How To Calculate Helical Length

A helix is defined as a spiral that also has a linear dependence upon a third dimension. Found both within nature and within the man-made world, examples of helices include springs, coils and spiral staircases. The length of a helix can be calculated using a simple formula.

Step 1

Write down the quantities that define the helix. A helix can be defined by three quantities: the radius, the rise of the helix in one revolution and the number of turns. For this example, we will define the following symbols:

$r=\text{ Radius}$
$H=\text{ Rise of helix in one revolution}$
$N=\text{ Number of turns}$

Step 2

Calculate the length associated with one turn within the helix. To do this use the following formula:

$L=\sqrt{H^2+C^2}$

In this nomenclature, H^2 means "H multiplied by H" or "H squared." C is the circumference of the circle and is equal to :

$C=2\times 3.145\times R$

For example, if a spiral staircase has a radius of 1 meter, then the circumference is equal to :

$C=2\times 3.145\times 1=6.29\text{ meters}$

If the staircase rises by approximately 2 meters after each turn (H = 2) then the length associated with one turn around the staircase is:

$L=\sqrt{2^2+{6.29}^2}=\sqrt{4+39.6}=6.60\text{ meters}$

Step 3

Calculate the total helical length (T). To do this use the formula:

$T=NL$

Following the example, if the staircase has 10 turns:

$T=10\times 6.60=66\text{ meters}$