Question

If the area of a regular hexagon is $54\sqrt{3}{\mathrm{cm}}^2$, then the length of its each side is
(a) 3 cm
(b) $2\sqrt{3}\mathrm{cm}$
(c) 6 cm
(d) $6\sqrt{3}\mathrm{cm}$

Answer 1

Given: The area of a regular hexagon is $54\sqrt{3}{\mathrm{cm}}^2$.

We know, a hexagon is formed by joining 6 equilateral triangles together.
Therefore, Area of 6 equilateral triangles = $54\sqrt{3}{\mathrm{cm}}^2$

$$\begin{gathered} \mathrm{Area}\mathrm{of}6\mathrm{equilateral}\mathrm{triangles}=6\times \frac{\sqrt{3}}{4}\times x^2 \\ \Rightarrow 54\sqrt{3}=6\times \frac{\sqrt{3}}{4}\times x^2 \\ \Rightarrow \frac{54\sqrt{3}\times 4}{6\times \sqrt{3}}=x^2 \\ \Rightarrow 9\times 4=x^2 \\ \Rightarrow 36=x^2 \\ \Rightarrow x=6\mathrm{cm} \end{gathered}$$

Hence, the correct option is (c).