Spiral Length Calculator

From the definition of a spiral, you should choose a central point and draw a curved pattern getting progressively farther away as it revolves around that point. The 3^rd century BC Greek mathematician Archimedes invented a spiral equation that can be described in polar coordinates:

where:

• aa – Parameter marking initial radius of the spiral and location of centerpoint;
• bb – Constant controlling distance between successive turns;
• rr – Distance from the beginning of the spiral; and
• θθ – Rotation angle from the start.

💡 Archimedean spiral was one of only a few curves beyond the straight line and the conic sections that were known in antiquity. Although Archimedes did not discover the spiral, he described it in his work On Spirals to determine the quadrature of the circle and triangle, so the spiral was named after him.

To keep things simple, we will use the approximate formula. Let's imagine the spiral as a series of concentric circles (one per each turn), where the radius increases by the thickness of the tape with every turn. Our simplified roll length formula will work if the thickness is small compared to the smaller diameter of the roll.

You can observe that the number of turns NN is just the difference between the two radii divided by the tape thickness tt. In terms of diameter, we will use the outer diameter DD of the spiral, inner diameter dd, and thickness tt to calculate the number of turns NN:

We can calculate the total roll length formula as follows:

Using another simple formula, we can quickly find how to calculate the spiral length of a helical item found in several applications such as handrails, stair stringers, or helical strakes. If we have a 360-degree helical curve wrapping around a cylinder, we can calculate the spiral length LL by knowing the height HH and diameter of the cylinder DD:

The diameter of the cylinder DD is connected with its circumference CC by the following equation:

🔎 If you expand the helically rolled cylinder, then the cylinder's height and circumference will form a right triangle with the spiral length being the hypotenuse. That's why you can use the Pythagorean theorem to find the length.

If you are interested in how to find the length of the Archimedean spiral from known dimensions, you just need to:

1. Enter the outer diameter of your spiral. This is the total length of a straight line running from one point on the outer circumference of the spiral to a point on the opposite side.

2. Enter the inner diameter of your spiral. This is the diameter of a circle in the center of the spiral. If you're wondering how to calculate the diameter of a circle, you should check our diameter of a circle calculator.

3. Determine the thickness of your spiral. This is the separation distance or distance between the arms of a spiral.

In response, you will receive the total length and number of rings in the spiral from the spiral length formula.

If you want to calculate the helix spiral length around a cylinder, you should enter the following in the second section of the calculator:

1. Cylinder height.
2. Cylinder circumference.

To find the exact length of the spiral, you need to integrate the polar equation of the spiral from the initial angle to the final angle.

If you want to calculate using polar coordinates, you should first convert the spiral equation using our polar to cartesian coordinates calculator. Due to this transformation, you can define a set of parameters for the spiral geometry.

The approximate formula used in this spiral length calculator is simple to use, even on a simple pocket calculator. The results obtained by this method differ from the exact equation by no more than 0.1%, which is negligible if we consider that when we measure the diameter, the precision is probably about 1 mm over 1 m (0.1%).