Question

The figure above shows a regular hexagon with sides of length $a$ and a square with sides of length $a$. If the area of the hexagon is $384\sqrt{3}$ square inches, what is the area, in square inches, of the square?

• A. $256$
• B. $192$
• C. $64\sqrt{3}$
• D. $16\sqrt{3}$

Answer 1

The correct option is A $256$

Area of a hexagon $=6\times$ area of an equilateral triangle of the same side as the hexagon $=6\times \frac{\sqrt{3}}{4}\times a^2=\frac{3\sqrt{3}{a}^2}{2}$

Since the area is given to be $384\sqrt{3}$, we equate the two areas and get

$384\sqrt{3}=\frac{3\sqrt{3}{a}^2}{2}$

$\Rightarrow a^2=\frac{384\times 2}{3}=256$

$\Rightarrow a=16$ inch

Area of square $=$ side${}^2=a^2=(16{)}^2=256$ square inch