How do I prove 3 circles with radius 5 can't cover a square of 10 by 10 completely, but 4 circles will?

Answer 1

Area of a circle is pir^2 if r=5, then the area is 25pi or 25x3.14159 = a little less than 80

Area of 3 such circles is less than 3x80= about 240 Area of 4 such circles is 4x8= 320

Area of 10x10 square is 100. Two circles would have covered that square, with about 160 Area

But you probably mean place the circles over the square overlapping in a manner that covers all of the square. One square covers everything in the square except 4 corners of about 5 Area each. Place a circle over each of those 4 corners, and circles will cover the square completely.

Put each of 4 circles over the square, but with just half circles from each. Two on the left and right sides, two up and down. That covers the square completely. 4 definitely works.

Problem is showing 3 won't work. If you put 2 circles in any manner over the square, it always leaves 2 areas disjointed, not covered, which are impossible to cover with just one circle. Try it. It just won't work. As for a formal proof. That's as close as I can get.