One printed diameter, read three ways — the pair's width, the ratio it adds to the outline, and the share of the jaw line it covers — beside the jaw-width ladder, from 0.66 to 1.14, that gives every row a millimetre width. Print it or keep it on your phone.
Your cheekbone width is the denominator of every ratio on this sheet; one ornament's printed diameter is w, and the pair's width, 2w, is what the two tables in sections 3 and 4 divide.
Two numbers are needed and both exist. The first is the jaw ratio — the run from one jaw corner to the other, divided by the widest horizontal measure across the cheekbones. The second is cheekbone width in millimetres, without which a ratio buys nothing. The published adult band runs 128.2 to 148.8 mm, and 136 mm is the working line used on two of this site's eyewear pages; the ladder in the next section prints all three.
A pair worn level with the jaw line adds twice its own diameter to that line and nothing to the cheekbone line above it, so the outline's ratio rises by the pair's width divided by your cheekbone width. Four millimetres of diameter at 136 mm is 8 ÷ 136 = 0.059; forty millimetres is 80 ÷ 136 = 0.588.
| Jaw ratio | at 128.2 mm | at 136 mm | at 136 mm, in inches | at 148.8 mm |
|---|---|---|---|---|
| 0.66 | 84.6 | 89.8 | 3.53 | 98.2 |
| 0.72 | 92.3 | 97.9 | 3.86 | 107.1 |
| 0.78 | 100.0 | 106.1 | 4.18 | 116.1 |
| 0.84 | 107.7 | 114.2 | 4.50 | 125.0 |
| 0.90 | 115.4 | 122.4 | 4.82 | 133.9 |
| 0.96 | 123.1 | 130.6 | 5.14 | 142.8 |
| 1.02 | 130.8 | 138.7 | 5.46 | 151.8 |
| 1.08 | 138.5 | 146.9 | 5.78 | 160.7 |
| 1.14 | 146.1 | 155.0 | 6.10 | 169.6 |
Every cell is one multiplication — ratio × cheekbone width — exact and rounded to one decimal, so any cell can be re-derived on the spot. The inch column takes the product before rounding and divides by 25.4: 0.66 × 136 = 89.76, and 89.76 ÷ 25.4 = 3.5339, so 3.53. The published band for this ratio on a soft tape is 0.72 to 1.08, with a published deviation of 0.12; the ladder runs 0.06 — half a deviation — past each published end, and that overhang is this site's own choice. Two rows sit exactly on the published ends, which is why the step is 0.06 and not 0.05. The top row at the wide column is a combination rather than a face: it implies a jaw of 169.6 mm. Read one row at your own width.
| Printed diameter | The pair, in mm | Ratio added at 128.2 mm | at 136 mm | at 148.8 mm | Deviations at 136 mm |
|---|---|---|---|---|---|
| 4 mm / 0.16 in | 8.0 | 0.062 | 0.059 | 0.054 | 0.49 |
| 6 mm / 0.24 in | 12.0 | 0.094 | 0.088 | 0.081 | 0.74 |
| 8 mm / 0.31 in | 16.0 | 0.125 | 0.118 | 0.108 | 0.98 |
| 10 mm / 0.39 in | 20.0 | 0.156 | 0.147 | 0.134 | 1.23 |
| 12 mm / 0.47 in | 24.0 | 0.187 | 0.176 | 0.161 | 1.47 |
| 14 mm / 0.55 in | 28.0 | 0.218 | 0.206 | 0.188 | 1.72 |
| 16 mm / 0.63 in | 32.0 | 0.250 | 0.235 | 0.215 | 1.96 |
| 20 mm / 0.79 in | 40.0 | 0.312 | 0.294 | 0.269 | 2.45 |
| 25 mm / 0.98 in | 50.0 | 0.390 | 0.368 | 0.336 | 3.06 |
| 30 mm / 1.18 in | 60.0 | 0.468 | 0.441 | 0.403 | 3.68 |
| 40 mm / 1.57 in | 80.0 | 0.624 | 0.588 | 0.538 | 4.90 |
| 50 mm / 1.97 in | 100.0 | 0.780 | 0.735 | 0.672 | 6.13 |
Cells are the pair — twice the printed diameter — divided by the cheekbone width, exact and then rounded to three decimals; the last column is the same quotient before rounding, at 136 mm, divided by the published deviation of the ratio, 0.12. That count is scale-invariant: the same quotient taken on the detector's own scale, 0.026 for the same quantity, gives the identical figure. The 8 mm row is 0.98 deviations, and one whole deviation at the working width is a diameter of 8.16 mm (0.32 in) — 7.69 mm at the narrow end of the published cheekbone band and 8.93 mm at the wide end. A pair of 40 mm hoops is 4.90 deviations, which is 1.63 times the full distance between the narrowest and widest published outline centres — 0.36, three deviations, or 24.5 mm of diameter. Every row is the full value, which assumes the ornament's widest level sits on the jaw line; what happens when it does not is in section 5.
