3 unit circles are tangential to each other. What is the perimeter of the green region contained within?
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When we join the centers of the three circles, we get an equilateral triangle. The green region is formed by three arcs of 6 0 ∘ sectors.
Since the radius of the unit circle is 1 , its perimeter is 2 π × 1 = 2 π . The arc length of a 6 0 ∘ sector of a unit circle is 3 6 0 ∘ 6 0 ∘ th the perimeter of a unit circle, which is 3 6 0 ∘ 6 0 ∘ × 2 π = 3 π . Since three such sectors form the border of the green region, the perimeter of the green region is 3 × 3 π = π .