A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be the area of the final square?
(a) 3/4 of original square
(b) 1/2 of original square
(c) 1/4 of original square
(d) 2/3 of original square.

Solution:

Let us represent the schematic arrangement of the problem as given below:

The outermost square is of side ‘s’.

The area of the outermost square is = s²

Hence the circle inscribed in it will be of diameter = s

We know that the diagonal of the square = √2 side

Thus side of the square = diagonal/√2 = s/√2

Hence the area of the innermost square is = (s/√2)² = s²/2

The area of the smaller square is half of the original square.

Answer: Correct answer is (b)

✦Try This: A square of maximum possible size is cut from a circular sheet of board. Subsequently, a circle of maximum possible size is cut from the resultant square. What will be the area of the final circle?

The schematic diagram of the bottle is given below:

Let the radius of the original circular sheet of board = r

Therefore the diameter of the circular sheet of board = 2r

To calculate the length of the side of the inscribed square we use the Pythagorean theorem:

If x is the side of the inscribed square then we get,

x² + x² = (2r)²

2x² = 4r²

x² = 2r²

x = r√2

The radius of the circle inscribed in the square will be = x/2

Since x = r√2, the radius will be

Radius of the new circle inscribed in the square = r√2/2 = r/√2

The area of the circle = π(radius)² = π( r/√2)² = πr²/2

The area of the original circle is = πr²

Hence the area of the new circle is half of the area of the original circle.

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 11

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NCERT Exemplar Class 8 Maths Chapter 11 Problem 3

A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be the area of the final square? (a) 3/4 of original square. (b) 1/2 of original square. (c) 1/4 of original square. (d) 2/3 of original square

Summary:

A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. The area of the final square will be 1/2 of the original square

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