Calculating 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 × 3.145 × R)

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

(C = 2 × 3.145 × 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 × 6.60 = 66 text{ meters})