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### Finding the Length of a Coil

```
Date: 07/07/99 at 14:19:09
From: Robert Johnson
Subject: Finding the length of a coil

I am looking for a formula to figure the length of a coil or spring
when given the diameter and pitch and number of turns.
```

```
Date: 07/07/99 at 17:32:09
From: Doctor Rick
Subject: Re: Finding the length of a coil

Hi, Robert.

Imagine that you take a paper triangle like this:

/|
/       |
/              |
L        /                     |
/                             | P*N
/                                    |
/|                                          |
/       P|                                          |
/_________pi*D____|__________________________________________|
pi*D*N

L = length of coil
D = diameter of coil
N = number of turns
P = pitch

Wrap it around a cylinder the diameter of the coil. It will wrap
around exactly N times, since Pi*D is the circumference of the
cylinder. At each turn, the diagonal edge (hypotenuse) of the triangle
is a distance P farther along the cylinder than on the previous turn;
this is the pitch. The hypotenuse follows the path of the coil, so its
length L is the length of the coil. You can find it from the
Pythagorean theorem:

L^2 = (pi*D*N)^2 + (P*N)^2
= N^2 * (pi*D)^2 + P^2)

L = N*sqrt((pi*D)^2 + P^2)

where sqrt() means the square root of the expression in parentheses,
and P^2 means P squared.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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