Question

What are the dimensions of rectangle A6 in the following image?

Assumptions
1. A sheet of A0 paper is a rectangle with an area of precisely one square meter. Its length (the long side) is twice its width.
2. When A0 paper is cut in half across its long side, two sheets of A1 paper are formed. Likewise, when A1 paper is cut in half across its long side, A2 paper is formed, and so on.
3. All rectangles in the series (A0, A1, A2, ...) are similar.

Answer 1

I think assumption 1 part 2 is not correct! The correct assumption is: the square of length is twice the square of width. Now the Answer:
Define variable a to be the width of paper size A6.
Then its length would be 1.4142 x a (square root of 2) which is equal to width of paper size A5
subsequently, for sizes A4, A3, A2, A1 and finally for paper size A0:
The width would be 8a and its length 8a x 1.4142

As we have the area of A0 as one square meter or 1000000 mm, therefor:
8a x 8a x 1.4142 = 1000000

a = 105 mm

the width of paper size A6 is 105 mm
and its length is 105 x 1.4142 = 148 mm