Question

find the ratio between area of square inscribed in a circle and equilateral triangle circumscribed about the same circle

Answer 1

To find :- find the ratio between area of square inscribed in a circle and equilateral triangle circumscribed about the same circle ?

Answer :-

Let radius of circle is r cm , side of square is s cm and side of equaliteral ∆ is a cm .

so,

→ Diagonal of square = diameter of circle .

→ √2s = 2r

→ s = (2r/√2) = √2r

also,

→ Inradius of equaliteral ∆ = a/2√3

→ r = a/2√3

→ a = 2√3•r

then,

→ Area of square : Area of Equaliteral ∆ = (s)² : (√3/4)a²

→ Area of square : Area of Equaliteral ∆ = (√2r)² : (√3/4)(2√3r)²

→ Area of square : Area of Equaliteral ∆ = 2r² : (√3/4)12r²

→ Area of square : Area of Equaliteral ∆ = 2r² : 3√2r²

→ Area of square : Area of Equaliteral ∆ = 2 : 3√2

→ Area of square : Area of Equaliteral ∆ = √2 : 3 (Ans.)

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