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A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. Then, find the ratio between the area of the square and the area of the equilateral triangle?

A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. Then, find the ratio between the area of the square and the area of the equilateral triangle? Correct Answer <span lang="EN-IN" style=" line-height: 107%; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">2√3 : 9</span>

Formula used:

Area of triangle = √3 / 4 × (side)2

Area of square = (side)2

Calculation:

Suppose the side of an equilateral triangle is 6a units.

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∴ Radius of circle = side / 2√3

⇒ 6a / 2√3

⇒ √3a unit

Now, diameter of the circle = 2 × √3a = 2√3a units

We know that

Diagonal of square = √2 × Side of square

⇒ 2√3a = √2 × side of square

⇒ Side of square = √6a units

Now, area of equilateral triangle = √3 / 4 × (side)2

⇒ √3 / 4 × (6a)2

⇒ 9√3 a2        ----(1)

 Area of square = (side)2

⇒ (√6a)2 = 6a2        ----(2)

According to question -

Required ratio = 6a2 : 9√3a2

⇒ 2√3 : 9

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