3D Container Packing

Video coming soon2 worked examples

Theory

The Golden Rule: Never calculate the total volume of the large container and divide it by the total volume of the small box. This assumes the items can be melted down like liquid! You must divide the Length by Length, Breadth by Breadth, and Height by Height.

1. The Packing Strategy

To find how many small boxes fit into a large container, you must assign items systematically to minimise unused space.

Find how many boxes fit along the length, breadth, and height independently.
Multiply these three numbers together to find the total for that orientation.

2. Orientations and Restrictions

If the question allows the boxes to be rotated, you must test at least two different arrangements (orientations) of the length and breadth to see which one fits more boxes inside. However, if a box has a "THIS WAY UP" symbol or must be placed with a "label facing upwards", the height of the small box is permanently locked and cannot be rotated.

⚠️ Common Examiner Traps

The "One Orientation Only" Trap: If there are no restrictions on how a box is packed, you must calculate multiple orientations. Many candidates lose marks by only attempting to find the number of boxes for a single orientation.
The "Unit Mismatch" Trap: The internal dimensions of a shipping crate are often given in metres (e.g., 4.10 m), while the small boxes are given in centimetres (e.g., 60 cm). You must convert the metres into centimetres before dividing.

Worked examples

Example 1

Tabitha is a van driver. Orders are packed into boxes.

The van has internal dimensions: 410 cm (L) $\times$ 190 cm (B) $\times$ 220 cm (H).
The boxes have dimensions: 60 cm $\times$ 40 cm $\times$ 20 cm.
The boxes are marked "THIS WAY UP" on the 20 cm vertical side.
All boxes must be aligned in the same direction.

Calculate the maximum number of boxes that will fit in the van. (3 marks)

Step 1: Lock the height. Because of the "THIS WAY UP" rule, the 20 cm side must point upwards.

220 \text{ cm (Van H)} \div 20 \text{ cm (Box H)} = 11 \text{ layers high}

Step 2 (Orientation 1): Fit the 60x40 base into the 410x190 van base.

Length: $410 \div 60 = 6.83 \rightarrow 6 \text{ boxes}$
Breadth: $190 \div 40 = 4.75 \rightarrow 4 \text{ boxes}$
Boxes per layer = $6 \times 4 = 24 \text{ boxes}$

Step 3 (Orientation 2): Rotate the base to fit 40x60 into 410x190.

Length: $410 \div 40 = 10.25 \rightarrow 10 \text{ boxes}$
Breadth: $190 \div 60 = 3.16 \rightarrow 3 \text{ boxes}$
Boxes per layer = $10 \times 3 = 30 \text{ boxes}$

Step 4: Choose the best orientation and multiply by the number of layers. Orientation 2 is better (30 per layer).

30 \times 11 = 330 \text{ boxes maximum}

Example 2

No Restriction (test every orientation)

A storage crate has internal dimensions 100 cm (L) $\times$ 70 cm (B) $\times$ 50 cm (H). Cartons measuring 30 cm $\times$ 20 cm $\times$ 20 cm may be packed in any orientation. Calculate the maximum number of cartons that will fit.

With no restriction, try each way of standing the carton and keep the best. Divide each dimension and round down every time.

Orientation A (30 along L, 20 along B, 20 up):

\lfloor 100/30 \rfloor \times \lfloor 70/20 \rfloor \times \lfloor 50/20 \rfloor = 3 \times 3 \times 2 = 18

Orientation B (20 along L, 30 along B, 20 up):

\lfloor 100/20 \rfloor \times \lfloor 70/30 \rfloor \times \lfloor 50/20 \rfloor = 5 \times 2 \times 2 = 20

Orientation C (20 along L, 20 along B, 30 up):

\lfloor 100/20 \rfloor \times \lfloor 70/20 \rfloor \times \lfloor 50/30 \rfloor = 5 \times 3 \times 1 = 15

Answer: Orientation B fits the most, so the maximum is 20 cartons.

Ready to practise?

10 3D Container Packing questions with answers and video solutions