Three circles of radius 3.5 cm are placed so that each circle touches the other two externally. Find the area of the region enclosed between the three circles (the curvilinear triangular gap). Use pi = 22/7 and give the answer in cm^2 (rounded suitably).

Introduction / Context:
This problem tests composite area reasoning with circles and an equilateral triangle. When three equal circles touch each other externally, the centers form an equilateral triangle whose side equals the diameter (2r). The gap enclosed between the circles is the area of that equilateral triangle minus the areas of the three 60-degree sectors (one sector from each circle) that lie inside the triangle.

Given Data / Assumptions:

• Radius r = 3.5 cm
• Diameter = 2r = 7 cm
• Centers form an equilateral triangle of side a = 7 cm
• Each sector angle inside the triangle is 60 degrees
• pi = 22/7

Concept / Approach:
Gap area = (area of equilateral triangle) - (area of 3 sectors of 60 degrees). For equilateral triangle: Area = (sqrt(3)/4)*a^2. For sector: (theta/360)*pi*r^2 with theta = 60 degrees. Use pi = 22/7 for the sector calculation and round the final result sensibly.

Step-by-Step Solution:
Side of equilateral triangle a = 2r = 7 cm Triangle area = (sqrt(3)/4) * a^2 = (sqrt(3)/4) * 49 (sqrt(3)/4)*49 ≈ 21.2176 cm^2 Area of one 60-degree sector = (60/360) * pi * r^2 = (1/6) * pi * (3.5^2) r^2 = 12.25, so sector = (1/6) * (22/7) * 12.25 12.25 = 49/4, so sector = (1/6) * (22/7) * (49/4) = (1/6) * (22*7/4) = (1/6) * (154/4) = 154/24 ≈ 6.4167 cm^2 Three sectors total = 3 * 6.4167 = 19.25 cm^2 (approx) Gap area ≈ 21.2176 - 19.25 = 1.9676 cm^2 Rounded suitably: 1.967 cm^2

Verification / Alternative check:
The gap must be small because three sectors occupy most of the triangle. The computed gap near 2 cm^2 is reasonable for circles of radius 3.5 cm, confirming the scale is correct.

Why Other Options Are Wrong:
1.867, 1.767, 1.567 cm^2: result from incorrect sector area or incorrect subtraction count. 2.067 cm^2: too large; typically happens if you under-calculate sector area or forget one sector.

Common Pitfalls:
Using diameter instead of radius in the sector area formula. Using 120 degrees instead of 60 degrees for the sector angle. Subtracting only one sector instead of three.

Final Answer:
Area of the enclosed region ≈ 1.967 cm^2