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. For a regular polygon with \(n = 15\) sides, the smallest angle of rotational symmetry is \(\frac{360^{\circ}}{15}=24^{\circ}\). Any multiple of \(24^{\circ}\) is also an angle of rotational symmetry.

Step2: Check each option

• For \(12^{\circ}\): \(24\div12 = 2\), \(360\div12=30\). Since \(15\) is not a factor of \(30\) (because \(30\div15 = 2\)), \(12^{\circ}\) is not an angle of rotational symmetry.
• For \(45^{\circ}\): \(360\div45 = 8\), and \(15\) is not a factor of \(8\).
• For \(72^{\circ}\): \(360\div72=5\), and \(15\div5 = 3\). So \(72^{\circ}\) is a multiple of the smallest angle of rotational symmetry (\(24^{\circ}\), since \(72\div24 = 3\)).
• For \(90^{\circ}\): \(360\div90 = 4\), and \(15\) is not a factor of \(4\).

Answer:

\(72^{\circ}\)