How the paper is drawn

What the generator does with your settings, the standards behind every page size and unit, how each kind of paper is laid out, and how to make sure the printout comes out at its true size.

The short version

Every sheet is drawn as vector lines whose positions are worked out in points (1/72 in) from the page size, margins and spacing you choose. The PDF stores those numbers, rounded to 0.001 pt (under half a micrometre), as drawing instructions rather than as a picture, so a printer set to 100% puts every line where it belongs at any resolution. Everything happens in your browser, and the same settings always make the same file, which is what lets a share link reproduce a sheet exactly.

Units

The inch is defined as exactly 25.4 mm, and a PDF measures pages in points of exactly 1/72 in, so every size converts without approximation: 1/4 in is 18 pt and 5 mm is 360/25.4 = 14.173 pt. Whatever you type (a fraction such as 1/4 or 1 1/4, a decimal, or a value with its own unit such as 6 mm in an inch box) is converted to points before any geometry is done, and the result is converted back only for display.

Page sizes

A sizes come from ISO 216. Letter and Tabloid are sizes A and B of the US drawing-sheet series in ASME Y14.1; Legal is the customary North American 8.5 × 14 in sheet, which neither standard defines. Landscape turns the same sheet sideways. A custom sheet can be anything from 2 in to 48 in on each side.

A sizes in inches and US sizes in millimetres are rounded here; the generator uses the exact values.

How a pattern fits the page

The margins mark out the area the pattern may use, measured from each edge of the paper. Only whole cells are drawn: whole squares, whole hexagons, whole rows of lines. Whatever is left over is split evenly between the two sides, so the pattern is centred and its outer lines can sit a little inside the margins you set. Every line is placed at a whole number of spacings from the first one, computed afresh rather than by adding the spacing over and over, so there is no rounding drift across the page.

With heavier lines turned on, the option to keep whole heavy squares trims the grid further, so that its outer edges are heavy lines too. Each line is centred on its exact position, so a line along the very edge of the area spreads half its width beyond it. Many printers can't print the outer few millimetres of a sheet, which is why the generator warns about any margin under 0.2 in (5 mm).

Each kind of paper

Graph paper and dot grids

Squares of the size you set. Heavier lines are counted from the first line, or from the axes when axes are drawn: the grid then has the same number of squares on each side of the axes, which are always heavy lines. Medium lines have to divide the heavy-line interval (5 inside 10, for example). Dot grid paper puts a filled dot of the size you choose, measured as a diameter in points, at every crossing of the same grid.

Isometric paper

A lattice of equilateral triangles with the side length you set: points sit one side apart along each column, and the columns are side × √3 ÷ 2 apart (0.866 of the side). Lines run at 30°, 90° and 150°, or at 0°, 60° and 120° when the lattice is turned. The sloping lines are cut off at the edges of the block of whole columns and rows. Isometric dot paper puts a dot on every lattice point instead.

Hexagon paper

Regular hexagons with side a measure a × √3 across the flats and 2a across the corners. With pointy tops, hexagons in a row are a × √3 apart and the rows are 1.5a apart, every other row shifted by half a hexagon; flat tops turn the same tiling sideways. You can give the size as the side or as the width across flats. Neighbouring hexagons share edges exactly, and each shared edge is drawn once so it isn't darker than the others.

Lined paper

Rules exactly one spacing apart across the full width, in whole rows centred top to bottom. The margin line is measured from the left edge of the paper, as notebook rulings are, not from the page margin. There is no binding standard for notebook rulings; the presets are the rulings commonly printed on notebooks and pads:

The US spacings (wide 11/32 in, college 9/32 in, narrow 1/4 in, a margin about 1 1/4 in from the left edge) and Gregg ruling's three lines to the inch with a centre line are as described in the reference overview of ruled paper listed below; 8 mm with a margin is the usual ruling of UK A4 refill pads, and 6 and 7 mm are the narrow ones. Manufacturers vary slightly, which is why any spacing can be typed in.

Handwriting paper

Each writing row has a top line, a midline halfway down and a baseline; the writing height is the distance from the top line to the baseline. The descender space, for the tails of g, j, p, q and y, is the gap between one row's baseline and the next row's top line. The baseline is drawn in the heavier line style. The presets set the descender space to half the writing height, the proportion of the school handwriting paper we checked (for example a 3/4 in writing space with a 3/8 in midline and skip space for kindergarten). For the height a particular class should use, follow the school's or the handwriting programme's own guidance.

Polar graph paper

Circles one spacing apart out from the centre, and rays every so many degrees, measured counter-clockwise from the 3 o'clock direction as in mathematics. The ray steps all divide 180°, so full and half circles both close evenly. Light rays start only where neighbouring rays are at least 4 pt apart, so the centre doesn't fill in with ink; heavier rays run all the way in. Degree labels sit just outside the outer circle.

Semi-log and log-log paper

A logarithmic axis with C cycles (decades) over a length L gives each decade the length D = L ÷ C, and puts the value v of decade d at D × (d + log₁₀ v) from the start of the axis. Each decade has lines at 1 to 9, with the 1 lines heavier, plus finer lines at steps of 0.1, 0.2 or 0.5 wherever they stay at least 3 pt (about 1 mm) apart. A decade must be at least 0.5 in long, which limits how many cycles fit on a page. Each decade is numbered 1 to 9, leaving out numbers where they would touch. The linear axis of semi-log paper is ordinary whole squares.

