A triangular signboard has sides of 9 cm, 40 cm and 41 cm. What is its area in square cm?

Notice that $9^2+40^2=81+1600=1681=41^2$, so the triangle is right-angled and the two shorter sides serve directly as base and height: area = $(1/2)\times 9\times 40=180$ sq cm. Heron'$s$ formula confirms it, with $s=45$ giving √$(45\times 36\times 5\times 4)$ = √32400 = 180. The trap $820=(1/2)\times 40\times 41$ pairs a leg with the HYPOTENUSE, which is never a height; 360 forgets the 1/2; 90 halves the answer twice.