A regular polygon has 16 lines of symmetry, as shown in the diagram. (a) Write down the order of rotational symmetry of this polygon. [1] (b) Calculate the ...

(a) Order of rotational symmetry. In a regular polygon the number of sides, the number of lines of symmetry and the order of rotational symmetry are all the same number, because every vertex is interchangeable with every other. With 16 lines of symmetry the polygon has 16 sides, so its order of rotational symmetry is 16 [B1]. Each rotation of \(360\div 16 = 22.5^\circ\) moves every vertex on to the next one.

(b) Each interior angle. The exterior angles of a polygon add to \(360^\circ\), so one exterior angle of this regular polygon is \(360\div 16\). The interior angle is the supplement of the exterior angle at the same vertex:

\( 180 - 360 \div 16 \) or equivalent [M1]

Each interior angle \(= 157.5^\circ\) [A1]

The alternative method, \((16-2)\times 180 \div 16 = 2520 \div 16 = 157.5\), gives the same value and is equally acceptable. Notice the answer is not a whole number of degrees; do not round it to \(158^\circ\), since no rounding is asked for and the exact value is available.