Question

consider the regular polygons.
octagon
dodecagon
which of the statements are true regarding the two regular polygons? check all that apply.
□ the order of rotational symmetry for the octagon is 8.
□ the dodecagon has a rotational symmetry of 180°.
□ the order of rotational symmetry for the dodecagon is 10.
□ the smallest angle of rotational symmetry for the octagon is 36°.
□ the smallest angle of rotational symmetry for the dodecagon is smaller than the one for the octagon.

Explanation:

Step1: Check octagon rotation order

The order of rotational symmetry of a regular n-gon equals n. For octagon (n=8), order is 8.

Step2: Check dodecagon 180° symmetry

A regular dodecagon (n=12) has 12 sides. Rotating by 180° maps it to itself (180 is a multiple of its smallest rotation angle).

Step3: Check dodecagon rotation order

For dodecagon (n=12), order of rotational symmetry is 12, not 10.

Step4: Calculate octagon smallest rotation angle

Smallest rotation angle = $\frac{360^\circ}{n}$. For octagon: $\frac{360^\circ}{8}=45^\circ$, not $36^\circ$.

Step5: Compare smallest rotation angles

Dodecagon smallest angle: $\frac{360^\circ}{12}=30^\circ$. Octagon smallest angle is $45^\circ$, so 30° < 45°.

Answer:

• The order of rotational symmetry for the octagon is 8.
• The dodecagon has a rotational symmetry of 180°.
• The smallest angle of rotational symmetry for the dodecagon is smaller than the one for the octagon.