Shortest Paths on a Cube

Transcript

Shortest Paths on a CubeWe know that the shortest path between two points in a flat surface is the straight line between them. But, what about in a 3D figure like the cube? Let’s take an example What is the shortest path for the ant to reach the laddu? The shortest path is The ant walks diagonally across the side wall, hits the top edge at an angle, and walks diagonally to the laddu. Let’s unfold the box (into a net) and see if we walked a straight line If we walked a straight line, then it is the shortest distance When you unfold the box flat, the red path becomes a perfectly straight line. Because it is a straight line on the flat net, it is mathematically guaranteed to be the shortest possible distance. Similarly, let’s do another question What is the shortest path for the ant to reach the laddu? Here, the shortest path is The ant walks straight up to the edge, turns exactly 90 degrees, and walks straight to the laddu. Like this Let’s unfold the box (into a net) and see if we walked a straight line The blue path, when unfolded, looks like a bent "V" shape. Since it's bent, it means the ant is walking farther than it needs to. That means, it is not the shortest distance Let’s do it again Hence, red path is the shortest distance Thus, by using a net, we convert the problem of finding the shortest path on a cuboid to the problem of finding the shortest path on the net. Let’s do some questions