To approach the problem of packing circles with a radius of 14 cm into a square and a hexagon, let's explore how many circles can be efficiently packed in each shape and the associated configurations.

1. Packing Circles in a Square

For a square, circles can be packed in a grid-like or staggered (hexagonal) arrangement.

• Side Length of Square: Let the side length of the square be $s$ cm.

Grid Arrangement:

• In a grid arrangement, the circles are packed directly next to each other.
• The number of circles along one side of the square: $\frac{s}{2r} = \frac{s}{28}$.
• Total number of circles: $\left(\frac{s}{28}\right)^2$.

Hexagonal Arrangement:

• The circles are packed in a staggered formation, allowing for a denser packing.
• Each row will have either $\frac{s}{28}$ circles or $\frac{s}{28} - 1$ circles (alternating).
• This arrangement allows you to fit more circles into the same area. The packing density is approximately $\frac{\pi}{\sqrt{12}} \approx 0.9069$, better than the grid's $\pi/4 \approx 0.7854$.

2. Packing Circles in a Hexagon

For a hexagon, circles can be packed more efficiently than in a square due to the shape's similarity to a circle.

• Side Length of Hexagon: Let the side length of the hexagon be $a$ cm.

Hexagonal Arrangement:

• The circles are packed in a hexagonal arrangement, which aligns with the hexagon's sides.
• For a hexagon with side length $a$, the number of circles along one side is $\frac{a}{2r} = \frac{a}{28}$.
• The total number of circles will be a combination of full and half circles fitting along the hexagon's side.

Calculations and Examples

To perform precise calculations, you'd need to specify the side lengths of the square and hexagon. Here are some general insights:

1. Hexagonal Packing in Square: If the side length of the square is an integer multiple of the diameter, the hexagonal packing will generally fit more circles.

2. Hexagonal Packing in Hexagon: This packing method is the most efficient, as it maximizes the use of space. If $a$ is a multiple of 28 cm, you can calculate the exact number of circles.

Summary:

• In a square: Hexagonal packing is more efficient.
• In a hexagon: Hexagonal packing naturally fits best due to the alignment of the circles with the hexagon's sides.

Would you like to dive into more specific calculations based on given side lengths, or have any questions?

Related Questions:

1. How does changing the radius of the circles affect packing density in different shapes?
2. What is the difference in the packing efficiency between a square and a hexagon?
3. How many circles can fit in a square with side length 100 cm?
4. How would irregular packing impact the number of circles that fit in a square?
5. What is the maximum number of circles that can be packed in a hexagon with a side length of 100 cm?

Tip: Remember, hexagonal packing generally offers the best density when arranging circles!