inscribing a regular hexagon
Of all the regular polygons, the hexagon is the construction that feels like a gift. To draw a regular six-sided figure inside a circle, you do not even need to change the compass setting — you simply 'walk' the radius around the rim, and it lands back where it started after exactly six steps. This is why hexagons show up in honeycombs and bolt heads: six is the friendliest number for a circle.
Here is why it works. Draw a circle of radius r and mark any point A on it. Without changing the compass width (still r), put the point at A and swing an arc cutting the circle at the next point B; step from B to C, and so on. Each chord you cut has length r. The reason is a hidden equilateral triangle: the centre O together with two neighbouring points, say O, A, B, has |OA| = |OB| = |AB| = r, so triangle OAB is equilateral and the central angle is 60 degrees. Six such 60-degree slices fill the full 360 degrees exactly, so the sixth step closes the figure. Joining the six points gives a regular hexagon; joining every other one gives an equilateral triangle.
The hexagon is the cleanest case of inscribing a regular polygon, and it is a stepping stone: bisecting its central angles gives a regular 12-gon, then a 24-gon, and so on, doubling forever. Squares and equilateral triangles are similarly constructible. The deeper question of WHICH regular n-gons can be built — answered by Gauss — is taken up under constructible regular polygons.
On a circle of radius r, keep the compass at r, start at A, and step around: A, B, C, D, E, F, back to A in six hops. Each chord is r because each central triangle (like OAB) is equilateral with a 60-degree central angle.
The radius steps around its own circle exactly six times — a hexagon's secret is the 60-degree equilateral triangle.
The radius equals the side of the inscribed regular hexagon, which is why the six-step walk closes perfectly. This easy case does not mean every regular polygon is constructible — the regular 7-gon and 9-gon, for instance, are not.