Pentagon & pentagram — golden ratio

What this is? A regular pentagon and its pentagram (star) drawn in SVG. The checker confirms the golden ratio relationships that arise: each diagonal is φ times a side, and the central pentagon is scaled by 1/φ.

Answer — drawing

Reason (why the golden ratio appears)

  1. In a regular pentagon, the diagonals form a pentagram. Each intersection creates similar isosceles triangles. The ratio of a diagonal to a side is the unique number φ satisfying φ = 1 + 1/φ, hence φ = (1+√5)/2.
  2. Because the small central pentagon is formed by the intersections of the diagonals, it is similar to the outer pentagon with linear scale 1/φ (and area scale 1/φ²).

Check

Automatic verification:

    We verify side equality, diagonal equality, diagonal/side ≈ φ, angle ≈ 108°, and the inner pentagon scale ≈ 1/φ.