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.

Circle packing efficiency refers to how effectively circles can be arranged within a given shape, such as a square or a hexagon, with minimal wasted space. Let’s go through both cases one by one.

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

#### Given:
- **Radius (r) = 14 cm**
- **Diameter (D) = 2r = 28 cm**

#### Packing in a Square:
- The most straightforward packing arrangement in a square grid has circles stacked directly above and beside each other.
- The packing density for circles in a square arrangement is approximately 78.54%, meaning 21.46% of the area will be wasted.
- This percentage arises from the fact that each circle occupies an area within a square with side length equal to the circle's diameter.

#### Packing in a Hexagon:
- Circles can also be packed in a hexagonal grid arrangement, which is more efficient than a square grid.
- The packing density for circles in a hexagonal arrangement is approximately 90.69%, meaning only 9.31% of the area is wasted.
- In this case, each circle fits more tightly around its neighbors.

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

#### Given:
- **Diameter (D) = 24 inches**
- **Radius (r) = D/2 = 12 inches**

#### Packing in a Square:
- Similar to the first problem, the packing density for circles in a square arrangement is approximately 78.54%.
- Each circle occupies a square of side length equal to its diameter.

#### Packing in a Hexagon:
- The hexagonal grid arrangement still provides better packing efficiency at about 90.69%.
- Circles fit more compactly in this configuration.

### Summary of Packing Densities:
- **Square grid packing**: 78.54% (21.46% wasted space)
- **Hexagonal grid packing**: 90.69% (9.31% wasted space)

In both cases, hexagonal packing is superior in terms of efficient use of space. However, the specific area and number of circles that can fit will depend on the dimensions of the square or hexagonal container in each case.

Would you like to dive deeper into the geometry or calculations of packing? Here are some follow-up questions:

1. How do we calculate the number of circles that fit into a specific area in each configuration?
2. What are the derivations behind the packing densities of 78.54% and 90.69%?
3. How does the size of the container affect circle packing efficiency?
4. How can circle packing be applied in real-world scenarios, like packaging or materials science?
5. What are the limitations of these packing methods when irregular shapes are involved?

**Tip:** When comparing packing densities, remember that hexagonal packing always offers the best density for circle packing in a two-dimensional plane.

Explore circle packing efficiency in both square and hexagonal grids for circles with a radius of 14 cm and diameter of 24 inches. Learn about packing densities, key formulas, and how hexagonal packing offers better space utilization. Ideal for students in Grades 9-12.

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