Ask Me Help Desk - Arc length on a helix

Arc length on a helix

Please give the formula for calculating the arc length on a helix

Represent the helix by $x=a\cdot cos(t), \;\ y=a\cdot sin(t), \;\ z=ct$

'a' is the radius and c is the length in one turn divided by 2Pi.

The formula is $L=2\pi\sqrt{a^{2}+c^{2}}$

EXAMPLE:

Suppose we have a string wrapped around a cylinder of radius 12 inches. The length that will make one complete turn in, say, 20 inches is given by:

$2\pi\sqrt{36+(\frac{10}{\pi})^{2}}=42.68 \;\ inches$

Quote:

Originally Posted by galactus

Represent the helix by $x=a\cdot cos(t), \;\ y=a\cdot sin(t), \;\ z=ct$

'a' is the radius and c is the length in one turn divided by 2Pi.

The formula is $L=2\pi\sqrt{a^{2}+c^{2}}$

EXAMPLE:

Suppose we have a string wrapped around a cylinder of radius 12 inches. The length that will make one complete turn in, say, 20 inches is given by:

$2\pi\sqrt{36+(\frac{10}{\pi})^{2}}=42.68 \;\ inches$

Thank you for that equation galactus - I waded through pages of calculus treatments of this question before finding your elegant solution.

But you made an error in your calculation when you squared half of the radius, rather than the whole radius. The correct answer is:

$2\pi\sqrt{144+(\frac{10}{\pi})^{2}}=78.00571860464 764 \;\ inches$

Since the length of a circle with a radius of 12 inches is

$2\pi\ r=75.39822\;\ inches$

the length of a helix of the same radius will go up from that number as the length of the helix increases.

I made a mistake. That should have said 12 inch diameter, not radius.

What I have is correct. It is a typo with the word radius. Thanks for the catch after all this time.