Why the blank is longer than the inside circumference
Bend a flat strip into a circle and the metal does not simply curl up unchanged. The outer surface is pushed onto a larger radius, so it has to stretch; the inner surface is squeezed onto a smaller radius, so it has to compress. Somewhere between the two there is a layer that does neither. That layer is the neutral axis, and its length before bending is the same as its length after. For a strip of uniform rectangular section bent to a generous radius, the neutral axis sits at the mid-thickness.
That single fact is the whole calculation. The finished ring has an inside diameter D, so the neutral axis lies on a circle of diameter D + t, where t is the stock thickness. Its circumference is π(D + t), and that is the length you cut.
The size of the correction surprises people. On a size 7 band in 1.2 mm stock the inside circumference is 54.42 mm but the blank is 58.19 mm — 3.77 mm longer, which is π times the thickness. Since one US size is 0.8128 mm of diameter and therefore 2.55 mm of circumference, cutting to the inside circumference makes the ring roughly a size and a half small. In 2 mm stock the error is 6.28 mm, close to two and a half sizes.
The correction depends only on thickness, never on ring size. A 2 mm thick band always needs 6.28 mm more than its inside circumference, whether it is a size 4 or a size 14.
The formula, and where the US size numbers come from
You need two things: the inside diameter of the finished ring, and the neutral-axis correction.
The US ring size scale is linear in diameter. Size 0 is defined as 0.458 inches of inside diameter and each whole size adds 0.032 inches. Convert to millimetres and you get the working form used here: D = 11.6332 + 0.8128 n. So a size 6 lands on 16.510 mm, which is exactly the 0.650 in the sizing charts print, and a size 7 lands on 17.3228 mm. Half sizes add 0.4064 mm; quarter sizes add 0.2032 mm.
The European and Japanese systems work differently. ISO 8653 designates a ring size by its inside circumference in whole millimetres, so an ISO size 54 measures 54 mm around the inside and its diameter is 54 ÷ π = 17.189 mm. That is why an ISO 54 and a US 7 are close but not identical — they are two independent scales that happen to overlap. Use the third input mode if your customer gives you an ISO or European number.
Once you have D, the blank length is π(D + t) plus whatever allowance your joint needs. A butt joint that you saw square and close tight needs nothing. If you cut both ends with a piercing saw from a longer strip and want the kerf to fall in the waste, add nothing to this length either — just saw on the waste side of your scribe line. Add an allowance only when you deliberately want extra metal to file back, or when you are leaving a visible solder gap to fill.
The weight comes from the same neutral length. The volume of a band of rectangular section is exactly π(D + t) · t · w — this is not an approximation, because the annulus area π(Ro2 − Ri2) factors precisely into π(D + t)t. Multiply by density and you have grams. If you want that figure for a piece that is not a plain band, work it out with the jewelry metal weight calculator instead.
Worked example: a size 7 sterling band, 1.2 mm thick and 4 mm wide
A customer wants a plain sterling band, US size 7, in 1.2 mm sheet, 4 mm wide. You are buying sheet at $1.20 a gram.
- Inside diameter. D = 11.6332 + 0.8128 × 7 = 11.6332 + 5.6896 = 17.3228 mm.
- Inside circumference. π × 17.3228 = 54.421 mm. This is what the finger feels, and it is not your cut length.
- Neutral-axis diameter. 17.3228 + 1.2 = 18.5228 mm.
- Blank length. π × 18.5228 = 58.191 mm. Cut the strip to 58.19 mm — call it 58.2 mm on a rule, or scribe at 2.291 in.
- Outside diameter. 17.3228 + 2 × 1.2 = 19.7228 mm. Worth checking against a bezel or a setting if the ring has to sit inside something.
- Volume. 58.191 × 1.2 × 4 = 279.32 mm³, or 0.27932 cm³.
- Weight. 0.27932 × 10.36 g/cm³ = 2.894 g.
- Metal cost. 2.894 × $1.20 = $3.47.
Now check the penalty for getting it wrong. Cut at 54.42 mm instead and the finished inside diameter would be 54.42 ÷ π − 1.2 = 16.12 mm, which is US size 5.52 — nearly a size and a half small, and a ring you cannot rescue without cutting it open.
US ring sizes, diameters and blank lengths
| US size | Inside dia (mm) | Inside dia (in) | Inside circ (mm) | Blank at 1.0 mm | Blank at 1.5 mm | Blank at 2.0 mm |
|---|---|---|---|---|---|---|
| 4 | 14.884 | 0.586 | 46.760 | 49.902 | 51.472 | 53.043 |
| 5 | 15.697 | 0.618 | 49.314 | 52.456 | 54.026 | 55.597 |
| 6 | 16.510 | 0.650 | 51.869 | 55.011 | 56.581 | 58.152 |
| 7 | 17.323 | 0.682 | 54.421 | 57.563 | 59.133 | 60.704 |
| 8 | 18.136 | 0.714 | 56.975 | 60.117 | 61.687 | 63.258 |
| 9 | 18.948 | 0.746 | 59.528 | 62.670 | 64.240 | 65.811 |
| 10 | 19.761 | 0.778 | 62.082 | 65.224 | 66.794 | 68.365 |
| 11 | 20.574 | 0.810 | 64.635 | 67.777 | 69.347 | 70.918 |
| 12 | 21.387 | 0.842 | 67.189 | 70.331 | 71.901 | 73.472 |
| 13 | 22.200 | 0.874 | 69.742 | 72.884 | 74.454 | 76.025 |
Every step of one whole size adds 0.8128 mm of diameter and 2.554 mm of blank length, at any thickness. Every 0.5 mm of extra thickness adds 1.571 mm of blank length, at any size.
