Question 1061891: Given six congruent circles drawn internally tangent to a circle of radius 18; each smaller circle is also tangent to each of its adjacent circles. Find the shaded area between the large circle and the six smaller circles.
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Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  + another small circle = 
The smaller circles' diameter is  of the diameter of the smaller circle.
Radius of the larger circle = 
Radius of each smaller circle = 
Area of each smaller circle = 
Since the larger circle is  times as wide,
its area,  , is  times larger:
 ,
The area between the large circle and the six smaller circles is
 .
NOTE:
You may be familiar with this arrangement of 6 circles around a central one,
because you can form it with 7 pennies (or seven coins, or right cylindrical objects of the same shape).
Do you also see it in "pilings od same sized cylindrical objects,
such as a flatbed truck cargo of metal pipes or plastic tubes.
Do you want a geometry proof?
Joining the centers of adjacent circles with line segments will give you a regular hexagon.
Adding 3 diameters of the outside circle passing through the hexagon vertices,
you have the hexagon split into 6 triangles.
The central angles measure  ,
and the triangle sides flanking those angles are congruent,
meaning that the triangles are isosceles, with a  vertex angle.
So, in the triangles, the base angles (adjacent to the hexagon sides)
must be congruent, and must measure
 .
So, those isosceles triangles (and all isoceles triangles with vertex angles) are equilateral triangles.
All sides of those triangles are congruent,
all measuring 2 times the radius of the smaller circles.
That means that there is definitely just enough room in the center for another smaller circle.
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