Question

A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is
A.\[\sqrt 3 :\sqrt 2 \]
B.\[\sqrt 3 :1\]
C.\[3\sqrt 3 :2\]
D.\[3\sqrt 2 \]

Answer 1

Hint: In this question, first we will find the radius of the triangle which will be equal to the radius of the circle and by using this radius we will then find the length of the diagonal of the square through which we will find the area of the square and hence we will find the ratio of the area of the triangle to the area of the square.

Complete step-by-step answer:
Let the radius of the circle be \[r\]and this radius \[r\]is also the radius of the equilateral triangle as shown in the figure below

Hence we can say \[r = \dfrac{a}{{2\sqrt 3 }} - - (i)\]
Where \[a\] is the length of the side of the equilateral triangle
Now consider the square which is inscribed in the circle, here we can say
Length of the diagonal of the circle = Diameter of the circle
Hence we can write length of the diagonal of the circle \[ = 2 \times r\]
Now substitute the value of r from the equation (i), hence we can write
\[d = 2 \times r = 2 \times \dfrac{a}{{2\sqrt 3 }} = \dfrac{a}{{\sqrt 3 }}\]
We know the area of a square of diagonal d is \[ = \dfrac{1}{2} \times d \times d\]
Hence by substituting the value of d in the above equation we can write
Area of a square \[ = \dfrac{1}{2} \times d \times d = \dfrac{1}{2} \times \dfrac{a}{{\sqrt 3 }} \times \dfrac{a}{{\sqrt 3 }} = \dfrac{{{a^2}}}{{2 \times 3}} = \dfrac{{{a^2}}}{6}\]
Now we know the area of an equilateral triangle \[ = \dfrac{{\sqrt 3 }}{4}{a^2}\]
Hence the ratio of the area of the triangle to the area of the square will be \[ = \dfrac{{\dfrac{{\sqrt 3 }}{4}{a^2}}}{{\dfrac{{{a^2}}}{6}}}\]
This is equal to \[ = \dfrac{{\dfrac{{\sqrt 3 }}{4}{a^2}}}{{\dfrac{{{a^2}}}{6}}} = \dfrac{{3\sqrt 3 }}{2}\]
Therefore the ratio of the area of the triangle to the area of the square is \[3\sqrt 3 :2\]
So, the correct answer is “Option C”.

Note: The figures or the mathematical shapes drawn inside another mathematical shape are denoted by the term ‘in’ while the mathematical shapes drawn outside of another mathematical shape are denoted by the term ‘circum’. Area of the equilateral triangle \[ = \dfrac{{\sqrt 3 }}{4}{a^2}\].