UPSC Prelims 2001 Q146: Science & Technology – Quantitative Aptitude
A circle of 1 m radius is drawn inside a square as shown in the figure given above. What is
the area of the shaded portion in m²?
- (a) (4 - π)
- (b) 1 - (π/2)
- (c) (1/4) - (π/4)
- (d) 1 - (π/4)
Answer: (d) 1 - (π/4)
Explanation
Area of the shaded portion between a circle of 1 m radius and the square
| Item | Working |
|---|---|
| Circle | Radius 1 m inside a square of side 2 m |
| Square area | 2 × 2 = 4 m² |
| Circle area | π × 1² = π m² |
| Answer given in the published key | 1 − π/4 m² = (4 − π) ÷ 4, the area of one of the four equal corners |
Given: a circle of 1 m radius drawn inside a square, with a shaded portion shown in a figure. The figure is not reproduced here. The space between the square and the circle is 4 − π m², split into four equal corners of (4 − π)/4 = 1 − π/4 m² each. The published key (Sarthaks, and our bank) gives 1 − π/4, the area of one such corner.
Answer: (d) 1 - (π/4).
Why each option is right or wrong
- (a) Incorrect. 4 − π is all four corners together, not the key's answer.
- (b) Incorrect. 1 − π/2 is negative, so it cannot be an area.
- (c) Incorrect. 1/4 − π/4 is negative.
- (d) CORRECT. The published key gives 1 − π/4.
Key point
Corner area outside a quarter circle of radius r = r² − πr²/4.
Remember
Square 4, circle π, one corner = 1 − π/4.
Common mistakes
- Taking all four corners (4 − π) as the answer.
- Using the radius as the square's side.
- Choosing a negative value for an area.
Exam tip
Check the sign: an area cannot be negative.
Sources
- NCERT Class X — Mathematics, chapter 'Areas Related to Circles'
- Sarthaks eConnect — UPSC 2001 question 'A circle of 1 m radius is drawn inside a square' (answer (d))