Exploring Some Geometric Themes · Lesson 8 of 13
Shortest Paths on a Cube and Cuboid
“Unfold a box surface to find straight routes, check their validity, and compare their lengths.”
Distinguish a surface route from a line passing through a solid. Use an unfolding to turn a route across faces into a plane problem. Calculate candidate lengths using right triangles. Check that a candidate stays on the net and compare different unfoldings.
An ant on a box wants to reach a laddu stuck to another face. It must crawl along the surface. It cannot take the straight line through the inside of the box, even if that would be the shortest journey through space.
The useful idea is to open the faces while keeping their sizes unchanged. A difficult-looking surface route becomes a line on a flat sheet. We can then draw, measure, and compare it using geometry we already know.
From Surface to Plane
Suppose the ant crosses from one flat face onto a neighbouring face. Rotate one face about the shared edge until the two lie in the same plane. This changes the angle between the faces, but it does not stretch the surface. Distances measured along a route drawn on it stay the same.
On the flattened pair of faces, a straight segment is the shortest route between its endpoints, provided it stays inside those faces. When you fold the model back, that segment becomes a surface path that bends in space at the shared edge. It need not look straight in the original perspective drawing.
Opening the faces crossed by a route into a plane while preserving distances on each face and keeping the required face joins connected.
Problem
In a valid two-face unfolding, the points are 4 cm apart perpendicular to the fold and 3 cm apart parallel to it. Find the shortest route within this unfolding.
- 1.The two perpendicular separations form the legs of a right triangle on the unfolded plane.
- 2.If the route length is d, then d² = 4² + 3² = 16 + 9 = 25.
- 3.Taking the positive square root gives d = 5 cm.
- 4.Check that the segment stays inside the two face rectangles and crosses their shared edge. It is the shortest route among paths using this unfolded region.
The Way We Unfold Matters
A full box net may contain gaps between parts of its outline. A line joining two marked points can cross such a gap. That line would pass over missing paper, so it cannot represent a continuous crawl on the box.
Changing the net changes where one endpoint appears relative to the other. The physical endpoints remain fixed on their original faces; only the way the faces open changes. Keep labels on both endpoints and on shared edges so that you do not accidentally compare routes to different points.
Straightness is not the only test. The segment must stay within the chosen face arrangement and connect the correct images of the physical endpoints. A diagonal drawn across empty space in a net is not a surface route.
Even when a segment is valid, a different sequence of faces may give a shorter one. Therefore distinguish two statements: “shortest within this unfolding” and “shortest among all allowed surface routes.” To establish the second, compare the relevant different unfoldings and check their validity.
Problem
For the same two fixed points, one valid unfolding gives a 42 cm straight segment. Another valid unfolding gives perpendicular separations of 24 cm and 32 cm. Which is shorter?
- 1.For the second route, use d² = 24² + 32².
- 2.Calculate 24² = 576 and 32² = 1024, so d² = 1600 and d = 40 cm.
- 3.Compare the two valid routes: 40 cm is shorter than 42 cm by 2 cm.
- 4.This comparison rules out the 42 cm route as the shortest. A claim of a global minimum still requires checking the other possible face sequences.
Opposite Vertices of a Cuboid
For opposite vertices of a rectangular cuboid, symmetry and the three edge directions give three standard candidate unfoldings. If the edge lengths are a, b, and c, two dimensions are added together when adjacent faces open into a rectangle; the third dimension forms its other side.
The three resulting rectangles have dimensions (a + b) by c, (a + c) by b, and (b + c) by a. Their diagonals connect opposite vertices on suitable unfolded face pairs. Compare the three diagonal lengths to obtain the shortest surface path in this opposite-vertex case. This specialised rule should not be applied unchanged to arbitrary points inside faces.
Problem
A cuboid measures 8 cm by 4 cm by 2 cm. Find the shortest surface distance between opposite vertices.
- 1.Combine 8 and 4: the first squared distance is 12² + 2² = 148.
- 2.Combine 8 and 2: the second squared distance is 10² + 4² = 116.
- 3.Combine 4 and 2: the third squared distance is 6² + 8² = 100.
- 4.The smallest squared distance is 100, so the shortest length is √100 = 10 cm. Comparing squares is enough because all lengths are nonnegative.
- 5.A path along three box edges would measure 8 + 4 + 2 = 14 cm, so restricting the ant to edges would miss the shorter face-crossing route.
For a cube of side s, the three candidates are equal: each is the diagonal of a rectangle 2s by s. The surface distance between opposite vertices is s√5. The straight line through the cube’s interior is s√3, which is shorter but unavailable to a surface traveller.
A Method You Can Reuse
Begin with the physical object and label the endpoints clearly. Choose a sequence of adjacent faces and unfold those faces. Locate the endpoints from their distances to the face edges, then draw the segment. A calculation only becomes meaningful after this geometric setup is correct.
For each candidate, record the dimensions used, its length, and whether it stays on the surface. This makes a comparison explainable. If some face sequences remain unchecked, state the result as the shortest route found so far rather than claiming a complete proof.
Trace a Route and Fold It Back
A paper cuboid can show why the unfolded line is useful. Make the model before drawing a route so that the face labels are unambiguous.
- Label two opposite vertices of a small cuboid and open it into a suitable net.
- Draw a straight candidate route and check that it stays within the relevant faces.
- Fold the model and follow the drawn route with a piece of thread.
- Compare it with a route along the edges and with a route from a different unfolding.
The route can look bent on the solid while remaining straight on the net. Folding preserves the surface length, so the flat calculation still describes the thread’s journey across the faces.
Quiz
Why can we calculate a surface route after unfolding?
What invalidates a straight candidate in a chosen net?
A valid unfolding gives perpendicular separations 24 and 32 cm. What is its route length?
What is the shortest opposite-vertex surface distance on an 8 by 4 by 2 cm cuboid?
What must follow finding a short straight route in one valid unfolding?
Practice Problems
- Find a valid unfolded route length with perpendicular separations 5 cm and 12 cm.
- Find the shortest opposite-vertex surface distance for a cube of side 6 cm.
- Compare opposite-vertex routes on a cuboid of sides 3, 4, and 6 cm.
- A line of length 9 cm leaves the chosen net; another of length 10 cm stays on its faces. Which is a usable candidate?
- Explain why the space diagonal of a solid box is not the answer to an ant’s surface-path problem.
1. Use d² = 25 + 144 = 169. 2. Therefore d = 13 cm. 3. This establishes the length for the stated valid unfolding.
Key Takeaways
Unfolding changes face angles while preserving surface distances. A straight segment is shortest within a valid unfolded region. A segment crossing missing parts of a net is invalid. Compare different face sequences before claiming the overall shortest path. For opposite cuboid vertices, compare the three standard unfolded rectangle diagonals.