My grandson posed an interesting problem to me the other day that involved geometry. He was trying to find the area between circles that touch each other at single points, sometimes called tangent circles. It has been a long time since I was in school and the same for studying geometry, so I went to the Internet to find a solution. At first glance, it seems to be a fairly involved problem since it involves partial curved areas. And the Internet solutions supported the point by showing rather involved geometric centered equilateral triangle solution with some calculus to handle the curved areas.
While studying these solutions, I kept thinking there is an easier way to find these areas; but I just can't quite see it yet. You know how your brain seems to know things before you do, well some part of my brain was saying "Hey dummy do it like this, it's easier!" So, after staring at this problem for some time, I finally realized the quick solution! What really surprised me is that I have not found this easy solution anywhere on the net! I know it is out there, but where it is I have no clue. So, I thought it would be worthwhile to compose an article about the solution. Maybe some college professor can show me what is wrong with my solution, but I can't see anything wrong with it.
Anyway, here is the answer. If you inscribe ( place inside ) a circle in a square you will note that the height and width of the square is equal to the diameter of the circle. So, simply, subtract the area of circle, which is 3.14 x radius x radius or pi R ^ 2 from the area of the square ( diameter of the circle squared ) and you get the four little areas we are looking for! If you divide the answer by four, you get the area of one of these little leftovers.
( R^2 pi x R^2) / 4 = 0.215R^2
Now, arrange any number of circles touching each other and encase them in squares to count the little areas. For instance, if you had four circles touching each other then you would have four little areas between them, so simply multiply our little area solution by four and you now have the total area between the circles. I think I came out with 4 x 0.215xR^2 = 0.86R^2. Simple, isn't it.
I'm going to place this article on the net and my old head on the block for some math student or professor to say "Hey dummy you messed up!" Have fun!
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