If PQRO is a square inside a Circle with centre at "O" and radius "a", what is the area of the shaded portion ?
So the square should have a diagonal equal to length of radius of circle. Let x be the side of square.
Hence diagonal of a square with side x= x root2
=> x root2 = a (radius of circle)
=>x= a/root 2
Hence area of square = (a/root 2)^2 = a^2/2.
Now the area of circular quadrant is (pi * a^2)/4
So shaded area = (pi * a^2)/4 - a^2/2, by simplifying
Hence Answer B.
Just a remark: For any quadrilateral with perpendicular diagonals (so obviously also for a square), the area is given by half the product of the diagonals.
(You can easily deduce it by expressing the areas of the triangles formed by the diagonals.)
So, when you know the diagonal of a square, you don't have to compute the side in order to find the area. You just have to square the diagonal and half it.
In the given question, the diagonal of the square is \(a\) (the radius of the circle), so the area of the square is \(a^2/2.\)
PhD in Applied Mathematics
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