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 isn’t. That’s the daily problem for anyone who folds. A4, US Letter, legal, and every other office paper format are rectangles, and almost every origami model assumes a square starting sheet. Convert one to the other without wasting material and you’ve turned a 500-sheet ream into 500 to 1500 origami starts.
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 87 mm wide. That’s enough to yield two more squares at 87 mm × 87 mm, plus a final waste strip of 87 × 36 mm. So one A4 sheet gives you one 210 mm square plus two 87 mm squares — three working squares from a single sheet if you want them at varied sizes.
Same-size multi-square yield.
To get multiple squares of equal size from one rectangle, find the largest square side S such that the rectangle can be tiled with that side:
- Squares across the long edge = floor(length / S)
- Squares across the short edge = floor(width / S)
- Total = (across long) × (across short)
For maximum yield, S is chosen so that both numbers are whole. For an A4 sheet (210 × 297):
- S = 210: 1 × 1 = 1 large square (210 mm), waste = 297 × 87 strip
- S = 105: 2 × 2 = 4 squares (105 mm each), waste = 105 × 297 strip after one row
- S = 100: 2 × 2 = 4 squares (100 mm), waste smaller but non-uniform
- S = 70: 3 × 4 = 12 squares (70 mm each), with small waste
- S = 50: 4 × 5 = 20 squares (50 mm each)
The yield grows quickly as you accept smaller squares.
Why A-series is friendly for origami.
A-series paper has a 1:√2 aspect ratio. That’s not square, but it has a useful property: each smaller A-size is created by halving the longer dimension. Two folded A4 squares neatly nest within an A3 square, and so on. For modular origami where you want stacked sizes, A-series is ideal.
US paper sizes lack this property. A US Letter sheet doesn’t fold down to another standard size cleanly.
Practical cutting tips.
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Fold-and-cut method. Fold the short edge over to align with the long edge — the diagonal fold marks where to cut. Crisp the fold, cut along it, you get a perfect square plus the waste strip.
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Don’t try to use the waste strip for tiny models. Below about 50 mm square, paper is fiddly for most folds. Save the waste strip for practice or test folds.
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Print first, cut after. If you’re printing patterned paper, print the whole sheet, then cut squares — better registration than printing on pre-cut squares.
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Modular projects. For modular origami (sonobe, kusudama, etc.) where you need 30 to 60 identical squares, optimize for highest count from one sheet rather than maximum size per piece. Cutting 12 squares from an A4 sheet costs you about 17% waste, but gets you a full kusudama from 5 sheets instead of buying 60 pre-cut origami sheets.
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Grain direction. Paper has a grain direction. For tight, precise folds, cut squares with the grain running parallel to a chosen edge — folds across the grain hold creases better than folds with the grain.
Quality of cut matters more than you’d think.
Crooked cuts cascade through complex models. A 1 mm error on a 70 mm square is 1.4% off — visible by the third fold. Use a metal ruler and a sharp craft knife, not scissors. Replace blades after about 20 sheets.