Question

The length of the side of a square is equal to the length of the side of an equilateral triangle. The ratio of their areas is
(a) 2 : 1
(b) $2:\sqrt{3}$
(c) 4 : 3
(d) $4:\sqrt{3}$

Answer 1

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

Let:
Length of the side of the square = Length of the side of the equilateral triangle = a unit

Now,

Area of the square$=a\times a=a^2{\mathrm{unit}}^2$

Area of the equilateral triangle$=\frac{\sqrt{3}}{4}{a}^2{\mathrm{unit}}^2$

$$\begin{gathered} \mathrm{Ratio}\mathrm{of}\mathrm{areas}=\frac{\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{square}}{\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{equilateral}\mathrm{triangle}} \\ =\frac{a^2}{{\displaystyle \frac{\sqrt{3}}{4}}{a}^2} \\ =\frac{4}{\sqrt{3}} \end{gathered}$$