Four equal circles touching pairwise; area of enclosed central region Four circular cardboard pieces of radius 7 cm are placed so that each piece touches two others, forming a square of centres. Find the area of the region enclosed by the four pieces (the curvilinear diamond at the centre).

Introduction / Context:When four equal circles of radius r touch pairwise, their centres form a square of side 2r. The central enclosed region’s area equals the square’s area minus four quarter-circles (i.e., one full circle).

Given Data / Assumptions:

• Radius r = 7 cm
• Centres form a square of side 2r = 14 cm
• π ≈ 22/7 or 3.14 (either yields same integer result here)

Concept / Approach:Area(enclosed) = Area(square of centres) − Area(4 quarter-circles) = (2r)^2 − πr^2.

Step-by-Step Solution:

Verification / Alternative check:Using π = 3.14: πr^2 = 3.14 * 49 = 153.86 → 196 − 153.86 ≈ 42.14 ≈ 42 (rounding consistent with option).

Why Other Options Are Wrong:12, 32, and 52 cm² do not match the precise geometric decomposition of the region.

Common Pitfalls:Confusing the arrangement (thinking of a ring or lens) or subtracting two circles instead of one full circle (formed by four quarters).

Final Answer:42 cm²