Quarter-inch graph paper on US Letter
Yes, and it is the reason imperial graph paper looks tidier than metric. US Letter is 8.5 by 11 inches; take a half-inch margin off each side and 7.5 by 10 inches of printable area remain, which quarter-inch squares divide exactly 30 and 40 times. Nothing is left over, so the first line sits exactly on the margin whether you centre the grid or push it to the corner. A heavy line every fourth square marks the inches.
The sheet
Vector, dimensioned in millimetres, so it prints at true scale on any printer. Print it from here to skip the page furniture, or save it and open it in anything.
Every line on the sheet
| Vertical | 0 | 12.7 | 0.5 | Heavy |
| Vertical | 1 | 19.05 | 0.25 | Fine |
| Vertical | 2 | 25.4 | 0.25 | Fine |
| Vertical | 3 | 31.75 | 0.25 | Fine |
| Vertical | 4 | 38.1 | 0.5 | Heavy |
| Vertical | 5 | 44.45 | 0.25 | Fine |
| Vertical | 6 | 50.8 | 0.25 | Fine |
| Vertical | 7 | 57.15 | 0.25 | Fine |
| Vertical | 8 | 63.5 | 0.5 | Heavy |
| Vertical | 9 | 69.85 | 0.25 | Fine |
| Vertical | 10 | 76.2 | 0.25 | Fine |
| Vertical | 11 | 82.55 | 0.25 | Fine |
Both US Letter dimensions are whole inches or halves of one, so any square size that divides a quarter of an inch fits the page without a remainder.
Because there is no remainder, the centred and top-left layouts produce identical sheets here. It is not the only combination that divides exactly, but it is the conventional one.
A heavy line every four squares turns the grid into inch blocks, which is what US ruled paper conventionally shows.
How it works
The formula
printableWidth = pageWidth - 2 x margin
printableHeight = pageHeight - 2 x margin
across = floor(printableWidth / spacing)
down = floor(printableHeight / spacing)
gridWidth = across x spacing
gridHeight = down x spacing
centred: firstLineX = margin + (printableWidth - gridWidth ) / 2
firstLineY = margin + (printableHeight - gridHeight) / 2
top-left: firstLineX = firstLineY = margin
verticalLines = across + 1
horizontalLines = down + 1
a line is heavy when every > 0 and index mod every = 0
heavyVertical = floor(across / every) + 1
lineLength = verticalLines x gridHeight + horizontalLines x gridWidth
Wv = total width of the vertical strokes, capped at gridWidth
Wh = total height of the horizontal strokes, capped at gridHeight
inkArea = Wv x gridHeight + Wh x gridWidth - Wv x Wh
coverage = inkArea / (pageWidth x pageHeight)
Every length above is held as a whole number of micrometres.
What it assumes
- Every length is held as a whole number of micrometres, a thousandth of a millimetre, because the answer turns on a floor division and a floating-point millimetre gets that wrong by an entire column. On A4 with a 6 mm margin and 4.4 mm squares, 198 divided by 4.4 evaluates to 44.99999999999999, so a naive implementation reports 44 squares where there are 45.
- Only whole squares are counted. A partial square in the leftover strip is not drawn and not included, because a grid that ends in a sliver is what makes printed graph paper look wrong.
- Ink coverage treats each stroke as centred on its line and entirely on the paper. At a zero margin the outermost line loses half its width over the edge, so the figure is a fraction of a per cent high there.
- Paper sizes are trimmed sizes. Your printer's own hardware margin, usually 4 to 6 mm, is not deducted, so a margin below about 5 mm may be clipped by the printer rather than by the page.
- Every millimetre here assumes the sheet is printed at 100%. Browsers and print dialogues default to fitting the page, which shrinks everything by two or three per cent and makes 5 mm squares measure 4.9.
Common questions
Does A4 take quarter-inch squares?
Not evenly. A4 is 210 mm wide, which is 33.07 quarter-inch squares, so you get 33 whole ones and a sliver. Quarter-inch is an imperial grid on an imperial sheet; on A4, 5 mm or 10 mm divides the page cleanly instead.
What about eighth-inch and half-inch?
Both divide US Letter exactly, for the same reason — 8.5 and 11 inches are whole numbers of eighths. Eighth-inch is fine for drafting and too dense to write in; half-inch is the usual choice for graphing by hand.
Sources
The full method, worked example and every assumption behind this figure are on Printable Graph Paper Planner.