A pool rack is an equilateral triangle, filled with 15 equal-sized balls. Seen from above, we'd see a triangle, with 15 circles within.
Imagine we used smaller and smaller balls. The more the balls, more area of the triangle would be covered.
In the limit, with infinite balls, would all of the triangle be covered?
(In reply to
re(3): Solution by Richard)
Yes, but in my opinion, that is easier to verify it oneself with paper/pencil than it is to follow someone else's explanation with html text. I think Bractal is justified in leaving the steps out.
If you are asking because you don't see how to actually derive it, take the centers of the three corner circles, and draw the (smaller) equilateral triangle with vertices at those points. Extend the sides of this smaller triangle to meet the sides of the larger triangle. Drop perpendiculars from the centers of two of these corner circles to the same edge of the outer triangle.
That divides the outer triangle's side length into 5 segments. Compute each with basic trig and sum.
re(4): Solution