Reading the result at the bench
Measure the blank on the strip, not on the mandrel. A steel rule with a knife-edge and a scribe gets you inside 0.2 mm, which is about a twelfth of a ring size — well inside what you can correct with a few strokes of a file. Digital calipers used as a depth gauge against a stop are better still.
Expect the first ring you close to come out slightly large rather than slightly small. Forming, planishing and truing on a mandrel all thin the metal a little, and thinner metal at the same length means a bigger inside diameter. If your shop consistently runs a quarter size big, subtract 0.64 mm from the blank as a standing correction and stop chasing it ring by ring.
The thickness you enter must be the thickness at the moment of bending, not the thickness of the sheet you bought. If you roll 1.5 mm sheet down to 1.2 mm to get the band you want, use 1.2. If you plan to sand 0.1 mm off the outside after soldering, use the pre-sanding thickness for the blank and accept that the finished ring will read about a tenth of a size large.
Very thick stock breaks the assumption. Once t exceeds roughly a quarter of the inside diameter — a 4.5 mm wall on a size 7, say — the neutral axis shifts inward toward the compression side, and the true blank is a little shorter than π(D + t). The calculator warns you when you cross that ratio. For heavy work, form a test ring in copper first.
Mistakes that cost a ring
- Cutting to the inside circumference. The single most common error, and it produces a ring roughly one and a half sizes small in 1.2 mm stock. The neutral axis, not the inside surface, is what keeps its length.
- Using the outside circumference instead. Overcorrecting by π·t in the other direction makes the ring the same amount too big. The correction is π·t, not 2π·t.
- Forgetting that the sheet has been rolled. Stock nominally sold as 1.2 mm can arrive at 1.15 mm, and rolling it yourself changes it further. Measure it with calipers before you calculate.
- Mixing scales. A customer who says "size 54" is quoting inside circumference under ISO 8653, not a US number. Feeding 54 into the US size field asks for a ring 55 mm in diameter.
- Adding a kerf allowance that is already accounted for. If you saw on the waste side of a scribed line, the kerf never comes out of your blank. Adding an allowance as well makes the ring large.
- Applying this to a tapered or comfort-fit band. A comfort-fit band has a domed inside surface, so its effective inside diameter is larger than a flat band of the same nominal size. Cut a comfort-fit blank about a quarter size shorter and confirm on a mandrel.
Sizing sets and mandrels do not always agree
Cheap sizing rings and cheap mandrels are frequently out by a quarter size or more, and they are often out by different amounts. Before you trust any calculated blank, check your own mandrel against a known-good sizing ring, or against the diameters in the table above with calipers. Every bench that produces consistent sizes has done this once and written the offset on the wall.
Key terms
- Neutral axis
- The layer within a bent section whose length is unchanged by the bend. For a strip of uniform rectangular section bent to a radius several times its thickness, it lies at the mid-thickness.
- Blank
- The flat strip or straight length of wire cut to size before forming. Its length is what this calculator gives you.
- Comfort fit
- A band whose inside surface is domed rather than flat, so only a narrow strip of metal touches the finger. It feels larger than a flat band of the same nominal size.
- ISO 8653
- The international standard that designates a ring size by its inside circumference in millimetres. A size 54 has a 54 mm inside circumference.
- Kerf
- The width of material a saw blade removes. A 2/0 jeweller's blade cuts a kerf of roughly 0.3 mm.
Where this sits among your other bench calculations
Blank length is the first of three numbers a ring job needs. The second is metal weight, which drives your material cost and, for gold, most of the price you quote — the metal weight calculator handles shapes this one does not, and the karat alloy calculator tells you what a given karat actually contains before you buy casting grain. The third is scrap recovery, since the offcut from every blank is worth real money; the scrap gold value calculator converts a jar of filings and sprue into a number.
The same neutral-axis idea shows up wherever you bend flat stock. Sheet metal workers call the correction the bend allowance and compute it with a K-factor — the fraction of the thickness at which the neutral axis sits — which is 0.5 for the generous radii used in ring making but drops toward 0.33 for tight bends in hard sheet. If you bend bezel wire, box corners or a bracelet blank around a former, the same formula applies with the former's diameter in place of D. Leatherworkers face the identical geometry when they wrap a strap around a buckle bar, which is one reason the leather hide yield calculator and this page get used by the same people.
What no formula covers is springback. Metal bent cold wants to open again, so the ring you form on a mandrel will always be a little larger than the mandrel until you close and solder it. That is a property of the alloy and its temper, not of the arithmetic, and the only cure is to anneal properly and to close the joint tight before soldering.