Paper sizes: the A series, and why a sheet has no size in pixels
A sheet of paper has a size in millimeters and no size at all in pixels. That is the whole of what follows, and it is the thing most often got wrong: an A4 is 210 by 297 millimeters always, and it becomes 2480 by 3508 pixels only once someone has chosen a resolution.
The A series starts from one square meter
The series is set by the standard ISO 216. A0 measures 841 by 1189 millimeters, which is one square meter to within a fraction of a percent. Every format after it is the previous one cut in half across its long side: A1 is half an A0, A2 half an A1, and so on down.
An A4 is therefore a sixteenth of a square meter, four halvings down from A0. That is not a coincidence to remember but the definition itself, and it is what makes the next two sections work.
The ratio of the square root of two, and why folding does not change it
The A series has a shape no other family has: cut it in half and you get the same shape again. Only one ratio does that, and it is the square root of two, 1.4142. A sheet 1 by 1.4142 cut across gives two sheets 0.7071 by 1, which is the same rectangle turned.
This is why a photocopier offers 141 percent and 71 percent. Going from A4 to A3 multiplies each side by the square root of two; going back divides by it. No margin has to be recomputed, no image gets cropped, and that is the entire point of the series.
The pixel trap
A paper size is a physical size. It has no pixels until a resolution is chosen, and the same A4 gives a different figure for every one of them: 2480 by 3508 pixels at 300 dots per inch, written 300 dpi, then 1240 by 1754 at 150 dpi, and 595 by 842 at 72 dpi.
That last pair is worth knowing for another reason: 595 by 842 is also the size of an A4 in PostScript points, the unit a PDF is laid out in, because a point is exactly a seventy-second of an inch. A file that says 595 by 842 is not low resolution, it is measured in points.
The B and C series, and the envelope that holds the sheet
B sits between two A formats and C between an A and a B, each series built on the same halving rule. The point of C is postal: a C4 envelope holds an unfolded A4, a C5 holds an A4 folded once, and a C6 holds one folded twice.
That is the whole logic of the naming. An envelope marked C5 does not tell you its millimeters, it tells you what fits inside it, which is the only thing the person at the counter needs to know.
US Letter belongs to no series
Letter measures 8.5 by 11 inches, that is 215.9 by 279.4 millimeters: wider than an A4 and shorter. Its ratio is 1.294, not the square root of two, so cutting it in half gives a rectangle of a different shape and the photocopier trick does not work.
This is why a document laid out for one prints badly on the other: the margins do not scale, and scaling to fit leaves an empty band on one edge. Neither format is wrong; they simply do not belong to the same family.
Grammage tells you what the sheet weighs
Paper is sold by mass per square meter, and ordinary office paper is 80 grams per square meter, written 80 gsm. Since an A4 is a sixteenth of a square meter, an A4 sheet of it weighs exactly five grams, which is the arithmetic behind every postage calculation.
Four sheets and an envelope come to just under twenty-five grams, which is why the first postal band is set where it is. The same reasoning gives 2.5 grams for an A5 and ten for an A3.
Why this page carries no converter
A format is not a unit. A4 is a name for a pair of lengths, not a quantity you can be given more or less of, and putting it in a converter would suggest a scale that does not exist.
The lengths themselves do convert, and the site does that elsewhere: millimeters to inches for the sides, square meters for the area. What belongs here is the rule that ties the formats together, and it is a rule about shape.
Length · Area · Weight & mass