How to Calculate a Spiral

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Spirals are one of nature's (and mathematics') more surprising and aesthetic phenomena. Their mathematical description may not be immediately apparent. But by counting a spiral's rings and making a few measurements, you can figure out some key properties of the spiral.

    Determine the number of rings in the spiral. This is the number of times the spiral curve wraps around the center point. Call this number of rings "​R.​"

    Determine the outer diameter of the spiral as a whole. This is the length of a straight line that runs from one point on the spiral's outer circumference to a point on the circumference's opposite end. Call this length "​D​."

    Determine the inner diameter of the spiral. This is the diameter of the circle formed by the innermost ring of the spiral. Call this length "​d​."

    Plug the numbers obtained in the first three steps into the following formula:

    L = \frac{3.14 × R × (D+d)}{2}

    For example, if you had a spiral with 10 rings, an outer diameter of 20 and an inner diameter of 5, you would plug these numbers into the formula to get:

    L = \frac{3.14 × 10 × (20+5)}{2}

    Solve for "​L​." The result is the length of the spiral. Using the example from the previous step:

    \begin{aligned} L &= \frac{3.14 × 10 × (20+5)}{2} \\ &= \frac{3.14 × 10 × 25}{2} \\ &= \frac{785}{2} \\ &= 392.5 \end{aligned}


    • Be sure to take all measurements of the spiral in the same units.

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