What Are The Dimensions Of A4, A3 And A2 Paper In Centimetres And Who Decided On Their Size?

A4 paper size, widely used in the United Kingdom and Europe, is measured as 210 mm by 297 mm. This is the size typically used for letters and other correspondence.
A3 paper is larger (yes, as the numbers get smaller, the sheets of paper get bigger), measuring 297 by 420 mm. And A2 paper is larger yet, measuring 420 x 594 mm, which few personal printers can handle.
You may notice that one edge of each paper is double the next smaller size. For instance, A4 measures 210 by 297mm, while A3 measures 420 mm on the long edge. 420 is double 210. That’s because each smaller sheet is calculated on half the dimensions of the larger.
For instance, fold an A3 sheet in half on the long edge (420 mm), and you’ll have a sheet of A4, measuring 210 mm by 297 mm. A0 paper, the paper from which all these other sizes are created, measures 841 mm by 1189 mm, which measures one square meter.

A3 size is 29.7cm in height and 42cm across

These paper sizes are defined by Standard 216 from the International Standards Office (ISO 216)
This standard allowed paper sizes to be standardized across the world (although most countries have their own variants).
The paper size is based on a sheet with an area of 1 square metre with a height to width ratio of 1.414 (= square root of 2). This is called A0
If you chop this sheet in half so that the longer edge is halved you get two sheets of A1. Chop an A1 sheet in half and you get two sheets of A2. etc.etc.
In reality it is usual for the measurements to be rounded to the nearest millimetre to keep the numbers neat and the ISO216 standard has a built in tolerance of between +/-1.5mm to +/-3mm depending on sheet size.
These are generally accepted dimensions for the sizes: A0 841 × 1189, A1 594 × 841, A2 420 × 594, A3 297 × 420, A4 210 × 297, A5 148 × 210, A6 105 × 148, A7 74 × 105

I don't know why the square root of 2 was chosen as the ratio… If anyone knows, please tell!