[Deejax]

It is a bit difficult to explain without drawings.

Look at a layer of coins. The closest you can get them together is by making rows of coins and placing them next to each other. Start with a horizontal row of coins. Make a second row above it, displaced half a coin to the right (or left). And so on. You get a layer of coins in alternating rows, each row displaced half a coin compared to the previous. If you were to draw lines between the centers of neighbouring coins you would get triangles. (i do not know the correct term, but the are triangles with 60 degree angles and equal side lengths.).

Now we need to draw a grid over the coins. Do this by drawing paralel horizontal lines through the centers of the coins. To draw the vertical lines take a single row and draw vertical line through the centers of every coin in that row. If you've done this correctly the vertical lines will not run through coins in the rows directly above and below this row.

If you look at the grid. every square should contain two quarters and one half coin, totalling one coin.

If we set the radius of the coins to 1, the calculation becomes easy.

Every square contains one coin.

The surface of a coin is Pi*Radius² = Pi*1² = Pi

The side of a square is two times the radius of a coin = 2. The total surface of a square is then 2*2 = 4

This means that the fraction of the surface covered with coin is Pi/4 = approx 78.5%


I hope you understand this. I don't know of a way to draw at IW. Would make it a LOT easier.