This method for constructing a regular heptagon using angle trisection was found by Plemelj in 1892 and published in 1912. The construction is taken from [1, pp. 183–184]. The approximation for was known to Abûl-Wefâ and Heron of Alexandria [1, p. 184].

Start with the trigonometric identity

.

Since

,

replace by and by in the trigonometric identity to get the cubic

.

Then

.

Since and , take .

Substitute to get

.

Substitute to get the Vieta form of the equation,

.

Set

to get the positive solutions

,

,

.

So

,

where

.

Then

.

Reference

[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.