An equilateral traffic sign has an altitude of 12√3 cm. What is its area, in square cm?

For an equilateral triangle of side a the altitude is (√3/2)a, so $a=2\times 12$√3/√3 = 24 cm and the area = (√3/4) $\times 24^2=144$√3 sq cm. The direct route agrees: $(1/2)\times 24\times 12$√3 = 144√3. The trap 108√3 = (√3/4)$(12\surd 3{)}^2$ puts the ALTITUDE into the side formula, which is the commonest slip here; 288√3 drops the 1/2 from (1/2) $x$ base $x$ height; 72√3 takes the side to be 12 cm, forgetting that the altitude is shorter than the side.