a) As presumably 5 of you know, the area of an A0 sheet of paper is a square metre. Size(A(n+1)) = 1/2 Size(A(n)). Therefore the size of an A4 sheet of paper is 1/16 of a square metre.
b) As mentioned in comments, the A series of paper sizes has the property that each size is got by halving the previous size along its long size. Therefore the ratio of the long side to the short side is sqrt(2) to 1.
c) Putting these facts together (viz the ratio and the fact that a sheet of A0 paper is a square metre) yields the sides of a sheet of A0 paper being 2^1/4 metres by 2^-1/4 metres (i.e. the square root of the square root of 2). Dividing these by 4 yields the size of an A4 sheet of paper.