A rectangular field has perimeter 100 $m$ and its diagonal is 40 $m$ long. Find its area in $m^2$.

Let the sides be $l$ and $b$. Then $l+b=50$ and $l^2+b^2=40^2=1600$. Squaring the first, $(l+b{)}^2=2500$, so $2\text{lb}=2500-1600=900$ and the area lb = $450{\text{m}}^2$. (Stopping at 2lb gives $900{\text{m}}^2$; assuming a square of side 25 m gives $625{\text{m}}^2$ but its diagonal would be 35.36 m, not 40 m; ½ × $40\times 40=800{\text{m}}^2$ treats the diagonal as if it cut the field into two right isosceles triangles.)