Circles Within Rectangle

Overview

Engineering reference data for Circles Within Rectangle in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Coordinate Table

The table below shows a sample of the circle center coordinates for the default rectangular grid pattern. The full pattern consists of 20×20 = 400 circles.

Packing Efficiency

The ratio of the total area of the circles to the area of the container rectangle is a key metric for packing efficiency.

$$\text{Area Ratio} = \frac{n \cdot \frac{\pi d^2}{4}}{w \cdot h} \times 100\%$$

For the default example ($n=400$, $d=0.5$ in, $w=h=10$ in), the efficiency is:

$$\text{Area Ratio} = \frac{400 \cdot \frac{\pi \cdot (0.5)^2}{4}}{10 \cdot 10} \times 100\% = 78.5\%$$

Note: This ratio does not equal the maximum packing density in the geometric sense, as it is based on the total container area, not the area optimally covered. The rectangular grid pattern shown has a theoretical maximum efficiency of approximately 78.54% ($\frac{\pi}{4}$) for circles of equal size with no spacing.

Hexagonal (Staggered) Packing

For denser packing, circles can be arranged in a hexagonal pattern, where rows are offset by half a circle diameter. This pattern generally allows more circles to fit within a given rectangle compared to the simple rectangular grid. The calculator provided uses a rectangular grid, but for engineering applications like optimal pipe routing, a hexagonal layout should be considered.

Interactive Charts

References

• Original Source