The diagonal of a square measures 18 cm. What is the area of the square?

A diagonal splits the square into two 45°–45°–90° triangles, so the side is 18/√2 = 9√2 cm and the area is $(9\surd 2{)}^2=162{\text{cm}}^2$. The same result follows from the shortcut area = $d^2/2=324/2=162{\text{cm}}^2$. Treating the diagonal as the side gives $324{\text{cm}}^2$, the usual slip.