# 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....

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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### 1 Answer

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**