1. Pack the circles with radius equal to 14cm using square and hexagon.


To address your problem of packing circles with a radius of 14 cm into both square and hexagonal arrangements, let’s explore both arrangements in detail.

### 1. **Square Packing Arrangement**

In square packing, the circles are arranged such that each circle is directly aligned with its neighbor both horizontally and vertically, forming a grid of circles inside a square.

- **Diameter of each circle** = \( 2 \times \text{radius} = 2 \times 14 \, \text{cm} = 28 \, \text{cm} \)

Thus, in square packing:
- The center-to-center distance between adjacent circles = Diameter = 28 cm.
  
Let’s now determine how efficiently the area is used by the circles.

- **Area of a square cell** (the area occupied by one circle) = \( 28^2 = 784 \, \text{cm}^2 \)
- **Area of one circle** = \( \pi \times (14^2) = 615.75 \, \text{cm}^2 \)

The packing efficiency (fraction of the square area covered by circles) is:

\[
\text{Efficiency} = \frac{\text{Area of one circle}}{\text{Area of one square cell}} = \frac{615.75}{784} \approx 0.785
\]

Thus, the efficiency of the square packing is about **78.5%**.

### 2. **Hexagonal Packing Arrangement**

In hexagonal packing, each circle is surrounded by six other circles in a hexagonal pattern. This arrangement is generally more efficient than square packing.

- **Distance between centers of adjacent circles** (which is still the diameter) = 28 cm.
  
In hexagonal packing, each circle is enclosed by a hexagonal unit cell. The height of the hexagon is equivalent to two circle radii, and the width is slightly larger due to the 60-degree angles between circle centers.

- **Area of the hexagonal cell** = \( \frac{3 \sqrt{3}}{2} \times (14^2) = 509.2 \, \text{cm}^2 \)
  
- **Area of one circle** = \( 615.75 \, \text{cm}^2 \)

The packing efficiency (fraction of the hexagonal area covered by circles) is:

\[
\text{Efficiency} = \frac{\text{Area of one circle}}{\text{Area of one hexagonal cell}} = \frac{615.75}{509.2} \approx 1.21
\]

Since the hexagonal cell is smaller, the packing efficiency is higher. Theoretically, the efficiency of hexagonal packing is about **90.69%**.

### Conclusion:

- **Square Packing Efficiency**: ~78.5%
- **Hexagonal Packing Efficiency**: ~90.69%

The hexagonal arrangement is a more efficient way to pack the circles than the square arrangement, as it uses more of the available space.

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Would you like further details on these calculations or visual representations of the arrangements?

Here are five follow-up questions you might be interested in:
1. How would the packing efficiencies change with different circle sizes?
2. How do these packing methods work in three dimensions?
3. What are real-world applications of circle packing in manufacturing?
4. Can you calculate the total number of circles that can fit in a given area?
5. How do irregular-shaped containers affect packing efficiency?

**Tip:** When packing objects, the goal is often to maximize space utilization, so understanding the geometry behind packing can help optimize real-world scenarios like shipping or design layouts.

Pack the circles with radius equal to 14cm using square and hexagon.

Explore how to pack circles with a 14 cm radius using both square and hexagonal arrangements. Learn the efficiency calculations, geometry concepts, and formulas used in determining the optimal packing methods. Suitable for high school students (Grades 10-12).

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