Mathematics Project on Efficiency in Packaging
Efficiency in Packaging
Hexagonal Packing
Objective:
To investigate the efficiency of packing of objects of different shapes in a cuboid box.
Efficiency is the percentage of box space occupied by the objects.
Description
- Took a certain number of cylindrical tins and packed them in a cuboid container.
- (a) For illustration I took 81 tins.
- (b) Second time I took 64 tins.
- © Third time I took 49 tins.
- The cylindrical tins can be placed in two different ways. These are:
- (a) Square packing
- (b) Hexagonal packing
- I wished to study which packing out of two is more efficient.
- To understand the difference between the two packing I have drawn figures on the left side pages.
Example 1 (81 Tins) - Calculation
Case 1: Square Packing
Each base circle is circumscribed by a square.
- Area of one circle = π R²
- Area of square = 4 R²
- Area of circle / area of square = π R² / 4 R²
This ratio will be evidently the same as the cross section of all the tins to the total base area.
Percentage efficiency = π / 4 × 100 = 78.5%
Therefore the efficiency in case of square packing is 78.5%.
Case 2: Hexagonal Packing
Here we determine the sides of the base of the container in terms of the radius of the cylindrical tin.
One side of the rectangular base i.e. BC = 18 × R.
To determine the other side, AB = 2 × R + 9 × h, where h is the altitude of the equilateral triangle formed by joining the centres of three adjacent circles.
h = 2R sin 60°
AB = 2R + 18R sin 60°
sin 60° = √3 / 2
Now AB = 2R + 18R × √3 / 2 = (2 + 9√3) R
Area of ABCD = 18R × (2 + 9√3) R = 18R² (2 + 9√3)
Percentage efficiency = 81π R² × 100 / 18R² (2 + 9√3) = 80.3%
Example 2 (64 Tins) - Calculation
Case 1: Square Packing
Each base circle is circumscribed by a square.
- Area of one circle = π R²
- Area of square = 4 R²
- Area of circle / area of square = π R² / 4 R²
This ratio will be evidently the same as the cross section of all the tins to the total base area.
Percentage efficiency = π / 4 × 100 = 78.5%
Therefore the efficiency in case of square packing is 78.5%.
Case 2: Hexagonal Packing
Here we determine the sides of the base of the container in terms of the radius of the cylindrical tin.
One side of the rectangular base i.e. BC = 16 × R.
To determine the other side, AB = 2 × R + 8 × h, where h is the altitude of the equilateral triangle formed by joining the centres of three adjacent circles.
h = 2R sin 60°
AB = 2R + 16R sin 60°
but sin 60° = √3 / 2
Now AB = 2R + 16R × √3 / 2 = 2R + 8R√3 = (1 + 4√3) 2R
Area of ABCD = 16R × (1 + 4√3) 2R = 32R² (1 + 4√3)
Percentage efficiency = 64π R² × 100 / 32R² (1 + 4√3) = 79.3%
Example 3 (49 Tins) - Calculation
Case 1: Square Packing
Each base circle is circumscribed by a square.
- Area of one circle = π R²
- Area of square = 4 R²
- Area of circle / area of square = π R² / 4 R²
This ratio will be evidently the same as the cross section of all the tins to the total base area.
Percentage efficiency = π / 4 × 100 = 78.5%
Therefore the efficiency in case of square packing is 78.5%.
Case 2: Hexagonal Packing
Here we determine the sides of the base of the container in terms of the radius of the cylindrical tin.
One side of the rectangular base i.e. BC = 14 × R.
To determine the other side, AB = 2 × R + 7 × h, where h is the altitude of the equilateral triangle formed by joining the centres of three adjacent circles.
h = 2R sin 60°
AB = 2R + 14R sin 60°
but sin 60° = √3 / 2
Now AB = 2R + 14R × √3 / 2 = 2R + 7R√3 = (2 + 7√3) R
Area of ABCD = 14R × (2 + 7√3) R = 14R² (2 + 7√3)
Percentage efficiency = 49π R² × 100 / 14R² (2 + 7√3) = 77.96%
Remarks
- In the calculations here the number of tins was fixed and the cuboid dimensions are variable. A similar exercise may be done with fixed cuboid dimensions and variable number of tins.
- We can also determine the efficiency for packing of spheres in a cuboid.
- Volume of sphere = 4⁄3 π R³
- Volume of cube = 8 R³
- Percentage efficiency = 4π R³ / (3 × 8 R³) = π / 6 = 52%
Note
When 81 and 64 tins were taken, hexagonal packing was more efficient, but in case of 49 tins, square packing was more efficient.