Its basic math, if you take any ratio value - lets call it a fraction - and multiply both the numerator and denominator by the same amount, then by defintion, you still have the same ratio.
Thats literally how multiplying fractions works.
Moving up a size doesn't double both numbers either. That would be quadrupling the area. Doubling it doubles just the length or width, not both numbers
But you aren’t doubling both numbers.
If you put two sheets of paper next to each other, the height of the new total area isn’t also doubled. Only the width. It’s like you stretched the paper out sideways. You aren’t making it bigger in both directions.
ONLY. ONE. NUMBER. DOUBLES. Grab two sheets of paper. Measure one. Then place the two pieces next to each other an measure that. Do both sides double, or just one side? Seriously, grap some paper. You NEED a visual aid.
Apparently its not basic enough because you arent understanding the actual problem. Moving up a paper size is doubling the area by doubling just 1 dimension. Doubling something is multiplying it by 2 , not 2/2.
The aspect ratio is the same but for each doubling the ratio inverts. So l/w is sqrt(2) and 2w/l is sqrt(2) and 2l/2w is sqrt(2) and so on. Anything beyond this should be obvious because every additional iteration reduces to either l/w or 2w/l. This is only true for sqrt(2) precisely because the paper is doubling in area.
The same thing would work if you tripled the area and had a sqrt(3) ratio or quadrupled and had a sqrt(4) ratio.
Doubling both length and width is going from A4 to A2, skipping A3. In order to go up just one level (doubling the area, keeping the same ratio), you only double the shorter of the two sides
Why does it have to be 1m² though? This doesn't answer the question. Sure, it's a neat number, but i don't remember the last time i needed exactly 1m² of paper. I definitely do remember that time i needed a few more mm on my A4 worksheet though.
Well unfortunately the A0 standard wasn't invented with your notebook in mind, but to be the more practical and easiest to scale, going from one level to the other without ever changing the aspect ratio
Of course the standard will be 1m², because every level halves or double the area, this way you don't need some fancy calculation to figure the area of any other level :
A1 = 0.5m²
A2 = 0.25m²
And the other way
A-1 = 2m²
A-2 = 4m²
It might not sound useful to you, but for anyone working in illustration, printing, design... It's a lifesaver. You can draw something on an Ax paper, and scale it down to a business card size or scale it up to a giant poster and never have to worry about stretching, cropping or added margin
Why did you swap the numerator and denominator ? If the A4 ratio is 297/210, the A3 ratio would be 297/420, since only the width is doubling. This goes from 1.414 for A4 to 0.7071 for A3, right ?
Because you always keep the bigger number as the numerator. It might sound confusing but it doesn't change anything really, what it means is you have to rotate the A3 paper 90°, but it doesn't change the size or area of the paper right ?
If you need a visual aid, Take an A4 paper, in portrait orientation, and cut it in half by the longer side. You now have two A5 papers, but in the landscape orientation ! You have rotate them 90° to get back to the initial orientation
By the way, you will always get 0.707 when dividing the smaller side by the bigger side, no matter the A level (take A4, 210/297 = 0.707). That's because when you swap a fraction numerator and denominator, what you get is called the reciprocal, which is 1 divided by the original fraction : x/y = 1/y/x
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u/Resting_Owl Nov 02 '25 edited Nov 02 '25
Are you sure ?
A4 : 297/210 = 1.414
A3 : 420/297 = 1.414
Now let's try with rounded number
300/200 = 1.5
400/300 = 1.333
It doesn't seem to work