Question

Arrange four quarter coins touching each other as a 2-D square lattice. Calculate the area of the air-space (dead-space). (Ignore the thickness of the quarters.) Repeat the problem for a triangular lattice.

Answer 1

The radius of a quarter is, 12.13 mm

In the first case for square lattice, the empty space inside the black square is the dead space.

So as the radius of the quarter is 12.13 mm, the length of the side of the black square is, 2x12.13mm=24.26mm=2.426cm

Now the area of the square is,

Now area covered by the circles is,

So the dead area is,

Now for the second case of the triangular lattice,

The inner black triangle is an equilateral triangle with side 2.426cm and the angles are 60degree

So the fraction of a quarter that is inside the triangle is 60/360=1/6

So there is total 3x1/6=1/2 quarter inside the triangle.

Now the area of the triangle is,

and area of the circles inside the triangle is,

So the dead space is,