Construction of Polygons [Using Ruler and Compass Only]

250. The Hexagon's Secret · Side length equals radius every time

The side length of the hexagon always equals the radius of the circumcircle.

In a regular hexagon the side length equals the radius of its circumscribed circle. Six radii cut the hexagon into six equilateral triangles, so each side matches the radius exactly.

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Selina ICSE: Construction of Polygons [Using Ruler and Compass Only]

What this lesson covers

Try to break it

Drag A around the circle to rotate the hexagon, or drag O to slide it. At every position, each side of the hexagon equals the circle's radius r — six equal arcs of radius r always close back to the start. Try to find a side different from r; impossible.

How you build it

Construct a regular hexagon from arcs on a circle.

Point tool: mark the centre point O.
Point tool: mark point A a little away from O — the first vertex.
Arc tool: click O, then A. This draws the circumcircle and locks the compass to radius OA — and a hexagon''s side equals its radius.
Arc tool: click A (same radius). The arc cuts the circle at the next vertex.
Point tool: mark B where the arc meets the circle. AB equals the radius.
Arc tool: click B (same radius).
Point tool: mark C where the arc meets the circle.
Arc tool: click C (same radius).
Point tool: mark D where the arc meets the circle.
Arc tool: click D (same radius).
Point tool: mark E where the arc meets the circle.
Point tool: mark F on the circle between E and A — the sixth vertex. All six are now one radius-step apart.
Segment tool: join A to B.
Segment tool: join B to C.
Segment tool: join C to D.
Segment tool: join D to E.
Segment tool: join E to F.
Segment tool: join F to A. All six sides equal the radius — a regular hexagon.

The proof, step by step

Prove that the side of the constructed regular hexagon equals the circumcircle radius.

OA = OB = r (Both are radii of the same circle).
AB = r (By construction, the arc was drawn with radius r).
Therefore, triangle OAB is an equilateral triangle.
Each angle in an equilateral triangle is 60°. So, ∠AOB = 60°.
Six such angles (6 × 60° = 360°) perfectly divide the circle, forming a regular hexagon.

Worked example

In the construction of a regular hexagon using Method II, the radius of the circumcircle is equal to:

Method II relies on the property that the side of a regular hexagon is exactly equal to the radius of its circumcircle. Each side subtends a 60° angle at the center, forming equilateral triangles.

Half the side length
The side length — correct
Twice the side length
The length of the longest diagonal