Star Polygon - Interiors of Star Polygons

Interiors of Star Polygons

Star polygons leave an ambiguity of interpretation for interiors. This diagram demonstrates three interpretations of a pentagram.

• The left-hand interpretation has the 5 vertices of a regular pentagon connected alternately on a cyclic path, skipping alternate vertices. The interior is everything immediately left (or right) from each edge (until the next intersection). This makes the core convex pentagonal region actually "outside", and in general you can determine inside by a binary even-odd rule of counting how many edges are intersected from a point along a ray to infinity.
• The middle interpretation also has the 5 vertices of a regular pentagon connected alternately on a cyclic path. The interior may be treated either:
  • as the inside of a simple 10-sided polygon perimeter boundary, as below.
  • with the central convex pentagonal region surrounded twice, because the starry perimeter winds around it twice.
• The right-hand interpretation creates new vertices at the intersections of the edges (5 in this case) and defines a new concave decagon (10-pointed polygon) formed by perimeter path of the middle interpretation; it is in fact no longer a pentagram.

What is the area inside the pentagram? Each interpretation leads to a different answer.