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A square is drawn inside the circle as shown in the figure above. If t

June 1, 2025 by rawan239

A square is drawn inside the circle as shown in the figure above. If the area of the shaded portion is 32/7 units then the radius of the circle is :

√2 units

2 units

3 units

4 units

This question was previously asked in

UPSC CAPF – 2009

Download PDF Attempt Online

The correct option is B) 2 units.

The problem involves a square inscribed in a circle. Let the radius of the circle be `r`. The diameter of the circle is `2r`. When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Let the side of the square be `s`. The diagonal of the square is `s√2`. Thus, `s√2 = 2r`, which means `s = 2r/√2 = r√2`. The area of the square is `s² = (r√2)² = 2r²`. The area of the circle is `πr²`. The shaded portion is the area of the circle minus the area of the square. Given the area of the shaded portion is 32/7 units, we have `πr² – 2r² = 32/7`. Factoring out `r²`, we get `r²(π – 2) = 32/7`. Using the approximation `π ≈ 22/7`, we have `r²(22/7 – 2) = 32/7`. This simplifies to `r²((22 – 14)/7) = 32/7`, which is `r²(8/7) = 32/7`. Multiplying both sides by 7/8, we get `r² = (32/7) * (7/8) = 32/8 = 4`. Taking the square root, `r = √4 = 2` (since radius must be positive). The radius of the circle is 2 units.

For a square inscribed in a circle of radius `r`, the side length is `r√2` and the area is `2r²`. The ratio of the area of the inscribed square to the area of the circle is `2r² / (πr²) = 2/π`. The shaded area calculation relies on the difference between the circle’s area and the square’s area.

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