How To Calculate A Spiral

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.

Step 1

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.​"

Step 2

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​."

Step 3

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​."

Step 4

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:

\(L = \frac{3.14 × 10 × (20+5)}{2}\)

Step 5

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

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