Question

Area of regular hexagon varies directly as the square of its side. A regular hexagon having side 4 cm. has area $24\sqrt{3}{\mathrm{cm}}^2$. Find the area of regular hexagon having side 6 cm.

Answer 1

$$\begin{gathered} \mathrm{Let}A\mathrm{and}S\mathrm{denote}\mathrm{the}\mathrm{area}\mathrm{and}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{side}\mathrm{of}\mathrm{hexagon},\mathrm{respectively}. \\ Thus,wehave: \\ A\alpha S^2 \\ \therefore A=k{S}^2\left(k\mathrm{is}\mathrm{constant}.\right) \\ 24\sqrt{3}=k\times 4^2 \\ \Rightarrow k=\frac{24\sqrt{3}}{16} \\ \Rightarrow k=\frac{3\sqrt{3}}{2} \\ \therefore A=\frac{3\sqrt{3}}{2}{S}^2\left(\mathrm{Equation}\mathrm{of}\mathrm{variation}\right) \\ \Rightarrow A=\frac{3\sqrt{3}}{2}\times 6^2 \\ \Rightarrow A=54\sqrt{3} \end{gathered}$$