Problem Overview

The area of the circumcircle is 200 cm². For a square with an incircle area of 100 cm², the incircle radius is approximately 5.64 cm, making the side length about 11.28 cm, and the circumcircle radius about 7.98 cm, yielding an area of roughly 200 cm² when rounded.

Step-by-Step Solution

The incircle touches all four sides of the square, so its diameter equals the side length s. Given the incircle area πr² = 100, solve for r = √(100/π). Thus, s = 2r = 2√(100/π)

The circumcircle passes through all four vertices, so its diameter equals the square’s diagonal s√2. The circumradius R = (s√2)/2 = r√2. The circumcircle area is πR² = π(r√2)² = πr² ⋅ 2 = 100 × 2 = 200 cm² exactly.

Introduction to Incircle and Circumcircle

In geometry problems for exams like CSIR NET, questions about a square’s incircle area 100 cm² and circumcircle area test radius relationships. The incircle fits inside touching all sides; the circumcircle passes through all vertices. Their areas relate by a factor of 2 due to the diagonal being s√2

Detailed Calculation Breakdown

Start with incircle: πr² = 100, so r = 100/π ≈ 5.64 cm. Side s = 2r ≈ 11.28 cm. Diagonal d = s√2 ≈ 15.96 cm, so circumradius R = d/2 ≈ 7.98 cm. Area πR² ≈ 200 cm² (exact: 2 × 100).

This skips approximations for exam efficiency: since R = r√2, area ratio (R/r)² = 2

Common Mistakes

• No options are provided in this fill-in-the-blank question, but common distractors might include 100 (same as incircle), 400 (using side as radius), or π × 100 (misapplying ratios).
• The ratio of circumcircle to incircle area is always 2:1 for squares.
• Confusing incircle (diameter = side) with circumcircle (diameter = diagonal).
• Forgetting √2 factor, yielding 100 cm².
• Using side as circumradius, giving 400/π ≈ 127 cm².

Concept Summary Table

Practice ratio-based shortcuts: circumcircle area is always twice the incircle area in squares. Verify with π cancellation.

Exam Tips for CSIR NET

Practice ratio-based shortcuts: circumcircle area is always twice the incircle area in squares. Verify with π cancellation.