## Ellipse Perimeter Calculations Tool

This tool does the calculations from above, but with more terms for the Series.

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## Approximating the Length of Oval Bezels

By Gerald A. Livings (2005)

P » 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 gauge 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.

So back to do some research on the internet again. What I needed to find was a formula that would work with a wide range of sizes and length-to-width ratios. And as you can imagine, this was a total failure. There is nothing like this at all. By expanding my search to ANYTHING that might be oval shaped, I quickly found links to websites that discussed the orbit of comets. With a bit of work I found a formula that not only worked for ovals of trillions of miles but also appeared to work for the fractions of millimeters that I work with daily.

I originally did this in excel as a fourmula and it seemed to work fine. But when I converted the math to this form, it gives me an error of slightly more than double. I have to start from scratch and figure out if the error was mine, being dyslexic, It could be. Or if the error is due to my being unfamiliar with javascript. So use this and let me know if it seems correct.

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

Stone Length (MM) | Stone Width (MM) | Stone Circumference (MM) | Bezel Length (MM) | IN2 | Actual IN2 Charge |
---|---|---|---|---|---|

5 | 3 | 13.21 | 16 | 0.0177 | .25 IN2 |

6 | 4 | 16.24 | 19 | 0.0283 | .25 IN2 |

7 | 5 | 19.32 | 22 | 0.0412 | .25 IN2 |

8 | 6 | 22.41 | 25 | 0.0565 | .25 IN2 |

9 | 7 | 25.52 | 28 | 0.0742 | .25 IN2 |

10 | 8 | 28.64 | 31 | 0.0942 | .25 IN2 |

12 | 10 | 34.89 | 37 | 0.1414 | .25 IN2 |

13 | 10 | 36.73 | 39 | 0.1532 | .25 IN2 |

14 | 10 | 38.63 | 41 | 0.1649 | .25 IN2 |

14.5 | 5 | 36.90 | 39 | 0.0854 | .25 IN2 |

15 | 11 | 41.73 | 44 | 0.1944 | .25 IN2 |

16 | 12 | 44.83 | 47 | 0.2262 | .25 IN2 |

18 | 13 | 49.84 | 52 | 0.2757 | .50 IN2 |

20 | 7 | 50.85 | 53 | 0.1649 | .50 IN2 |

20 | 15 | 56.03 | 59 | 0.3534 | .50 IN2 |

22 | 8 | 55.84 | 58 | 0.2073 | .50 IN2 |

25 | 18 | 69.15 | 72 | 0.5301 | .75 IN2 |

30 | 22 | 83.45 | 86 | 0.7775 | 1.0 IN2 |

38 | 30 | 108.32 | 111 | 1.3430 | 1.5 IN2 |

40 | 30 | 112.06 | 115 | 1.4137 | 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.

Stone Size (MM) | Bezel Length (MM) | IN^{2} Charge |
---|---|---|

5 X 3 | 16 | .25 IN^{2} |

6 X 4 | 19 | .25 IN^{2} |

7 X 5 | 22 | .25 IN^{2} |

8 X 6 | 25 | .25 IN^{2} |

9 X 7 | 28 | .25 IN^{2} |

10 X 8 | 31 | .25 IN^{2} |

12 X 10 | 37 | .25 IN^{2} |

13 X 10 | 39 | .25 IN^{2} |

14 X 10 | 41 | .25 IN^{2} |

14.5 X 5 | 39 | .25 IN^{2} |

15 X 11 | 44 | .25 IN^{2} |

16 X 12 | 47 | .25 IN^{2} |

18 X 13 | 52 | .50 IN^{2} |

20 X 7 | 53 | .50 IN^{2} |

20 X 15 | 59 | .50 IN^{2} |

22 X 8 | 58 | .50 IN^{2} |

25 X 18 | 72 | .75 IN^{2} |

30 X 22 | 86 | 1.0 IN^{2} |

38 X 30 | 111 | 1.5 IN^{2} |

40 X 30 | 115 | 1.75 IN^{2} |

### Notes:

- On average, my filing tends to remove up to 2MM of material from the ends of the bezel before soldering. This may be different for each jeweler but this is fairly accurate for most of the bench jewelers in the trade that I know. The bezel length column in the is rounded up to the nearest whole number and 1MM is added for this reason.
- This is for ovals with uniform curves; bezels on stones that are not uniform ovals will need to be estimated with the old masking tape and measure method.
- A simple bezel needs only the charge for the appropriate bezel length.
- Pre-made Step bezel needs only the charge for the appropriate bezel length.
- When making your own step bezel then charge for the appropriate bezel length twice.
- Closed back bezels generally require backing material as well as the charge for the appropriate bezel length. So the last column shows what to also charge for the backing.
- If the stone you are measuring is not listed below then use the closest size with measurements above the stone you have. Also let your client know that this is an estimate and may change when the jeweler figures the correct length.

## More about the equation

P ≈ 2*a*[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.