Shortest Path on a Cylinder (Unfolded Net Method)

Grades 10-11

Question: Solid geometry: shortest path on an unfolded cylinder

To find the shortest path between two points on a cylinder's surface, we need to "unfold" its lateral surface into a flat shape. Here's the core mathematical logic:

1. The Unfolding Principle

A cylinder's lateral surface is curved, but geometrically it's a "developable surface". If we cut it along one straight line (a generator) and lay it flat, it becomes a rectangle.

The rectangle's height equals the cylinder's height $h$ .
The rectangle's width equals the base circle's circumference $C = 2\pi r$ .

2. Mapping the Path Between Two Points

Suppose start point

A

lies on the bottom circle and end point

B

lies on the top circle, directly opposite

A

(i.e., halfway around the cylinder):

In 3D space, this path looks like a helix spiraling up the cylinder (the red curve).
On the unfolded rectangular grid, the positions of A and B change:
- The horizontal distance becomes half the base circumference, $\pi r$ .
- The vertical distance is still the cylinder's height, $h$ .

3. Computing the Shortest Path

Since "a straight line is the shortest distance between two points" in plane geometry, the straight segment connecting

A

and

B

on the unfolded diagram is the shortest path. Using the Pythagorean theorem, we can compute the path length

L

\begin{align*} L^2 &= (\text{horizontal displacement})^2 + (\text{vertical displacement})^2 \\ L &= \sqrt{(\pi r)^2 + h^2} \end{align*}

Diagram Key

Left, 3D model (blue): Shows the cylinder and the helical path on its surface. Drag the rotation slider to see the relative positions of $A$ and $B$ .
Right, unfolded diagram (yellow): Shows the rectangle formed by flattening the lateral surface. The red straight segment makes the nature of the shortest path immediately visible.
Key points $A$ and $B$ : In the 3D view they sit on opposite sides of the cylinder; in the unfolded view they're connected across the rectangle's diagonal.

Looking at the unfolded diagram, you can see that any path that looks complicated on a curved surface reduces to the simplest possible straight-line logic once you "flatten" it.