Packing Circular Discs: Area Calculation

This solution calculates the ratio of uncovered area to total area for discs packed on an integer coordinate grid.

Analyzing the Disc Packing

We are given identical circular discs, each with a radius $r = \frac{1}{2}$. The centers of these discs are located at all integer coordinate points $(x, y)$, such as $(0,0), (1,0), (0,1), (1,1)$, and so on. Let's focus on a single unit square cell in the coordinate plane, for example, the square with vertices at $(0,0), (1,0), (1,1), (0,1)$. The area of this unit square is $A_{cell} = 1 \times 1 = 1$.

Calculating Covered Area in the Unit Cell

Within this unit square cell, portions of four discs centered at its corners contribute to the covered area. These discs are centered at $(0,0), (1,0), (0,1),$ and $(1,1)$.

• The area of one complete disc is calculated using the formula $A_{disc} = \pi r^2$. With $r = \frac{1}{2}$, the area is $A_{disc} = \pi (\frac{1}{2})^2 = \frac{\pi}{4}$.
• The portion of each disc that lies within the unit square cell is exactly a quarter-circle, as the center is at a corner and the radius is $\frac{1}{2}$.
• The area of one quarter-circle segment within the square is $\frac{1}{4} A_{disc} = \frac{1}{4} \times \frac{\pi}{4} = \frac{\pi}{16}$.
• Since four such quarter-circles (one from each of the four corner discs) lie within the unit square, the total area covered by the discs inside this cell is the sum of these four segments: $ A_{covered} = 4 \times \frac{\pi}{16} = \frac{\pi}{4} $

Finding the Uncovered Area Ratio

The total area of the unit cell is $A_{cell} = 1$. The area covered by the discs within this cell is $A_{covered} = \frac{\pi}{4}$. The area of the uncovered patches within the unit cell is the difference between the total cell area and the covered area: $ A_{uncovered} = A_{cell} - A_{covered} = 1 - \frac{\pi}{4} $ The ratio of the uncovered area to the total area is: $ \text{Ratio} = \frac{A_{uncovered}}{A_{cell}} = \frac{1 - \frac{\pi}{4}}{1} = 1 - \frac{\pi}{4} $