How to calculate helical length

Written by c. taylor
How to calculate helical length
A spiral staircase forms a helical pattern, for which you could calculate the helical length of the handrail. (spiral stairs image by Aaron Kohr from

A helix forms a spiral pattern, with which you may already be familiar. A spiral staircase, a coiled spring or a twisted wire all form a helical pattern. The helical length could correspond to the length of the spiral staircase's handrail, the full length of the coil or one of the wire's length. To calculate the full helical length, it is useful to first calculate the helical length of a single rotation. Armed with that calculation, the rest becomes considerably easier.

Measure the radius of the helix. This will be the distance from the exact centre to the outer edge of the helix, or spiral. In some cases, it might be easier to measure the horizontal diameter across the helix, and divide by two to determine the radius.

As an example, you might have a spiral staircase without a clear centre, such as a staircase spiralling up the inside wall of an empty lighthouse. The handrail is attached to the wall. Although a little awkward to measure, you could hold the tape measure straight out to measure 14 feet across the centre from inner wall to inner wall. By dividing by two, you know the radius is 7 feet.

Measure the height of one full rotation. This will be the measurement from the top edge of one spiral down to the top edge of the next lower spiral. As an example, if you were measuring a handrail, you could hold the end of the tape measure on the top of the handrail and let gravity drop the rest of the tape measure straight down until it passes the next, lower handrail below it. This distance would be the height of one rotation.

In the case of the lighthouse handrail, the tape measure would probably be obstructed by the stairs if you used that method. However, since the distance from the handrail to the stairs are uniform, you could measure from the top, inside edge of the stairs instead, and the height will still be the same. For instance, you might measure 10 feet from top of the one stair to the top of the next, lower stair. The height of one rotation, for both the stair and handrail, will be 10 feet.

Calculate the helical length of one rotation using the formula:

Length = Squareroot of [Height^2+ (2 * 3.14 * Radius)^2]

The notation "^2" means to square the preceding number or parenthesized calculation.

In the example, you would calculate:

Length = Squareroot of [10^2+ (2 * 3.14 * 7)^2]

Length = Squareroot of [100+ 43.96^2]

Length = Squareroot of [100+ 1,932]

Length = Squareroot of [2,032]

Length = 45 feet per rotation

Divide the length of one rotation by the height of one rotation to calculate the length/height ratio. In the example, the length is 45 feet for every 10 feet in height. Therefore, the ratio of length to height is 45 divided by 10, or 4.5. This tells you that the length is 4.5 times the height measurement. You can use this to calculate the total length.

Measure the total height of the helix. This will be the vertical distance from the top edge of the spiral, from beginning to end. If there is a fraction of a rotation involved, such as a stairway that rotates 3.5 times, these positions may not align vertically, but you will have to do the best you can. In the lighthouse example, you might be able to measure vertically from the top stair to the floor and find it is 100 feet. This should also be the total height of the handrail, since it stays at a fixed distance above the stairs.

Multiply the length/height ratio by the total height to calculate total helical length. In the example, 100 feet times 4.5 would give you a handrail length of 450 feet.

Things you need

  • Tape measure or ruler

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