3 circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them?

1 Answer
Feb 7, 2018

Area of the circles and the region bounded by them

#A = color(green)(12.204# sq units

Explanation:

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Three circles are touching each other and have a radius of 2.

To find the area of the three circles + the colored portion in between the circles.

Construction : Join A, B, C. It forms an equilateral triangle with sides a = 2r = 4 & #hatA = hatB = hatC = 60^0#

Area of #Delta# ABC #A_t= (sqrt3/4) a^2 = sqrt3 = 1.732#

Now we will find the area of the long sectors EF in circle A, DF in circle B and DE in circle C. All the three areas are equal as the circles have the same radius 2 and have a center angle of (360-60 = #300^0#

area of each sector

#A_s = (theta/360) pi r^2 = (300/360) * pi 2^2 = 10.472#

Area of the circles and the region bounded by them

#A = 3 * A_s + A_t = 3 * 10.472 + 1.732 = color(green)(12.204# sq units