Corner-Cut Box Size Calculator
The corner square to cut from a sheet for the largest folded box — exactly a sixth of the side on a square, and a flat peak either way.
Units are whatever you typed — the whole problem is scale-free, so inches in gives cubic inches out and centimetres give cubic centimetres. This is the bare geometry, with nothing taken off for glue tabs, overlapping corners or the thickness of the card. Nothing is uploaded.
The best cut has an exact answer, and on a square it is beautiful
The volume is the cut times what is left of the base — x(L−2x)(W−2x) — and setting its derivative to zero gives a quadratic with one root inside the sheet. On a square sheet the whole thing collapses: cut exactly one sixth of the side, and the box holds exactly two twenty-sevenths of the cube on that side.
| Corner cut | Base left | Volume | Of the best |
|---|---|---|---|
| 0.50 | 10.0 × 7.5 | 37.5 | 57% |
| 1.00 | 9.0 × 6.5 | 58.5 | 88% |
| 1.50 | 8.0 × 5.5 | 66.0 | 100% |
| 1.59 | 7.8 × 5.3 | 66.1 | 100% |
| 2.00 | 7.0 × 4.5 | 63.0 | 95% |
| 2.50 | 6.0 × 3.5 | 52.5 | 79% |
| 3.00 | 5.0 × 2.5 | 37.5 | 57% |
| 3.50 | 4.0 × 1.5 | 21.0 | 32% |
For a 10 by 10 square that is a cut of 1.667 and a box of 74.07 — and both figures are exact, not rounded. Cut nothing and you have a flat sheet; cut half the short side and you have no base left. The answer is always somewhere strictly between.
But the peak is flat, so the habit is the thing to fix
On a 11 by 8.5 sheet, anything between 1.04 and 2.20 still gets nine tenths of the best possible box — a band well over an inch wide. So measuring the optimum to three decimal places is wasted effort. What is not wasted is noticing that a one-inch corner falls outside that band as soon as the sheet gets long.
| Sheet | Best cut | Best volume | A 1-inch corner gets |
|---|---|---|---|
| A square (10 × 10) | 1.67 | 74.1 | 86% |
| Letter (11 × 8.5) | 1.59 | 66.1 | 88% |
| A4-ish (11.7 × 8.3) | 1.60 | 69.4 | 88% |
| Tabloid (17 × 11) | 2.18 | 183.0 | 74% |
| A long strip (30 × 10) | 2.26 | 315.6 | 71% |
On letter paper the habitual corner gets 88% of the best box, which is fine. On a long strip it gets 71%, because the right cut there is 2.26 and a fixed habit cannot know that. The optimal cut does grow with the sheet — but far more slowly than the sheet does, which is exactly why guessing it by eye goes wrong in one direction and not the other.
How to use
- Enter the length and width of your sheet.
- Enter the corner square you were going to cut.
- Compare it against the exact optimum.
- Check the ladder to see how flat the peak is.
Frequently asked questions
What size corner should I cut?
It depends on the sheet, and there is an exact answer. The volume is the cut times what is left of the base, and setting the derivative to zero gives a quadratic whose smaller root is the cut you want. The tool solves it directly rather than searching.
Is there a simple rule for a square sheet?
Yes, and it is unusually tidy: cut exactly one sixth of the side. The resulting box holds exactly two twenty-sevenths of the cube on that side. Both of those are exact, not rounded, which is rare enough to be worth knowing.
How precise do I need to be?
Far less than you would think. On a letter sheet anything between about 1.04 and 2.20 inches still gets nine tenths of the largest possible box — a band well over an inch wide. Measuring the optimum to three decimals is wasted effort.
So does the cut matter at all?
Yes, but as a habit rather than a measurement. A one-inch corner gets 88% of the best box on letter paper and only 71% on a long strip, because the right cut there is over two inches. A fixed habit cannot know the sheet it is on.
Does a bigger sheet want a bigger cut?
Yes, but far less than proportionately. Tripling the length of a sheet does not triple the cut — it moves it a little. That is exactly why guessing by eye goes wrong in one direction, always too small, and never too large.
What units does this use?
Whichever you type. The problem is scale-free in length, so inches give cubic inches and centimetres give cubic centimetres. Double every dimension of the sheet and the best cut doubles while the volume goes up eightfold.
Does this allow for glue tabs or card thickness?
No — it is the bare geometry of cutting four squares and folding four flaps up. Real card has thickness that eats into the base, and tabs need material that this does not account for, so treat the volume as an upper bound.
Does this send anything anywhere?
No. Every figure is computed in your browser, and nothing is uploaded or stored.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.