A3 has exactly twice the area of A4. That is the defining property of the ISO 216 series: each size is exactly half the one above, and the halving preserves the shape.
About the units
The A series is built on a single geometric idea. If a rectangle has sides in the ratio 1 to √2, then cutting it in half across the long side produces two rectangles with the same ratio. That is the only aspect ratio for which this works, and ISO 216 uses it throughout. A0 is defined as the sheet with that ratio and an area of exactly one square metre, giving 841 × 1189 mm. A1 is half of A0, A2 half of A1, and so on down to A10. The system was proposed by the German scientist Georg Christoph Lichtenberg in 1786 and standardised as DIN 476 in 1922.
The exact factor
The area factor is exactly 2, and the linear factor is √2 = 1.414. A4 is 210 × 297 mm and A3 is 297 × 420 mm — the long side of one becomes the short side of the next, which is the halving made visible. Scaling A4 artwork to A3 means enlarging linear dimensions by 141 per cent, which is why photocopiers have a 141% button. Going down, the reduction is 71 per cent. Those two numbers appear on every office copier in the world outside North America, and they exist solely because of the √2 ratio.
Where you meet this conversion
Printing and reprographics. Enlarging a document from A4 to A3 for presentation, or imposing two A4 pages side by side on an A3 sheet, both rely on the exact doubling. Architectural and engineering drawings are issued in A-series sizes precisely because a drawing can be reduced or enlarged between sizes without changing its proportions or invalidating a printed scale bar. Poster printing, plan sets and booklet imposition all use the property.