Geometry Practice: Circle, square area

This aptitude question is a geometry question. Concept: Ratio of areas of circle inscribed in a square and a square inscribed in the circle.

Question

A circle of maximum possible size is cut from a square sheet. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be area of the final square?

1. 75% of the size of the original square
2. 50% of the size of the original square
3. 75% of the size of the circle
4. 25% of the size of the original square

  Correct Answer      Choice (2). 50% of the size of the original square.

Explanatory Answer

Let the side of the original square be 'a' units. Therefore, the area of the original square = a² units.

Concept 1: The diameter of the circle of maximum possible dimension that is cut from the square will be the side of the square. So, its diameter will be 'a' units.
Concept 2:The diagonal of the square of maximum possible dimension that can be cut from the circle will be the diameter of the circle. So, the diagonal of the final square will be 'a' units.

If the diagonal of the final square is 'a' units, then its area = \\frac{a^2}{2}) units.
Therefore, the area of the new square will be 50% of the area of the original square.

  The correct answer is Choice (2).