage:

Subtract the table percentage from 100 and divide the result by 2.
Divide the crown height percentage by the result from step 1.
Push the tan button to obtain the crown angle.

Crown angle. If you can measure the table percentage but not the crown height percentage, the crown angle can be obtained by measuring the length of the kite-shaped bezel facet. Divide its length by the diamond diameter times 100 to get the bezel length percentage, then:

Subtract the table percentage from 100 and divide the result by 2.
Divide the result from step 1 by the bezel length percentage.
Push the cos button to obtain the crown angle.

Crown height. If you have measured the table percentage and the bezel percentage, the crown height percentage can be obtained using the following formula:

Subtract the table percentage from 100 and divide the result by 2.
Square the bezel percentage and subtract the square of the result from step 1.
Take the square root of the result from step 2 to obtain the crown height percentage.

Michael Cowing, MS, FGA, is a gem expert and appraiser in Crownsville, Md. For more information on diamond cut, visit Cowing’s Web site at www.acagemlab.com.