Question

a regular polygon has 15 sides. which is a possible angle of rotational symmetry for the figure? 12° 45° 72° 90°

Explanation:

Step1: Recall the formula for rotational symmetry angle

The formula for the angle of rotational symmetry of a regular polygon is $\frac{360^{\circ}}{n}$, where $n$ is the number of sides. Also, any multiple of $\frac{360^{\circ}}{n}$ that is less than or equal to $360^{\circ}$ is also an angle of rotational symmetry. For a 15 - sided polygon, $\frac{360^{\circ}}{15}=24^{\circ}$.

Step2: Check the given options

We need to check which of the given angles is a factor of $360^{\circ}$ and can be written as $k\times\frac{360^{\circ}}{15}$ ($k\in\mathbb{Z}$).

• For $12^{\circ}$: Since $24^{\circ}\div2 = 12^{\circ}$ and $12^{\circ}\times30=360^{\circ}$.
• For $45^{\circ}$: $360\div45 = 8$, but $45$ is not a multiple of $24$.
• For $72^{\circ}$: $360\div72=5$, but $72$ is not a multiple of $24$.
• For $90^{\circ}$: $360\div90 = 4$, but $90$ is not a multiple of $24$.

Answer:

$12^{\circ}$