Circle Packing

First of all we should take the no of unittouching circles touching the big circle

So if we draw a circle with same center but rad of 2.5 cm than center of all unit circle touching the main circle would lie on it Then, No. Of unit circle < (2πr)/d Here r=2.5. , d=diameter=2 Now, 2πr/d = 7.8 Hence, no. of unit circle touching big circle = 7 Now, only one circle can be inserted between 7 circles Hence total no of circles is 8

Consider two small circles touching the outer circle at periphery and also touching each other. Connect the centers of these two small circles to each other and to the center of big circle. We have an isosceles triangle with 2 equal sides a and b of 2.5 and base c= 2.

We want to find out the angle C subtended between the equal sides of this isosceles triangle. By law of cosines,

c^2 = a^2 + b^2 - 2abcos C.

Solving, Cos C = 0.68. Angle C = 47 degrees aprox.

360 degrees/47 degrees = 7.659.

Thus maximum of 7 circles at periphery and of course 1 circle in the center.

There is a logical solution to this question also. Here... Area of big circle=Πr^2=Π(3.5)^2. Area of unit circle=Π. Thus no. of circles=(area of big circle)/(area of unit circle)=(3.5)^2=12.25. But this cannot be the answer because the circles can't overlap each other so a lot of space will be wasted. Thus answer is less than 12. Now looking at the given diagram and the numbers...we draw 3 unit circles in a row touching each other such that if I draw a line passing through their centres, it will be parallel to the diameter of big circle. A similar row of 3 unit circles is possible on the other side of the diameter of big circle. Thus we have drawn 6 unit circles. Now using a little visual imagination you will find that one circle can be fitted on top of the row of 3 unit circles. Similarly one more at the bottom of the lower row of unit circles. Thus we have 2 more circles. Total=6+2=8.

Draw tangents to the unit circle from the center of the big circle.
The angle these tangent will sustain =2 * ArcSin(1/2..5) = A
The number of unit circles would be = 360/A = 7.
In the center a space left is a circle dia= 3. Only one unit circle here.

Total 7 + 1 = 8.