A4 paper size

Let's start off with three basic rules:

1) The width:length ratio is constant for all paper sizes
2) Halving a paper along the long side will give you two transposed pages of the next smaller page size.
3) The area of a A0 page is 1m²

We can calculate the width:length ratio as follows:
Take a landcape A4 page, and dividing it in half will give you two portrait A5 pages. The width of the A5 page (w₅) will be the half of the length of the A4 page (l₄), and the length of the A5 paper (h₅) will be equal to the width of the A4 page w₄). Because the width:length ratio of both pages is the same, we can write:

w₄ / l₄ = w₅ / l₅
w₄ / l₄ = l₄ / (2w₄)
2w₄² = l₄²
l₄ = sqrt(2) x w₄

Using the result above, the width:lenght ratio can be written as:
w₄ / l₄ = 1 / sqrt(2)

The area of an A0 page is 1m², A1 is 0.5m², etc. The area of a An page can therefore be written as 1/2ⁿ m². Because the area is the product of the height and length of the page, we get the following formula:

w_nh_n = 1/2ⁿ

Using the width:length ratio, this can be simplified to:
w_n = 2^{-n/2 - 1/4}
h_n = 2^{-n/2 + 1/4}

So for a A4 paper, we get:
w₄ = 2^-2.25 = 0.210 m
h₄ = 2^-1.75 = 0.297 m