A single vertex on a circle does not create a polygon.
The Euclidean Steiner minimal tree connecting to vertices is simply a line segment.
The minimum Steiner tree in an equilateral triangle connects each corner with the unique triangle center, the single Steiner point of the tree.
The shortest connection of the four corners of a unit square measures 1 + √3. It has two Steiner points.
The Euclidean Steiner minimal tree which links the five vertices of a regular pentagon has three Steiner points where three line segments of the tree meet at 120° angles. The regular pentagon is the biggest regular polygon whose Steiner minimal tree segments are not on the sides of the polygon.
The shortest path connecting all corners of a hexagon runs along five of its six edges.
The shortest Euclidean Steiner tree in a regular n-gon with n > 5 covers n − 1 sides of the polygon.