A regular hexagon ABCDEF is given. Find FC if AB = 7

1) Hexagon ABCDEF is regular, so all its angles are equal.

2) The sum of the angles of the hexagon (n = 6) is found by the formula (n-2) * 180 ° = (6 – 2) * 180 ° = 720 °.

3) This means one angle of the hexagon is 720 ° / 6 = 120 °.

4) The center O of the circumcircle ABCDEF is in the middle of the diagonal FC, since it must be equidistant from all vertices of the hexagon.

5) Consider a triangle OAF: OA = OF = FC / 2 -> angle OAF = angle OFA. Angle AOF = 360 ° / 6 = 60 °. 6) it follows from item 5 that the triangle OAF is equilateral (since all angles are 60 °).

7) Therefore OF = 7. FC = 2 * OF = 14.

Answer: 14