A steel bar is a prism 20 cm long whose cross-section is a regular hexagon of side 6 cm. Find its volume in ${\text{cm}}^3$. Take √3 = 1.73.

A regular hexagon of side a splits into six equilateral triangles, so its area = 6 × (√3/4)$a^2$ = (3√3/2)$a^2=1.5\times 1.73\times 36=93.42{\text{cm}}^2$. Volume of the prism = $93.42\times 20=1868.4{\text{cm}}^3$. (Treating each of the six triangles as a right triangle of legs 6 and 6 gives an area of $108{\text{cm}}^2$ and a volume of $2160{\text{cm}}^3$; using (3√3/4)$a^2$ halves the hexagon and gives $934.2{\text{cm}}^3$; 720 is the curved-in lateral surface, perimeter × length = $36\times 20$, an area in ${\text{cm}}^2$ rather than a volume.)