What a half square triangle is and why its cut size is odd
A half square triangle is a patchwork square made of two right triangles joined along their diagonal. It is the most-used unit in quilting: pinwheels, flying geese variants, sawtooth stars, broken dishes and hundreds of traditional blocks are built from nothing else. You make it by sewing two squares together along a diagonal line, then cutting them apart.
The awkward part is that you cannot cut a square the finished size plus two seam allowances, the way you would for a plain patch. The diagonal seam runs across the square at 45 degrees, and a seam allowance measured perpendicular to that diagonal costs more of the square's side length than a seam on a straight edge does — by a factor of √2, because that is the ratio between a square's diagonal and its side.
Work that through and the 2-at-a-time allowance is the seam allowance times (2 + √2): two straight-edge allowances plus one diagonal one. At the standard quarter-inch seam that is 0.854 inches, which quilters round up to the familiar seven eighths. The 7/8 rule is not arbitrary and it is not exact; it is 0.854 rounded to the nearest useful ruler mark, and the rounding is deliberately upward so the unit comes out a shade large rather than a shade small.
This is why the chart everyone memorises breaks the moment you sew at anything other than a quarter inch. Sew a scant quarter, sew a 3/8-inch seam because your machine has no quarter-inch foot, or work in metric with a 7.5 mm allowance, and the number changes. Deriving it from the seam allowance keeps it right.
Two, four and eight at a time
The three methods start from different squares and yield different numbers of units, and each has a distinct trade-off in accuracy.
Two at a time is the classic. Cut two squares the finished size plus 7/8 inch, draw a diagonal, sew a quarter inch either side of the line, cut on the line and press. Both units come out with the outer edges on the straight grain and only the diagonal on the bias, which makes them the most stable of the three. It also uses the least fabric per unit.
Eight at a time cuts one large square from each fabric, at twice the finished size plus 1¾ inches, marks both diagonals, sews on either side of both lines, then cuts the square in quarters and on both diagonals. Eight units fall out at once. It is dramatically faster for a quilt that needs dozens of matching units and it wastes no more fabric than the two-at-a-time route, because the arithmetic is exactly double in each direction. Its weakness is that a single inaccuracy in marking or sewing multiplies across all eight.
Four at a time stacks two squares right sides together, sews all four sides, then cuts both diagonals. It needs no marking at all, which is its appeal. But every unit comes out with two bias outer edges rather than none, so the units stretch as you handle them, and the starting square — the finished size times √2 plus 1½ inches — is bigger than the geometry strictly requires precisely so you have room to trim the distortion away.
Whichever you use, the unfinished size is the same: the finished size plus two seam allowances. A 3-inch finished HST is 3½ inches unfinished at a quarter-inch seam, and that is the number to trim to.
Worked example: 40 units finishing at 3 inches, eight at a time
Your pattern needs 40 half square triangles finishing at 3 inches, in two fabrics, sewn at a quarter-inch seam. You choose the eight-at-a-time method and are cutting from 42-inch fabric.
- Diagonal allowance. 0.25 × (2 + 1.41421) = 0.85355 in.
- Two-at-a-time square, for reference. 3 + 0.85355 = 3.854, rounded up to the next eighth = 3⅞ in.
- Eight-at-a-time square. (2 × 3) + (2 × 0.85355) = 7.707, rounded up = 7¾ in.
- Squares needed. Each pair of squares yields 8 units, so 40 ÷ 8 = 5 squares of each fabric, giving exactly 40 units.
- Trim size. 3 + (2 × 0.25) = 3½ in unfinished. Every unit should measure this before it goes into a block.
- Fabric. On 42-inch fabric with a half inch trimmed each side you have 41 usable inches, so 41 ÷ 7.75 = 5 squares across in a single row 7¾ inches deep. That is 7.75 ÷ 36 = 0.22 yd of each fabric — a quarter yard buys it.
- Quarter square triangle, if you needed one instead. 3 + (2 × 0.25) + (2 × 0.25 × 1.41421) = 4.207, rounded up = 4¼ in.
Compare that with the two-at-a-time route for the same 40 units: 20 squares of 3⅞ inches from each fabric, ten across the fabric in two rows, 7¾ inches of fabric depth. Identical fabric, one twentieth of the marking.
Accuracy: why the units come out small, and what to do about it
The single most common complaint about half square triangles is that they finish undersized. There are three causes and they compound. First, a seam allowance that is a hair over a quarter inch takes a hair off every unit; over the two seams in an HST and the four seams around a block, small errors accumulate fast. Second, pressing the diagonal seam to one side rolls a little fabric into the fold — the turn of cloth — which shortens the unit further. Third, the bias diagonal distorts under the iron if you push rather than press.