Headers and split pages

A header takes a band 26 pt tall across the top of the area inside the margins, plus a 10 pt gap before the pattern, with up to 4 labelled write-on lines; the first label gets twice the room when there are several. A split page divides the area into two parts, top and bottom or side by side, with a 1/4 in gap between them, and each part is fitted with whole cells of its own pattern as above.

The calibration ruler

The ruler is a 4 in bar ticked every 1/8 in for inch paper, or a 10 cm bar ticked every millimetre for metric paper, printed with the sheet. Lay a good ruler along it after printing: if the bar is exactly 4 in (or 10 cm), every square on the sheet is the size you asked for. If it is 3 7/8 in, the printer shrank the page to about 97%, and every square shrank with it. The ruler sits in the bottom margin when that margin is at least 0.43 in (11 mm), which keeps the bar at least 0.2 in (5.1 mm) from the paper's edge, the same allowance as the margin warning; otherwise it sits just inside the margin and the pattern area is a little shorter. On a very narrow page the bar is shortened to whole inches or centimetres.

Printing at actual size

Print dialogs often shrink a page to fit the printer's margins, which is the usual reason printed graph paper comes out a few percent small. In the print dialog, choose Actual size or a scale of 100% (in Chrome, Default), and turn off any "Fit to page", "Fit to printable area" or "Shrink oversized pages" option. Load paper of the same size as the sheet; printing a Letter sheet on A4 paper, or the reverse, makes the printer scale or crop it. Even at 100%, an office printer's own mechanical tolerance and the paper's response to heat and humidity can leave an error of up to about 1%. The guide to printing at actual size gives the settings for each browser and PDF reader, with its own sources.

The PDF asks viewers to preselect no scaling, using the PrintScaling viewer preference of the PDF standard (ISO 32000-1, table 150), and to pick the paper tray by page size. Adobe Acrobat and Acrobat Reader honour it; other viewers, including the PDF viewers built into browsers, may not, so check the dialog anyway. The Print button on the page prints the sheet at its physical size in CSS units, in which 1 in is 96 px and 1 pt is 1/72 in, on a print page set to the sheet's exact size with no page margin; keep the scale in the dialog at 100%.

The files

• PDF: one page per sheet, up to 50, all identical. Lines, rings and rays are vector paths, dots are filled circles, and the few labels (header labels, axis numbers, degree and decade numbers, the ruler's text) use Helvetica, one of the standard fonts every PDF reader has built in. There are no images and no watermark.
• SVG: one sheet, with its width and height set to the physical page size, for drawing and cutting programs.

Limits

• Spacing from 1 mm to 6 in.
• Up to 80,000 dots or hexagon edges per sheet, so a 1 mm dot grid fits Letter and A4 but not A3.
• Up to 6 cycles on each logarithmic axis, as long as each decade gets at least 0.5 in.
• Up to 4 header labels of up to 24 characters each.
• Line weights from 0.1 pt to 3 pt (heavy lines to 5 pt), dots from 0.3 pt to 8 pt across (bigger dots to 10 pt).

The figures on these pages

Every count quoted in the paper pages and guides, such as squares, dots, lines and rows per page, hexagons, rings and decade lengths, is worked out when the site is built by the same code that draws the PDF, for the page size and margins stated beside it. If the generator ever changed how a pattern is laid out, those figures would change with it rather than go stale. Conversions between inches and millimetres use the exact 25.4 mm inch.

Sources

Standards

• ISO 216:2007, Writing paper and certain classes of printed matter: trimmed sizes, A and B series (A3, A4 and A5).
• ASME Y14.1-2020, Drawing Sheet Size and Format, American Society of Mechanical Engineers (size A, 8.5 × 11 in, and size B, 11 × 17 in).
• NIST Office of Weights and Measures, SI units: length (the inch is exactly 25.4 mm).
• ISO 32000-1:2008, Portable document format, part 1: PDF 1.7 (user space units of 1/72 in, the standard fonts, and the PrintScaling and PickTrayByPDFSize viewer preferences).
• CSS Values and Units Module Level 3, absolute lengths (1 in = 96 px, 1 pt = 1/72 in).
• CSS Paged Media Module Level 3 (the print page size used by the Print button).
• ISO 5456-3:1996, Technical drawings, projection methods, part 3: axonometric representations (isometric drawing).

Paper conventions (no binding standard)

• Wikipedia, Ruled paper: overview of US rulings (wide, college, narrow, Gregg) and national rulings, with its references.
• Ampad quadrille pad listing: 4 squares per inch.
• TOPS engineering computation pad listing: 5 squares per inch printed on the back, green tint.
• Rapid A4 graph paper, 1:5:10 mm and Classmates 2, 10 and 20 mm squared paper listings.
• Pacon kindergarten handwriting paper, 3/4 in and 3/8 in ruling listing.
• Leuchtturm1917 A5 dot grid notebook listing: 5 mm dot spacing.

Printing

• The browser and PDF-reader documentation and source code, printer manuals and the printer-accuracy patent cited in the printing guide's sources.

Corrections

If a sheet printed at the wrong size even at 100%, or a size or convention here disagrees with its source, please get in touch with the share link and what the ruler measured. Corrections are recorded in the changelog.