Circle Packing — Definition, Formula & Examples

Circle packing is the arrangement of circles within a given region so that no two circles overlap. The goal is typically to fit as many circles as possible or to cover the greatest fraction of the available area.

A circle packing is a configuration of circles in a bounded region of the plane such that the interiors of any two distinct circles are disjoint. The packing density is defined as the ratio of the total area covered by the circles to the area of the containing region. For identical circles in the plane, the densest possible packing is the hexagonal arrangement, which achieves a density of $\frac{\pi}{2\sqrt{3}} \approx 0.9069$ .

Key Formula

\eta = \frac{n \cdot \pi r^2}{A_{\text{container}}}

Where:

$\eta$ = Packing efficiency (fraction of area covered by circles)
$n$ = Number of circles packed
$r$ = Radius of each circle
$A_{\text{container}}$ = Area of the containing region

How It Works

The simplest circle packing problem asks: how many circles of radius

r

fit inside a larger shape? For a square packing (grid arrangement), circles sit in aligned rows and columns. For hexagonal packing, every other row is offset by one radius, letting circles nestle into the gaps of the row below. The hexagonal arrangement always covers more area than the square arrangement for identical circles. You calculate packing efficiency by dividing the total circle area by the container area.

Worked Example

Problem: You pack 16 circles, each with radius 1 cm, in a square grid arrangement inside an 8 cm × 8 cm square. What is the packing efficiency?

Step 1: Find the total area covered by the 16 circles.

A_{\text{circles}} = 16 \cdot \pi (1)^2 = 16\pi \approx 50.27 \text{ cm}^2

Step 2: Find the area of the square container.

A_{\text{container}} = 8 \times 8 = 64 \text{ cm}^2

Step 3: Divide to get the packing efficiency.

\eta = \frac{16\pi}{64} = \frac{\pi}{4} \approx 0.7854

Answer: The packing efficiency is about 78.5%. This matches the known density of a square grid packing, which is always

\frac{\pi}{4}

. A hexagonal arrangement of the same circles would reach roughly 90.7%.

Why It Matters

Circle packing appears in materials science (arranging atoms in crystal lattices), logistics (fitting cylindrical objects into shipping containers), and telecommunications (positioning cell towers to maximize coverage). Understanding packing density connects geometry to real optimization problems you may encounter in engineering or computer science courses.

Common Mistakes

Mistake: Assuming a square grid is the most efficient way to pack identical circles.

Correction: A hexagonal (honeycomb) arrangement is denser, covering about 90.7% of the plane versus 78.5% for a square grid. Always consider offset rows.