344. The Hexagon's Circle · circumscribing a regular hexagon

Try to break it

Drag A around the circle. The regular hexagon rotates with A, and every vertex stays on the circle. Try to find a position where any vertex drifts off the circumscribing circle; impossible — six equal arcs of radius r partition the circle into exactly six vertices.

How you build it

Construct a circle around a regular hexagon.

• Place point A as the first vertex of the hexagon.
• Place point B as the second vertex of the hexagon.
• Place point C as the third vertex of the hexagon.
• Place point D as the fourth vertex of the hexagon.
• Place point E as the fifth vertex of the hexagon.
• Place point F as the sixth vertex of the hexagon.
• Draw side AB of the hexagon.
• Draw side BC of the hexagon.
• Draw side CD of the hexagon.
• Draw side DE of the hexagon.
• Draw side EF of the hexagon.
• Draw side FA to complete the regular hexagon.
• Draw the diagonal AD connecting opposite vertices.
• Draw the diagonal CF connecting opposite vertices.
• Mark the intersection of the diagonals as point O.
• Draw the circumscribed circle with center O and radius OA.

The proof, step by step

Prove that the constructed regular hexagon has all six vertices on the circle.

• In a regular hexagon, all sides and angles are equal.
• The main diagonals intersect at the center of symmetry, point O.
• Every vertex is equidistant from O, so a circle centered at O passes through all vertices.
• The radius of the circumscribed circle equals the side length of the hexagon.

Worked example

A regular hexagon ABCDEF has a side length of 6 cm. What is the radius of its circumscribed circle?

In a regular hexagon, the radius of the circumscribed circle is always equal to the length of its side. Since the side is 6 cm, the radius is also 6 cm.

• 3 cm
• 6 cm — correct
• 9 cm
• 12 cm