Let represent the projected arclength in the - plane of the corkscrew from to . Let represent the length along the axis over the same range of . Refer to figure 2.7. Since the axis length varies linearly as the arclength along the corkscrew, and therefore varies linearly as the projected arclength along the corkscrew,^{2.7} the following can be shown.

The range of is entirely within the circular region of the corkscrew. Its projected arclength in the - plane can therefore be determined from the following.

From the figure a relationship can be found for in terms of .

Making substitutions from above determines the total axis length.

The shift angle of a corkscrew, , is analogous to that of a loop. Figure
2.8 depicts the arclength of the corkscrew as the hypotenuse of the triangle,
while the lower leg represents the projected length in the - plane. The length of the axis is
the length of the right leg.

The arclength of the corkscrew is given by the Pythagorean Theorem.