I calculated that given a rectangle with 16:9 dimensions and one with 4:3 that the 4:3 one will have 12 percent more area given the same diagonal length. If anyone's interested, please check my result.

Here's my work:

(16x)^2+(9x)^2=d^2

(4y)^2+(3y)^2=d^2

A(d)=16x*9x=144/337*d^2 and the other: A(d)=4y*3y=12/25*d^2

thus the ratio = 12/25*337/144 ~ 1.12