Have we now completely analysed the problem of finding the shortest path between two points on a cuboid?

Question

Homi · Accepted Answer

Understand the Question

To find the shortest path between two points on the surface of a 3D cuboid (without moving through the inside), we unfold the 3D surface into a flat 2D net.
On a flat plane, the shortest path between any two points is a straight line, whose length can be found using the Pythagoras theorem.
Because a cuboid can be unfolded in multiple ways, we must test each relevant unfolding and choose the path with the smallest length.

Step 1 · The Unfolding Method

Since the path must remain entirely on the cuboid's surface, we flatten the 3D shape into a 2D net. On a flat 2D surface, the shortest distance between two points is a straight line. Diagram 1

Step 2 · Paths Between Adjacent Faces

When point $A$ is on one face (e.g., top face) and point $B$ is on an adjacent face (e.g., side face), unfolding the two adjacent faces creates a larger rectangle. The straight-line segment joining $A$ and $B$ across this rectangle gives the shortest path for that specific unfolding. Diagram 2

Step 3 · Paths Between Opposite Faces

When points $A$ and $B$ lie on opposite faces (e.g., top and bottom), the path must cross one or more intermediate faces (front, back, left, or right). Each distinct unfolding produces a different candidate straight-line path. Diagram 3

Step 4 · Complete Analysis and Comparison

A complete analysis requires evaluating the straight-line length using the Pythagoras theorem for all possible relevant unfoldings and selecting the minimum value:

$\text{Shortest Path} = \min(\text{Path}_1, \text{Path}_2, \dots, \text{Path}_n)$

Answer

Yes, the problem is completely analysed by systematically unfolding the cuboid into all possible 2D nets, finding the straight-line distance for each using the Pythagoras theorem, and selecting the minimum length among them.

Common Mistakes

Single Net Assumption: Assuming there is only one way to unfold the faces between two points. A cuboid has multiple unfolding configurations, and the direct line on one net might be longer than on another.
Measuring Across 3D Space: Measuring the direct Euclidean distance through the interior of the cuboid rather than along the unfolded surface.

Question 26

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