The professional answer is to cut oversize and trim. Add a quarter inch to the cut square for units up to about three inches and half an inch for larger ones, sew, press, then square each unit to its unfinished size with a ruler that has a 45-degree line you can lay along the seam. Every unit then measures exactly right and the diagonal runs exactly corner to corner, which is what makes points meet in the finished block.
The cost is real: the oversize allowance is fabric you cut off and throw away, and trimming forty units takes time. For a scrappy quilt where blocks meet no points, cut to exact size and accept the variation. For a sampler or anything with sharp points, oversize and trim. This calculator's yardage figure already includes whatever oversize you select, so you can see the price of the decision before you commit.
Starch is the other lever. A fabric pressed with spray starch before cutting behaves far more like paper and far less like cloth, and the bias edges stop stretching under the presser foot. It matters most on the four-at-a-time method, where two of every unit's outer edges are bias.
Half and quarter square triangle cut sizes at a quarter-inch seam
| Finished size | 2 at a time | 4 at a time | 8 at a time | Quarter square triangle | Unfinished size |
|---|---|---|---|---|---|
| 1 in | 1⅞ | 3 | 3¾ | 2¼ | 1½ |
| 1½ in | 2⅜ | 3⅝ | 4¾ | 2¾ | 2 |
| 2 in | 2⅞ | 4⅜ | 5¾ | 3¼ | 2½ |
| 2½ in | 3⅜ | 5⅛ | 6¾ | 3¾ | 3 |
| 3 in | 3⅞ | 5¾ | 7¾ | 4¼ | 3½ |
| 3½ in | 4⅜ | 6½ | 8¾ | 4¾ | 4 |
| 4 in | 4⅞ | 7¼ | 9¾ | 5¼ | 4½ |
| 4½ in | 5⅜ | 7⅞ | 10¾ | 5¾ | 5 |
| 5 in | 5⅞ | 8⅝ | 11¾ | 6¼ | 5½ |
| 6 in | 6⅞ | 10 | 13¾ | 7¼ | 6½ |
Every value is the formula on this page evaluated at a 0.25 in seam allowance. Add your oversize allowance on top if you intend to trim. Note that the 2-at-a-time column is always the finished size plus 7/8, and the 8-at-a-time column is always twice the finished size plus 1¾.
Pitfalls and what this calculator does not model
- A scant quarter inch is not a quarter inch. Many quilters deliberately sew a thread narrower to compensate for the turn of cloth. If you do, enter what you sew, not what your foot is called.
- The 7/8 rule is a rounding. The exact allowance at a quarter-inch seam is 0.854 in. Rounding up to 0.875 is what gives you the tiny surplus that trimming removes.
- Yield comes in fixed multiples. Four- and eight-at-a-time methods produce units in fours and eights, so an odd quantity always leaves a surplus. That is arithmetic, not waste you can plan away.
- The fabric figure assumes squares are cut in rows across the width. It ignores the offcut down the side of each row, which can often supply smaller units elsewhere in the quilt.
- Directional prints change everything. The four- and eight-at-a-time methods produce units in several orientations at once, so a one-way print will end up sideways in half of them.
- Bias edges stretch. The two-at-a-time method leaves all outer edges on the straight grain; the four-at-a-time method leaves two on the bias. Starch, and press rather than iron.
- Quarter square triangles are a different unit. They have four triangles, the outer edges are on the straight grain, and the cut allowance is 1¼ in at a quarter-inch seam rather than 7/8.
Where HSTs fit in the rest of the quilt's arithmetic
The cut size is the first number in a chain. Once you know the square you are cutting and how many you need, the amount of fabric to buy is a nesting problem, which the fabric yardage calculator handles directly — enter the cut square as both the piece width and length, with a zero seam allowance since the HST cut size already contains it.
The finished unit size then determines the block size, and the block size determines the quilt, which determines the backing and the binding. A quilt of 6-inch blocks each made of four 3-inch HSTs, set ten by twelve, finishes 60 by 72 inches before borders; from there the quilt backing yardage calculator gives the backing and the quilt binding calculator gives the strips. Working the chain forwards before you cut anything is how you avoid buying backing twice.
Quarter square triangles — sometimes called hourglass units — are the natural companion, and the calculator gives their cut size alongside. They are made by pairing two finished HSTs and cutting them again on the opposite diagonal, or by cutting a square in quarters diagonally. The allowance is larger, at the finished size plus 1¼ inches at a quarter-inch seam, because both diagonals now carry seams while the outer edges stay on the straight grain.
Finally, if a pattern gives you a cut size but not a finished size, you can run the formula backwards: subtract 7/8 from a two-at-a-time cut square and you have what it finishes at. That is often the quickest way to work out whether a vintage pattern was written for a quarter-inch seam or something else entirely.