Circle Packing Density Calculator

Formula

The formula to calculate the Circle Packing Density (D) is:

$D = \frac{N \times A_c}{A_t}$

Where:

$D$ is the density
$N$ is the number of circles
$A_c$ is the area of one circle
$A_t$ is the area of the container

What is Circle Packing Density?

Circle packing density refers to the proportion of a given area that is occupied by circles when they are packed together as closely as possible. This concept is often used in various fields such as material science, logistics, and mathematics to optimize space and resource utilization. The density is a measure of how efficiently the circles fill the container area, and it is expressed as a ratio or percentage. Higher density values indicate more efficient packing, while lower values suggest that there is more unused space within the container.

Example

Let's say the number of circles (N) is 10, the area of one circle (A_c) is 3 square units, and the area of the container (A_t) is 50 square units. Using the formula:

$D = \frac{10 \times 3}{50} = 0.6$

So, the circle packing density (D) is 0.6 or 60%.

How to Calculate Packing Density

Formula: $\eta = \frac{V_{packed}}{V_{total}}$

$\eta$ : Packing density
$V_{packed}$ : Volume of packed objects
$V_{total}$ : Total volume

Example: $\eta = \frac{500}{1000}$

Volume of packed objects: 500 cm³
Total volume: 1000 cm³

Linear Packing Density Formula

Formula: $\eta = \frac{L_{packed}}{L_{total}}$

$\eta$ : Linear packing density
$L_{packed}$ : Length of packed objects
$L_{total}$ : Total length

Example: $\eta = \frac{30}{50}$

Length of packed objects: 30 cm
Total length: 50 cm

Density of a Circle

Formula: $\rho = \frac{m}{A}$

$\rho$ : Density
$m$ : Mass
$A$ : Area

Example: $\rho = \frac{10}{3.14}$

Mass: 10 grams
Area: 3.14 cm²

Packing Density of Spheres

Formula: $\eta = \frac{\pi}{3\sqrt{2}}$

$\eta$ : Packing density

Example: $\eta = \frac{3.14}{3\sqrt{2}}$

Packing Density vs Bulk Density

Formula: $\eta = \frac{\rho_{packed}}{\rho_{bulk}}$

$\eta$ : Packing density
$\rho_{packed}$ : Density of packed objects
$\rho_{bulk}$ : Bulk density

Example: $\eta = \frac{2.5}{1.2}$

Density of packed objects: 2.5 g/cm³
Bulk density: 1.2 g/cm³

Circle Packing in Rectangle Calculator

Formula: $\eta = \frac{n \cdot A_{circle}}{A_{rectangle}}$

$\eta$ : Packing density
$n$ : Number of circles
$A_{circle}$ : Area of one circle
$A_{rectangle}$ : Area of the rectangle

Example: $\eta = \frac{5 \cdot 3.14}{20}$

Number of circles: 5
Area of one circle: 3.14 cm²
Area of the rectangle: 20 cm²

Packing Efficiency of Circles

Formula: $\eta = \frac{A_{packed}}{A_{total}}$

$\eta$ : Packing efficiency
$A_{packed}$ : Area of packed circles
$A_{total}$ : Total area

Example: $\eta = \frac{15}{25}$

Area of packed circles: 15 cm²
Total area: 25 cm²