If the length of a diagonal of a square is (a +b), then the area of the square is :

$\frac{1}{2}(a^2+b^2) +ab$

We know that,

Diagonal of a square = \(\sqrt {2}\) Side

Area of a square = (Side)²

We havem,

The length of a diagonal of a square = (a + b)

Side of the square = \(\frac{ Diagonal of a square }{\sqrt {2}}\)

⇒ Side of the square = \(\frac{ (a + b)}{\sqrt {2}}\)

The area of the square = [\(\frac{ (a + b)}{\sqrt {2}}\) ]²

= \(\frac{ (a + b)^2}{2}\)

=\(\frac{ ((a^2 + b^2 + 2ab))}{2}\) = \(\frac{ ((a^2 + b^2))}{2}\) + ab