Shortest path on a cuboid

The idea

Paths on a cuboid's surface become paths of the same length on its net, so the shortest path is a straight line on a net that stays inside it; different unfoldings give different lengths, so all must be compared.

A hungry ant

A hungry ant is on a cuboid and there is a laddu on the other side of it. The ant must walk on the surface: it cannot fly or dig through the box.

How can we find the shortest path for the ant?

• Measure every possible path with a thread
• Unfold the cuboid and draw a straight line
• We can never be sure

Unfold the box

The ant is on the top of a glass box and the laddu is on the bottom. Tap a way to unfold the faces the ant walks over. The red line on the flat net is the path. Drag the box to turn it.

Unfold, then draw a straight line

A path on the surface of a cuboid becomes a path of the same length on its net. The shortest path is a straight line on a net, provided the line stays inside the net. Different unfoldings give different lengths, so we must compare all of them.

In Way 2 the straight line is the hypotenuse of a right triangle with legs 24 and 32. By the Baudhayana Theorem, d² = 24² + 32² = 1600, so d = 40 cm.

• Way | Faces the ant crosses | Straight line
• 1 | top, front, bottom | 53.7 cm
• 2 | top, back, bottom | 40 cm
• 3 | top, left, bottom | 52.2 cm
• 4 | top, right, bottom | 60.1 cm
• 5 | top, right, back, bottom | outside the net

Notes

To find the shortest path between two points on a cuboid, unfold it into a net and draw a straight line. A path on the cuboid and its picture on the net have the same length. The line must stay inside the net, and different unfoldings give different lengths, so compare all the unfoldings.

Check yourself

On a net, the straight line from the ant to the laddu is the hypotenuse of a right triangle with legs 9 cm and 12 cm. How long is this path?

Answer: 15 cm

d² = 9² + 12² = 81 + 144 = 225, so d = 15 cm.

On this unfolding, part of the straight line from the ant to the laddu is outside the net. What does that tell us?

An ant walks a path 52 cm long on the surface of a cuboid. The faces are unfolded into a net and the path is drawn on it. How long is the path on the net?

Ravi unfolds a cuboid in just one way. The straight line stays inside the net and is 53.7 cm long. He says, "This is the shortest path." What is wrong?

• It is a path on the cuboid, but a long one. Outside the net there is no cuboid, so the ant cannot walk there. This is not a path at all.
• It is the shortest path, because a straight line is always the shortest. A straight line is shortest on a flat page, but this line is not a path on the cuboid. Part of it is not on any face.
• It is not a path on the cuboid, so we must unfold the cuboid in another way — correct. Yes! The line must stay inside the net. Here it does not, so we unfold the box differently.
• 52 cm: unfolding does not change lengths — correct. Yes! A path on the cuboid and its picture on the net have the same length. Unfolding only turns the faces flat.
• Less than 52 cm, because the net is flat. Flattening does not shrink anything. The faces keep their sizes and the path keeps its length.
• More than 52 cm, because the faces are spread out. Spreading the faces out does not stretch them. Every piece of the path keeps its length.
• A straight line inside the net is not a path on the cuboid. It is: a line inside the net is a real path of the same length on the cuboid.
• Another unfolding may give a shorter path, so he must compare all the unfoldings — correct. Yes! In the toy, Way 1 gave 53.7 cm, but Way 2 gave only 40 cm.
• A path on a net cannot be measured. It can: it is a straight line, and we can measure it or use the Baudhayana Theorem.