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

2. Pack the circles with diameter equal to 24 inches using square and hexagon.

When packing circles in a square or hexagonal pattern, the goal is to fit as many circles as possible into a given area efficiently. Let's explore each case separately for square and hexagonal packing.

### 1. **Pack Circles with Radius 14 cm using Square and Hexagonal Patterns**

**Given:**
- Circle radius: \( r = 14 \) cm
- Circle diameter: \( d = 2r = 28 \) cm

#### (a) **Square Packing**

In a square packing arrangement, circles are arranged such that the centers form a square grid. The distance between the centers of two adjacent circles is equal to the diameter of the circles.

The packing efficiency (fraction of the area occupied by the circles) is lower in square packing compared to hexagonal packing. The efficiency for square packing is about **78.54%**.

For an area covered by a square pattern:

\[
\text{Area of a single square grid cell} = d^2 = 28^2 = 784 \, \text{cm}^2
\]
\[
\text{Area of one circle} = \pi r^2 = \pi (14^2) = 615.75 \, \text{cm}^2
\]
\[
\text{Efficiency} = \frac{\text{Area of the circle}}{\text{Area of the square grid cell}} \approx \frac{615.75}{784} \approx 0.7854
\]

#### (b) **Hexagonal Packing**

In hexagonal packing, circles are arranged such that their centers form a hexagonal grid. This pattern is denser, and the packing efficiency is about **90.69%**.

The distance between the centers of adjacent circles is still the diameter, but circles are arranged so that they form equilateral triangles.

For an area covered by a hexagonal pattern:

\[
\text{Efficiency} = 0.9069
\]

---

### 2. **Pack Circles with Diameter 24 Inches using Square and Hexagonal Patterns**

**Given:**
- Circle diameter: \( d = 24 \) inches
- Circle radius: \( r = \frac{24}{2} = 12 \) inches

#### (a) **Square Packing**

In square packing:

\[
\text{Area of one square grid cell} = d^2 = 24^2 = 576 \, \text{square inches}
\]
\[
\text{Area of one circle} = \pi r^2 = \pi (12^2) = 452.39 \, \text{square inches}
\]
\[
\text{Efficiency} = \frac{\text{Area of the circle}}{\text{Area of the square grid cell}} \approx \frac{452.39}{576} \approx 0.7854
\]

#### (b) **Hexagonal Packing**

In hexagonal packing:

\[
\text{Efficiency} = 0.9069
\]

---

### Summary of Efficiencies
1. **For circles with radius 14 cm:**
   - Square packing efficiency ≈ 78.54%
   - Hexagonal packing efficiency ≈ 90.69%

2. **For circles with diameter 24 inches:**
   - Square packing efficiency ≈ 78.54%
   - Hexagonal packing efficiency ≈ 90.69%

### Next steps:
- Do you need more detailed calculations for a specific packing arrangement?
- Should I proceed with visualizing the packing patterns?

---

**Relative Questions:**
1. How does circle size impact packing density in larger grids?
2. How would irregular circle sizes affect the efficiency of packing in these patterns?
3. Could you apply the same packing principles to spheres in 3D space?
4. How does packing efficiency change with non-circular shapes, like ellipses?
5. What are real-world applications of these packing methods?

**Tip:** Hexagonal packing always maximizes the use of space compared to square packing for circular objects.

1. Pack the circles with radius equal to 14 cm using square and hexagon. 2. Pack the circles with diameter equal to 24 inches using square and hexagon.

Explore the efficiency of packing circles with a radius of 14 cm and a diameter of 24 inches in both square and hexagonal patterns. Understand key geometric concepts and formulas, such as area calculations, packing efficiency, and the Hexagonal Packing Theorem. Suitable for high school students studying advanced geometry.

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