Helix / Spring Volume

Aug 2007
Hello, I need some help learning how to calculate the volume of a Spring. The only resource i can find is on wikipedia, but i am having trouble understanding it.

Please see : Spring (mathematics) - Wikipedia, the free encyclopedia

This is the forumula they give.

R is the distance from the center of the tube to the center of the helix,
r is the radius of the tube,
P is the speed of the movement along the z axis.
Ok, the helix will be 400 wide, so R = 200 units.
The radius of the tube will be 30, so r = 30 units.

Now this is where im stuck, they ask for P for the speed it moves upwards / downwards. How should i get this? For example if i want my spring to have 2 complete revolutions and have a total height of 500 units, how should i know what P is?

I also need to know n, which is not explained very well what n is on the page.

Many thanks if you can help me understand.


MHF Hall of Honor
Jul 2006
Chaneysville, PA
Your helix has radius 200 and the tubing as radius 30. Correct?. This conflicts with R which says they're both 200.

In this event R+r=230. That's what I'll use.

Assume you're wrapping a tube of radius 30 around a helix of radius 200,

This forms a helix. A helix can be represented by \(\displaystyle x=acos(t), \;\ y=asin(t), \;\ z=ct\)

a in this case is R+r=230. The radius of the helix + radius of tubing.

Since you want two revolutions when z=500 units, you have \(\displaystyle c=\frac{500}{4\pi}=\frac{125}{\pi}\)

You have the helix: \(\displaystyle (230cos(t))i+(230sin(t))j+(\frac{125}{\pi}t)k\)

It's length in making two revs in a distance of 500 units is given by

\(\displaystyle 2\pi\sqrt{230^{2}+(\frac{125}{\pi})^{2}}\approx{1466.6}\)

Now, this is a cylinder of radius 30 with height 1466.6

\(\displaystyle {\pi}900(1466.6)=4,146,713.8 \;\ \)

Check my calculations out.
Aug 2007
Wow thanks for taking the time to explain that to me more clearly.

Hopefully your message will help others aswell if they find this through a search engine.
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