An hexagon is a $6$ -sided polygon. The most commonly quoted hexagon is a regular hexagon, having congruent sides and congruent interior angles. Below is an example of a regular hexagon:

Below are some properties of regular hexagons in Euclidean geometry:

• The measure of any interior angle of a regular hexagon is $120^{\circ}$ .
• The smallest $n$ for which a regular $n$ -gon has diagonals which are not congruent is $n=6$ . For example, in the regular hexagon below, the diagonal drawn in blue and the one drawn in red are not congruent.
  \begin{pspicture}(0,0)(6,5.2) \psline[linecolor=blue](1.5,0)(1.5,5.196) \psline[linecolor=red](4.5,0)(1.5,5.196) \pspolygon(0,2.598)(1.5,0)(4.5,0)(6,2.598)(4.5,5.196)(1.5,5.196) \psdots(0,2.598)(1.5,0)(4.5,0)(6,2.598)(4.5,5.196)(1.5,5.196) \end{pspicture}
• The side of a regular hexagon has the same length as the radius of the circle circumscribing it. This fact is illustrated below.
  \begin{pspicture}(0,-0.2)(6,5.2) \pscircle[linecolor=cyan](3,2.598){3} \psline[linecolor=cyan](3,2.598)(1.5,5.196) \pspolygon(0,2.598)(1.5,0)(4.5,0)(6,2.598)(4.5,5.196)(1.5,5.196) \psdots(3,2.598)(0,2.598)(1.5,0)(4.5,0)(6,2.598)(4.5,5.196)(1.5,5.196) \end{pspicture}

From the last remark, it is easy to see that a regular hexagon is constructible using compass and straightedge.