How Long is a Piece of String?
Geometry & Trigonometry
Unroll a helix of wound string into a flat triangle and calculate the total length used to wind a real spool, ball or coil.
Introduction
String, thread, cable and rope are often wound in a helix around a cylindrical core, and the total length used is larger than you would guess from just the core's height. Unrolling one turn of the helix into a flat right-angled triangle turns the problem into ordinary trigonometry: the string's length in one turn is the hypotenuse of a triangle whose sides are the circumference of the core and the height gained in that turn. This exploration finds the total string length for a real wound object, and asks how the answer changes as each new layer sits on a slightly larger radius.
Guiding Questions
- String wound around a cylinder makes a helix. How long is one complete turn? Unroll it into a triangle.
- Find the total string length for a real wound object and check by unwinding and measuring.
- How does the answer change as the core radius grows with each layer?
- Connect your result to a spiral ramp or a screw thread. What else does the same mathematics describe?