Contained Region

Geometry Level 2

3 unit circles are tangential to each other. What is the perimeter of the green region contained within?

3 2 π \frac{3}{2} \pi 2 π 2 \pi 1 2 π \frac{1}{2} \pi π \pi

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2 solutions

Pranshu Gaba
Nov 29, 2016

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 60 ^\circ sectors.

Since the radius of the unit circle is 1 1 , its perimeter is 2 π × 1 = 2 π 2\pi \times 1 = 2 \pi . The arc length of a 6 0 60^\circ sector of a unit circle is 6 0 36 0 \dfrac{60^\circ}{360^\circ} th the perimeter of a unit circle, which is 6 0 36 0 × 2 π = π 3 \dfrac{60^\circ}{360^\circ} \times 2\pi = \dfrac\pi3 . Since three such sectors form the border of the green region, the perimeter of the green region is 3 × π 3 = π 3 \times \dfrac\pi3 = \pi .

Srinivas Modem
Nov 23, 2016

perimeter of the green region=3*2 pi r/6=pi

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