| Printed diameter | One against a 0.72 jaw | against a 0.90 jaw | against a 1.08 jaw | The pair at 0.90 | The pair at 1.08 |
|---|---|---|---|---|---|
| 4 mm / 0.16 in | 4.1% | 3.3% | 2.7% | 6.5% | 5.4% |
| 6 mm / 0.24 in | 6.1% | 4.9% | 4.1% | 9.8% | 8.2% |
| 8 mm / 0.31 in | 8.2% | 6.5% | 5.4% | 13.1% | 10.9% |
| 10 mm / 0.39 in | 10.2% | 8.2% | 6.8% | 16.3% | 13.6% |
| 12 mm / 0.47 in | 12.3% | 9.8% | 8.2% | 19.6% | 16.3% |
| 14 mm / 0.55 in | 14.3% | 11.4% | 9.5% | 22.9% | 19.1% |
| 16 mm / 0.63 in | 16.3% | 13.1% | 10.9% | 26.1% | 21.8% |
| 20 mm / 0.79 in | 20.4% | 16.3% | 13.6% | 32.7% | 27.2% |
| 25 mm / 0.98 in | 25.5% | 20.4% | 17.0% | 40.8% | 34.0% |
| 30 mm / 1.18 in | 30.6% | 24.5% | 20.4% | 49.0% | 40.8% |
| 40 mm / 1.57 in | 40.8% | 32.7% | 27.2% | 65.4% | 54.5% |
| 50 mm / 1.97 in | 51.1% | 40.8% | 34.0% | 81.7% | 68.1% |
Jaws are 0.72 × 136 = 97.9 mm, 0.90 × 136 = 122.4 mm and 1.08 × 136 = 146.9 mm — the published lower end, the middle row and the published upper end of the ladder in section 2 — and each cell is the diameter divided by the jaw, or twice the diameter for the two pair columns, exact and then rounded to one decimal. Against the trade's own hoop ladder: a 12 to 14 mm hoop is 9.8 to 11.4 per cent of the middle row, and its pair covers a fifth to nearly a quarter of that line; a 30 to 40 mm hoop is 24.5 to 32.7 per cent and its pair 49.0 to 65.4 per cent; a pair of 50 mm hoops spans 81.7 per cent — within nineteen points of covering the line corner to corner. Going the other way, a pair exactly equal to the middle-row line needs a diameter of 61.2 mm (2.41 in). Against a cheekbone width other than 136 mm every cell scales by 136 ÷ 128.2 = 1.061 or by 136 ÷ 148.8 = 0.914.
The ratio addition assumes the widest level of the ornament sits on the line between your jaw corners. The offsets are stated at 0, 5 and 10 millimetres rather than folded into one tidy figure, because no position in the comparison measures the level at which an ornament is widest. Treated as a circle of the printed diameter, a 20 mm hoop offset by 5 mm contributes 34.6 mm rather than 40.0 — 2.12 deviations instead of 2.45 — and the same hoop offset by its own radius contributes nothing at that level.
A stud whose widest level sits at the ear rather than the jaw is off that line by more than its own radius for most diameters sold, so at jaw level it adds nothing. What survives is the share table: a 4 mm stud is 3.3 per cent of a middle-row line — a comparison, not an addition.
Read the same rule from the other line and the sign turns over: an ornament whose widest level sits at the cheekbones is added to the denominator, and there a 12 mm pair moves the highest centre that rests on photographs — 0.9508 — down to 0.8082, which is 1.19 deviations in the direction of narrower. Both of those numbers describe an outline, not a face.
The deviations are estimated quantities. They were pooled from the four groups in the calibration set with five or more photographs each — 18, 12, 10 and 6, that is 46 faces and 42 degrees of freedom — and the published deviation of the ratio is 0.12 on a tape. Two of the seven published centres are shared and three rest on one, three and zero photographs respectively and are ordered along the literature rather than measured: 0.72, 0.72, 0.80, 0.85, 0.92, 0.95, 1.08. That is why these tables spend the deviations on a diameter and not on a diagnosis.
Nothing here answers whether a face looks smaller beside a large hoop. That question is not arithmetic, and the direction of the quantity these tables do carry is the opposite one: the added width goes into the outline rather than out of it.
The cheekbone band is a band: 128.2 to 148.8 mm is one deviation either side of the measured means this site publishes, and 136 mm is a working line rather than a person. If your own width is outside the band the tables still work, because every cell is one multiplication or one division away from the row it prints.