If the length of a [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a [square](https://artofproblemsolving.com/wiki/index.php/Square) is $a + b$, then the area of the square is:
$\mathrm{(A) \ (a+b)^2 } \qquad \mathrm{(B) \ \frac{1}{2}(a+b)^2 } \qquad \mathrm{(C) \ a^2+b^2 } \qquad \mathrm{(D) \ \frac {1}{2}(a^2+b^2) } \qquad \mathrm{(E) \ \text{none of these} }$
Let a side be $s$; then by the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem), $s^2 + s^2 = 2s^2 = (a+b)^2$. The area of a square is $s^2 = \frac{(a+b)^2}{2} \Rightarrow \mathrm{(B)}$.
Alternatively, using the area formula for a [kite](https://artofproblemsolving.com/wiki/index.php/Kite), the area is $\frac{1}{2}d_1d_2 = \frac{1}{2}(a+b)^2$.