Find the area of the larger circle in the following case:
Three identical smaller circles and a larger circle are tangent to each other. The smaller circles are inscribed inside the larger circle. If the radius of the smaller circle is "r", find the area of the larger circle.
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There is one large circle and three smaller circles with a radius r inside it such that all four circles are tangent to each other.
The inner circles will be arranged in a configuration such that a triangle can be drawn joining their centers. The triangle is equilateral with the length of the sides equal to 2*r.
The length of a median of the triangle is sqrt((2r)^2 - r^2) = sqrt(4r^2 - r^2) = sqrt(3r^2). The centroid intersects the median in the ratio 2:1. The distance of the centroid from any of the vertices is (2/3)*sqrt(3r^2). To this, if we add the radius of the smaller circle (r) we get the radius of the larger circle.
The radius of the larger circle is (2/3)*(sqrt 3)*r + r.
The area of the larger circle is pi*[(2/3)*(sqrt 3)*r + r]^2
= pi*r^2(1 + 2/sqrt 3)^2
=> pi*r^2(1 + 4/3 + 4/sqrt 3)
=> pi*r^2(3 + 4 + 4(sqrt 3))/3
=> pi*r^2(7+4*sqrt 3)/3
The area of the larger circle is pi*r^2(7+4*sqrt 3)/3
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