Origami Square from Rectangle Calculator
Calculate the largest square from a rectangular sheet, plus how many same-size squares one sheet yields.
Turn printer paper into origami without waste.
Origami paper is sold square. Your printer paper is not. That is the daily problem for anyone who folds. A4, US Letter, legal, and every other office format is a rectangle, and almost every model assumes a square. Convert one to the other without wasting material and a 500-sheet ream becomes anywhere from 500 to 10,000 origami starts, depending on how small you are willing to go.
The basic geometry.
The largest square cut from a rectangle has side length equal to the rectangle’s shorter side:
Square side = min(length, width)
That’s it for one square. The interesting math is when you optimize for multiple squares from one sheet.
Single-square yield from common paper:
| Paper format | Dimensions | Square side | Waste strip |
|---|---|---|---|
| A4 | 210 × 297 mm | 210 mm | 210 × 87 mm |
| A3 | 297 × 420 mm | 297 mm | 297 × 123 mm |
| A5 | 148 × 210 mm | 148 mm | 148 × 62 mm |
| US Letter | 216 × 279 mm | 216 mm | 216 × 63 mm |
| US Legal | 216 × 356 mm | 216 mm | 216 × 140 mm |
| Tabloid | 279 × 432 mm | 279 mm | 279 × 153 mm |
The waste strip from an A4 sheet is 210 mm long and 87 mm wide. Two 87 mm squares come out of it, leaving 87 × 36 mm of genuine scrap. So one A4 sheet yields one 210 mm square plus two 87 mm squares if you want them at mixed sizes.
Same-size multi-square yield.
To get several equal squares from one rectangle, pick a side S and tile:
- Squares along the long edge = floor(length / S)
- Squares along the short edge = floor(width / S)
- Total = the two multiplied
Yield only jumps when S crosses a threshold that adds a whole row or column, which is why the numbers below move in steps rather than smoothly. For A4 (297 × 210):
| Square side | Grid | Squares | Paper used | Waste |
|---|---|---|---|---|
| 210 mm | 1 × 1 | 1 | 70.7% | 29.3% |
| 105 mm | 2 × 2 | 4 | 70.7% | 29.3% |
| 100 mm | 2 × 2 | 4 | 64.1% | 35.9% |
| 74 mm | 4 × 2 | 8 | 70.2% | 29.8% |
| 70 mm | 4 × 3 | 12 | 94.3% | 5.7% |
| 65 mm | 4 × 3 | 12 | 81.3% | 18.7% |
| 50 mm | 5 × 4 | 20 | 80.2% | 19.8% |
| 42 mm | 7 × 5 | 35 | 99.0% | 1.0% |
Two things worth noticing. 70 mm is the sweet spot for A4, wasting under 6%, because 4 × 70 = 280 and 3 × 70 = 210 both land close to the sheet. And dropping from 70 to 65 mm gives you the same twelve squares while throwing away three times as much paper. Smaller is not automatically more efficient; it is efficient only when the new size divides the sheet cleanly.
Why A-series is friendly for origami.
A-series paper has a 1:√2 aspect ratio, and the useful consequence is that halving the long side gives you the next size down with the same proportions. That carries through to the squares: the biggest square from A4 is 210 mm, from A3 it is 297 mm, from A5 it is 148 mm. Each step is a factor of √2, so a set of A-sizes gives you a neat geometric ladder of square sizes without measuring anything.
Note what this does not mean. Two A4 squares do not fit inside an A3 square, though the areas are equal: two 210 mm squares side by side need 420 mm and A3 is only 297 mm across. The nesting property belongs to the sheets, not to the squares cut from them.
US paper sizes have no such relationship. Letter is 8.5 × 11 and Legal is 8.5 × 14, which are not proportional, so cutting one down never lands on the other.
Practical cutting tips.
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Fold-and-cut method. Fold the short edge over to lie along the long edge. The diagonal fold marks your cut line. Crisp it, cut along it, and you have a perfect square plus the waste strip, with no measuring at all.
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Don’t bother with the waste strip for tiny models. Below about 50 mm, paper gets fiddly for most folds. Keep the strips for practice folds and for testing whether a sequence works before you commit good paper to it.
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Print first, cut after. Printing patterned paper on a whole sheet and then cutting gives far better registration than trying to feed pre-cut squares through a printer.
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Modular projects. For sonobe or kusudama work, where you want 30 to 60 identical squares, optimise for count rather than size. Twelve 70 mm squares come off an A4 sheet at under 6% waste, so a 30-unit ball takes three sheets of ordinary printer paper instead of a packet of pre-cut origami paper.
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Grain direction. Paper has a grain, and it bends more willingly along it. For crisp work, cut your squares so the grain runs the same way on every one, or the units in a modular set will not behave alike.
Quality of cut matters more than you would expect.
Crooked cuts compound. A 1 mm error on a 70 mm square is 1.4% off, and by the third fold that error has been doubled twice and is plainly visible. Use a metal ruler and a sharp craft knife rather than scissors, and change the blade every twenty sheets or so. A dull blade drags the fibres and gives you an edge that is straight to the eye and ragged to the fold.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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