Inscribe a regular polygon, step by step

The radius of a circle steps round it exactly six times.

The question

Inscribe a regular hexagon.

compass+straightedge

Step 1→Keep the compass set to the circle's radius for the whole construction

Step 2→Put the point anywhere on the circle and mark an arc across it

Step 3→Move the point to that mark and repeat, all the way round — you get six marks

Step 4→Join the six marks in order

arcs=the evidence

Inscribe a regular hexagon. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
1. Keep the compass set to the circle's radius for the whole construction. The compass stays at the circle's radius for the whole construction. That single fact is what makes this work.
2. Put the point anywhere on the circle and mark an arc across it. Stepping the radius round from any point marks off a chord equal to the radius, and a chord equal to the radius subtends 60° at the centre.
3. Move the point to that mark and repeat, all the way round — you get six marks. Six lots of 60° is a full turn, so the sixth step lands exactly back at the start. If yours does not, the opening drifted.
4. Join the six marks in order. Joining the marks gives a regular hexagon; joining every other one gives an equilateral triangle, because it takes 120° each time.
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.

How it works.

01
Inscribe a regular hexagon. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
02
1. Keep the compass set to the circle's radius for the whole construction. The compass stays at the circle's radius for the whole construction. That single fact is what makes this work.
03
2. Put the point anywhere on the circle and mark an arc across it. Stepping the radius round from any point marks off a chord equal to the radius, and a chord equal to the radius subtends 60° at the centre.
04
3. Move the point to that mark and repeat, all the way round — you get six marks. Six lots of 60° is a full turn, so the sixth step lands exactly back at the start. If yours does not, the opening drifted.
05
4. Join the six marks in order. Joining the marks gives a regular hexagon; joining every other one gives an equilateral triangle, because it takes 120° each time.
06
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.