A circle is drawn inside a square having side as 'a' touching all the four sides of the square. Area of that circle will be

Question

A circle is drawn inside a square having side as 'a' touching all the four sides of the square. Area of that circle will be

1. A. πa²
2. B. πa²/2
3. C. πa²/4 (Correct answer)
4. D. 4πa²

Correct Answer

Option C — πa²/4

Detailed Solution & Explanation

The correct answer is πa²/4.

Key Points

• A circle inscribed in a square touches all four sides, so its diameter equals the side of the square: d = a.
• The radius is therefore r = a/2.
• Area = πr² = π(a/2)² = πa²/4.

Additional Information

• The ratio of the inscribed circle's area to the square's is π/4 ≈ 0.785, so the circle occupies about 78.5% of the square.
• Contrast the circumscribed case: a circle drawn *through* the square's corners has diameter equal to the diagonal, a√2, so radius a/√2 and area πa²/2.
• Recognising which of the two is described — touching the sides, or passing through the corners — is what the question actually tests.