In a regular hexagon ABCDEF, all angles are 120 degrees. Connect AD, BE and CF: you've divided the hexagon into six equilateral triangles (they must be equilateral, because you've divided each 120 degree angle exactly in half, so two angles in each triangle are certainly 60 degrees). Look at the line AD; from A to the centre of the hexagon is one side of one equilateral triangle, and from the centre to D is another side. Each side of each equilateral is as long as one side of the hexagon, and one side in the hexagon is P/6 in length, so AD, BE and CF are each 2*P/6 = P/3 in length.

Much easier to see if you draw the picture!