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Approximating the Length of Oval Bezels

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By Gerald A. Livings (2005)


P   ยป  2a [ 2 + (pi - 2) (b/a) 1.456 ]
where A = length and B= width


WARNING!

I am publishing this but it does not work quite yet. The numbers are not quite correct. I will have it working soon!


Why I did this...

I have found over the course of many years in the jewelry trade that jobs requiring oval bezels are consistently under-priced by sales floor associates. What normally happens is the sales floor associates take in the repair or custom jobs as an estimate so the jeweler can tell them the cost. Jobs tend to take longer to complete while waiting for contacting the client and approvals on the estimate amounts. Many times there would be errors in communication between everybody involved that would be less then pleasant.

The method of making bezels that most jewelers are taught is to wrap bezel wire around the stone and then cut, file and solder the bezel. To find the correct price for the bezel I always wrapped the stone first with a small strip of masking tape, sliced it with a razor, and then measured the length of the tape to find the length of the bezel wire I needed. This is what I used to determine cost.

I thought there must be a better way for sales floor associates and jewelers to reduce the number of estimates, get the jobs back to the client quicker, reduce errors in communication and reduce the amount of time and paperwork involved.

I started to research ovals in books and on the internet. Many jewelry-related websites, books and reference materials have charts and formulas for finding the length of oval bezels so by comparing them I thought I would find the one that worked for a wide range of sized.

First I combined all of the charts I could find to show the numbers side by side for the most common oval cabochons

Second, I put all of the formulas I could find on the chart to see the answer for the most common oval cabochons.

Third, I took cabochons of different sizes and wrapped different gauges of bezel wire around them and put this on the chart after cutting the bezel wire only. (After cutting and before filing and squaring off the ends, I found the bezel length for a particular cabochon size remained consistent despite the gauge of the bezel.)

When done with all of this I found that most charts only covered a small number of bezel sizes and were inaccurate at best, and most formulas were very simple and only accurate for bezels of a particular length-to-width ratio.

I did some research on the internet to find a formula that would work with a wide range of sizes and length-to-width ratios. This quickly led me to websites that discussed the orbit of comets. I found a formula that not only worked for ovals of millions of miles but also appeared to work for the fractions of millimeters that I work with daily.

I have this chart hanging near my bench and it has helped me reduce waste and work faster.

Table 1
 Stone Length (MM)  Stone Width (MM)  Stone Circumference (MM)  Bezel Length (MM)      IN2       Actual IN2 Charge 
5 3 13.21160.0177 .25 IN2
6 4 16.24190.0283 .25 IN2
7 5 19.32220.0412 .25 IN2
8 6 22.41250.0565 .25 IN2
9 7 25.52280.0742 .25 IN2
10 8 28.64310.0942 .25 IN2
12 10 34.89370.1414 .25 IN2
13 10 36.73390.1532 .25 IN2
14 10 38.63410.1649 .25 IN2
14.5 5 36.90390.0854 .25 IN2
15 11 41.73440.1944 .25 IN2
16 12 44.83470.2262 .25 IN2
18 13 49.84520.2757 .50 IN2
20 7 50.85530.1649 .50 IN2
20 15 56.03590.3534 .50 IN2
22 8 55.84580.2073 .50 IN2
25 18 69.15720.5301 .75 IN2
30 22 83.45860.7775 1.0 IN2
38 30 108.321111.3430 1.5 IN2
40 30 112.061151.4137 1.75 IN2

I have also made a simple chart that may be used in front of the client that has only 3 columns for pricing.

Table 2
  Stone Size (MM)  Bezel Length (MM)  IN2 Charge  
5 X 3 16 .25 IN2
6 X 419 .25 IN2
7 X 522 .25 IN2
8 X 625 .25 IN2
9 X 728 .25 IN2
10 X 831 .25 IN2
12 X 1037 .25 IN2
13 X 1039 .25 IN2
14 X 1041 .25 IN2
14.5 X 539 .25 IN2
15 X 1144 .25 IN2
16 X 1247 .25 IN2
18 X 1352 .50 IN2
20 X 753 .50 IN2
20 X 1559 .50 IN2
22 X 858 .50 IN2
25 X 1872 .75 IN2
30 X 2286 1.0 IN2
38 X 30111 1.5 IN2
40 X 30115 1.75 IN2

Notes:

More about the equation

P ≈ 2a[2+(pi-2)(b∕a)1.456] where A = length and B= width

I found this equation at www.numericana.com. It was on a web page that described the different formulas for aproximating the "Perimeter of an Ellipse", (http://www.numericana.com/answer/ellipse.htm#rivera). David F. Rivera had submitted this approximation to the perimeter of an ellipse to the errata of the 30th edition of the Standard Mathematical Tables and Formulae (CRC Press, http://www.mathtable.com/errata/smtf30_errata_p2/index.html). This formula then made its way to the numericana.com website.