The A series paper sizes are defined by the ISO 216 international standard and are based on a few key principles:
①. A0 Starting Point: The largest size, A0, has an area of 1 square meter (1 m²).
②. Constant Aspect Ratio: All sizes maintain the same aspect ratio (width:height) of 1:√2 (approximately 1:1.4142). This is the unique ratio where halving the longer side produces the next smaller size with the same ratio.
③. Halving Rule: Each subsequent size (A1, A2, A3, etc.) is defined by halving the longer side of the previous size:
A1 is half of A0; A2 is half of A1; A3 is half of A2; ...and so on.

①. A0 Dimensions: To get a sheet with an area of 1 m² and a ratio of 1:√2:
Width (w) x Height (h) = 1 m²
h = w * √2
Substitute: w * (w * √2) = 1 => w² * √2 = 1 => w² = 1/√2
Solving gives: Width (w) = 841 mm (rounded to nearest mm); Height (h) = 1189 mm (rounded to nearest mm); A0 = 841 mm × 1189 mm
②. Deriving Smaller Sizes: Halve the longer side (height) of A0 to get A1:
A1 height = A0 width = 841 mm
A1 width = A0 height / 2 = 1189 / 2 = 594.5 mm ≈ 594 mm (rounded)
A1 = 594 mm × 841 mm
③. Continue Halving:
A2 height = A1 width = 594 mm; A2 width = A1 height / 2 = 841 / 2 = 420.5 mm ≈ 420 mm; A2 = 420 mm × 594 mm
A3 height = A2 width = 420 mm; A3 width = A2 height / 2 = 594 / 2 = 297 mm; A3 = 297 mm × 420 mm
A4 height = A3 width = 297 mm; A4 width = A3 height / 2 = 420 / 2 = 210 mm; A4 = 210 mm × 297 mm
...and so on down to A10.

Scalability: The constant aspect ratio means you can perfectly scale content between sizes (e.g., enlarge A4 to A3 without cropping or distorting) and easily fit smaller sizes onto larger ones without wasted space or complex calculations.
Efficiency: The halving rule simplifies production, trimming, and folding. Two A4 sheets fit perfectly onto one A3 sheet, four onto one A2, etc.
International Standardization: Used in most countries worldwide (except primarily the US, Canada, Mexico, and the Philippines), ensuring compatibility for printing, copying, envelopes, and office supplies.
Tolerances: ISO 216 specifies allowable dimensional tolerances (typically ±1.0 mm or ±1.5 mm for dimensions over 150 mm, ±0.5 mm for smaller dimensions) and tolerances for squareness and edge straightness.
Rounding: Dimensions are always specified in whole millimeters (mm).

Origin: The underlying principle (the 1:√2 aspect ratio) was first proposed by Georg Christoph Lichtenberg in 1786. France adopted it as a standard in the late 18th/early 19th century, and it was later formalized internationally as DIN 476 (Germany) in 1922 and then ISO 216 in 1975.
This elegant mathematical definition creates a highly practical and efficient system for paper sizes.