Half-Square Triangles & Flying Geese
The ⅞-inch rule is 0.853553 rounded up. That slack is for trimming — skip the trim and 47 units gain a full inch.
Where the ⅞ inch actually comes from
"Cut your squares seven-eighths of an inch bigger than the finished size" is repeated everywhere and derived nowhere. It is one piece of geometry plus a deliberate fudge, and the fudge is the interesting half.
A right isosceles triangle with legs L has a hypotenuse of L√2, and moving that hypotenuse inward by a perpendicular distance t shortens each leg by exactly t√2. So a quarter-inch seam taken on the diagonal costs 0.25 × √2 = 0.353553″ of leg, and the two ordinary seam allowances on the outer edges cost 0.5″. Add them:
| Two outer seam allowances | 0.500000″ |
| The diagonal seam, t√2 | 0.353553″ |
| Exact allowance | 0.853553″ |
| The published rule | 0.875″ (⅞) |
| Rounded up by | 0.021447″, about one forty-seventh of an inch |
The rule rounds up on purpose. Every unit comes out a shade large so it can be trimmed square, which is why "cut generously and trim" is standard advice rather than an admission of sloppiness. The rule is working as designed.
But it only works if you actually trim
The slack is per unit, so across a row it adds up in a straight line. This is the arithmetic behind blocks that will not fit their neighbours — and it is not your tension, your machine, or your quarter-inch foot.
| Untrimmed units in a row | Row comes out over by |
|---|---|
| 4 | 0.086″ |
| 8 | 0.172″ |
| 12 | 0.257″ |
| 16 | 0.343″ |
| 24 | 0.515″ |
| 36 | 0.772″ |
| 47 | 1.008″ — a full inch |
A twelve-unit row — an ordinary sampler block — is already a quarter of an inch out, which is more than enough to fight with the block beside it. Forty-seven units and you have gained a full inch of quilt from nothing but rounding.
Flying geese are the same argument, run twice
The no-waste method crosses the large square with two diagonal seams rather than one, so the exact allowance is 0.5 + 2 × 0.353553 = 1.207107″ against a published rule of 1.25″. Double the diagonals, double the slack — 0.042893″ this time, for exactly the same reason and to exactly the same end.
The four small squares are half-square triangles in disguise, so they use the plain ⅞ rule. And a goose always finishes twice as wide as it is tall — that ratio is fixed by the construction rather than being a style choice, which is why this tool asks only for the width.
Cut sizes at a glance
| Finished HST | Cut squares | Unfinished | Exact would be |
|---|---|---|---|
| 2" | 2 7/8" | 2 1/2" | 2.854″ |
| 2 1/2" | 3 3/8" | 3" | 3.354″ |
| 3" | 3 7/8" | 3 1/2" | 3.854″ |
| 3 1/2" | 4 3/8" | 4" | 4.354″ |
| 4" | 4 7/8" | 4 1/2" | 4.854″ |
| 4 1/2" | 5 3/8" | 5" | 5.354″ |
| 5" | 5 7/8" | 5 1/2" | 5.854″ |
| 6" | 6 7/8" | 6 1/2" | 6.854″ |
The exact column is there to be looked at, not cut to — you cannot mark 0.854″ on a ruler. That is precisely why the rule exists in eighths.
Notes
- All of this assumes a scant quarter-inch seam. The whole derivation is built on ¼″, so if your seam runs wide the units come out small and no cutting rule will rescue them. Sew two strips together and measure before cutting a whole quilt's worth.
- Trim to the unfinished size, not the finished one. The unfinished size includes the seam allowances that will disappear when the unit is joined.
- Press before trimming. An unpressed unit measures smaller than it is and you will trim away real fabric chasing a number.
- The bias edge is the fragile one. Every one of these units has a raw bias edge somewhere, and bias stretches. Handle gently, and starch if you find units going out of square between the iron and the ruler.
How to use
- Pick half-square triangles or flying geese and enter the finished size.
- Cut the squares at the size shown and sew a scant quarter-inch seam.
- Trim each unit to the unfinished size — the rule leaves slack for it.
- Check the drift figure to see what skipping the trim would cost.
Frequently asked questions
What size squares do I cut for a half-square triangle?
The finished size plus ⅞ inch, for the two-at-a-time method. A 3-inch finished HST needs 3⅞-inch squares, which give two units that trim to 3½ inches unfinished.
Where does the ⅞ inch come from?
Two ordinary quarter-inch seam allowances on the outer edges (½ inch) plus the diagonal seam. A right isosceles triangle loses exactly t×√2 off each leg when its hypotenuse moves in by t, so a quarter-inch diagonal seam costs 0.353553 inches. The exact total is 0.853553 — and ⅞ is that rounded up.
Is the ⅞ rule exact?
No, and it is not meant to be. It rounds up by 0.021447 inches so every unit comes out slightly large and can be trimmed square. That is why "cut generously and trim" is the standard advice — it is the rule working as designed rather than an admission of imprecision.
Why do my HST blocks come out too big?
Almost certainly because you are not trimming. The rule builds in about a fiftieth of an inch per unit, which sounds like nothing but adds up in a straight line: twelve units in a row are already a quarter-inch over, and forty-seven units gain a full inch. It is not your tension or your machine.
What size squares for flying geese?
By the no-waste method, one large square at the finished width plus 1¼ inches and four small squares at the finished height plus ⅞ inch. That makes four geese. The large square gets crossed by two diagonal seams rather than one, which is exactly why its allowance is bigger.
Why is the flying-geese allowance 1¼ instead of ⅞?
Because two diagonal seams cross the large square, so you pay the 0.353553 diagonal cost twice. The exact figure is 1.207107 and the published rule rounds it up to 1.25 — double the slack of the HST rule, for the same reason and to the same end.
Can flying geese be any proportion?
Not by this method. A goose finishes twice as wide as it is tall, fixed by the construction rather than chosen, which is why the tool asks only for the width. Other proportions need a different technique.
Does my seam allowance matter that much?
It is the foundation of the whole calculation. Every figure here assumes a scant quarter inch, so if your seam runs wide the units come out small and no cutting rule will save them. Sew two strips together and measure the result before cutting a quilt's worth of fabric.
Should I trim before or after pressing?
After. An unpressed unit measures smaller than it really is, so trimming first means cutting away real fabric to chase a number. Press, then trim to the unfinished size, and handle the bias edge gently while you do it.
Does this send anything anywhere?
No. Everything is calculated in your browser and nothing is stored or uploaded.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.