The total area of the four equal circles in the figure above is 36π, a

Question

https://gmatclub.com/forum/download/file.php?id=45919 The total area of the four equal circles in the figure above is 36π, and the circles are all tangent to one another. What is the diameter of the small circle? A. 6\sqrt{2} B. 6+\sqrt{2} C. 3\sqrt{2}-3 D. 6\sqrt{2}-6 E. 6\sqrt{2}+6 2019-01-10_1416.png

GMATinsight · Answer

Area of four equal circles = 36π

i.e. Area of each bigger circle = 9π = πr^2

i.e. r = 3

Make a triangle by joining centres of circles A, B and C

the Right triangle formed has legs = 2*r = 6 units long

i.e. Hypotenuse of triangle 6√2 = 2r + Diameter of smaller circle

Diameter of smaller circle = 6√2 - 6

Answer: Option D

Archit3110 · Answer

Posted from my mobile device

Solution attached.
IMO D

ksrutisagar · Answer

The combined area of four equal circles is 36 \pi

Each circle has area 9 \pi

So, radius of each of the equal circles is 3 = r

Now, draw a square connecting the centers of the four circles.

Side of the square = 2r = 6

Let the diameter of the inner circle be x

Then, diagonal of the Square = 6 \sqrt{2}  = 2r +x

6 \sqrt{2}  = 6 + x

x = 6 \sqrt{2}  - 6

Choice D

Romik · Answer

If the total area of four circles is 36π, then the area of one circle is 9π.

9π=πR^2
R=3

Then we can circumscribe a square about one of the bigger circles with the side equal to the 2R, and one of the vertex of this square will be the center of the inner circle.

Then we can draw a triangle through the centers of bigger and smaller circles and the intersection between one side of the square and circumference of the inscribed circle. Therefore the legs of the triangle is R, and the hypotenuse is the distance between centers of bigger and smaller circles. 
Let's call it x. 
x^2=2*3^2
x=√(18)=3√(2)

Let's call a radius of smaller circle as r.
r=3√(2)-3

Then diameter of smaller circle is 2*(3√(2)-3)=6√(2)-6

Answer is D

(Unfortunately, I cannot attach a drawing, I made a perfect one))

The total area of the four equal circles in the figure above is 36π, a

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The total area of the four equal circles in the figure above is 36π, a

Prep Toolkit

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