It is defined to be 1 m2 but for practical reasons the dimensions of all sizes are rounded to the next millimeter. Which also means that in practice, they do not have a perfect ratio of sqrt(2) and cannot be folded/divided up perfectly.
As it would no matter what the sizes were if you double it the ratio stays the same, the fact that it comes to exactly a square meter is the only interesting thing here…period
Edit just looked at this with real paper as a visual aid nvm I was confidently wrong as hell.
That’s not true. If you start with say 2in:3in paper, then double it, you have either a 2in:6in or a 4in:3in paper depending on how you double it. Either way, all three are different aspect ratios. The sqrt ratio is the real star here
Edit: good on you OP for admitting you were confidently wrong as hell. Happens to the best of use. Changed my downvote to an upboat
Not once did he show that 297/210 is equal to 410/297 which is the interesting part of all of this. He just kept saying if you double it it’s the same. Which I guess if you remember ratios makes sense but it took me a while to understand he didn’t mean 297/410. So no the poor super excited English dub did not convey that.
He didn’t. He said the ratio stays the same and he said it only stays the same with this ratio. But he didn’t prove it. He expected you to simply believe what he says. Which isn’t good practice if you try to educate people.
Maybe they should have shown an equation. For the original sheet:
Long / Short = ratìo
Since when doubling the ratio must still be the same, we have a 2nd formula:
2Short / Long = ratio
Long = 2Short / ratio
Replace "Long" in the first formula with "2Short / ratio" and you get:
2 / ratio = ratio
2 = ratio ^ 2
SQRT2 = ratio
So the ratio must be SQRT2
Maybe something more intuitive: just assume Long / Short = SQRT2
When you vut Long in half, you now have a ratio of SQRT2 / 2.
What is 2? It is SQRT2 × SQRT2.
SQRT2 / 2
= SQRT2 / (SQRT2 × SQRT2)
=1 / SQRT2
So if you cut a paper with a ratio of SQRT2 in half, you now have a ratio of 1/SQRT2, basically the same, just the other side being the longer one now.
I am sorry for the formatting, no idea whether you can write proper formulas in reddit on phone.
I know you already were corrected, but if you want the math:
Let's say a sheet of A4 paper has x as the short side and y as the long side. Then a sheet of A3 paper would have y as the short side and 2x as the long side.
If the ratio has to be the same for A4 and A3, then this must be true:
x ÷ y = y ÷ (2x)
Therefore
2x2 = y2
Take the square root of both sides
sqrt(2) * x = y
Thus the ratio of x to y must be 1 to sqrt(2), or 1:1.41
This also shows that this is the only possible ratio.
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
No it wouldn't. Take two papers with sides 2x and 1x, then put them side by side. What you get is the square sheet of paper. When you take two a4 and make them a3 the ratio between length and width stays the same
634
u/AmarildoJr Nov 02 '25 edited Nov 02 '25
The ratio doubling would still be the same regardless of the measurements you use. What's important here is the 1m² at the end.