Question

A square and a regular hexagon have equal perimeters. Their areas are in the ratio:

• A. $2:1$
• B. $2\sqrt{3}:1$
• C. $\sqrt{3}:2$
• D. $3:2$

Answer 1

The correct option is C $\sqrt{3}:2$

Let the side of the square $=a$.
Perimeter of the square $=4a$

Let the side of the hexagon $=b$
Perimeter of the hexagon $=6b$
Given, $4a=6b=>a=\frac{3}{2}b$
Area of the square $=a^2$
Area of the hexagon $=\frac{3\sqrt{3}}{2}{b}^2$
Now, Area of square $:$ Area of hexagon $=$ $a^2:\frac{3\sqrt{3}}{2}{b}^2$
$\Rightarrow {\left(\frac{3}{2}b\right)}^2:\frac{3\sqrt{3}}{2}{b}^2$
$\Rightarrow \frac{9}{4}{b}^2:\frac{3\sqrt{3}}{2}{b}^2$
$\Rightarrow \frac{9}{4}:\frac{3\sqrt{3}}{2}$
$\Rightarrow \sqrt{3}:2$