Constructing an Equilateral Triangle and a Regular Hexagon

Start with the two-arc method for an equilateral triangle. Draw a segment, then set the compass to the segment's length and swing an arc from each endpoint. The arcs cross at a point that is that same distance from both endpoints, so all three sides match. That one idea, radius equals side length, is the engine behind everything else in this topic.

Now put a circle into the picture. If you set the compass to the circle's own radius and swing an arc from any point on the circle, the arc crosses the circle at a point exactly one radius away. Connect that point to the center and to the original point, and you get a triangle with all three sides equal to the radius: equilateral, every angle 60 degrees. Since the angle at the center is 60 degrees and a full turn is 360 degrees, stepping that same radius around the circle again and again lands exactly six times before returning to the start, 360 divided by 60. Mark all six points and connect each to its neighbor, and the chords form a regular hexagon, because every side equals the radius and every central angle is 60 degrees.

This construction connects directly to inscribed polygons in general. Skip every other one of the six marks and the remaining three points sit 120 degrees apart, exactly a third of the circle apart, which rebuilds the equilateral triangle but now inscribed in the circle rather than built from scratch. One compass setting, one circle, two different regular polygons.