Question

The ratio of the area of a square of side a and that of an equilateral triangle of side a, is

(a) 2 : 1
(b) $2:\sqrt{3}$
(c) 4 : 3
(d) $4:\sqrt{3}$

Answer 1

(d) $4:\sqrt{3}$

Let one side of the square and that of an equilateral triangle be the same, i.e. a units.
Then, Area of the square = (Side)² = (a)²
Area of the equilateral triangle = $\frac{\sqrt{3}}{4}{\left(\mathrm{Side}\right)}^2$ = $\frac{\sqrt{3}}{4}{\left(\mathrm{a}\right)}^2$
∴ Required ratio = $\frac{a^2}{{\displaystyle \frac{\sqrt{3}}{4}}{a}^2}=\frac{4}{\sqrt{3}}=4:\sqrt{3}$