What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?                                                                                                                   [Page No. 94]

A figure has rotational symmetry when it looks exactly the same after being turned about its centre by some angle. For a regular polygon with $n$ sides, the figure maps onto itself after a rotation of $\frac{360^{\circ}}{n}$, and repeating this gives $n$ rotations in all (counting the full $360^{\circ}$ turn). Such a polygon also has exactly $n$ lines of reflection symmetry.

Square: It has rotational symmetry of order 4, it returns to itself under rotations of $90^\circ$, $180^\circ$, $270^\circ$ and $360^\circ$ about its centre. It has 4 lines of reflection symmetry: the two diagonals and the two lines joining the midpoints of opposite sides.

Regular pentagon: rotational symmetry of order 5 (rotations of $72^\circ$, $144^\circ$, $216^\circ$, $288^\circ$ and $360^\circ$).

It has 5 lines of reflection symmetry, each passing through one vertex and the midpoint of the opposite side.

Regular hexagon: rotational symmetry of order 6 (rotations of $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$ and $360^\circ$). It has 6 lines of reflection symmetry, three joining opposite vertices and three joining the midpoints of opposite sides