Question

a regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. how many sides does the polygon have?
10
12
18
20

Explanation:

Step1: Recall the formula for rotational symmetry angle

The angle of rotational symmetry of a regular polygon is given by \(\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. So \(n=\frac{360^{\circ}}{\text{angle of rotational symmetry}}\).

Step2: Check divisibility

We need to find \(n\) such that \(360\) is divisible by \(n\). Let's check each option:

• For \(n = 10\), \(\frac{360}{10}=36\). The angles of rotational symmetry would be factors of \(36\), but \(20,40,80\) are not factors of \(36\).
• For \(n = 12\), \(\frac{360}{12} = 30\). The angles of rotational symmetry would be factors of \(30\), but \(20,40,80\) are not factors of \(30\).
• For \(n=18\), \(\frac{360}{18}=20\). The angles of rotational symmetry are factors of \(20\) (i.e., \(20,40(2\times20),80(4\times20)\) since \(360\div20 = 18\)).
• For \(n = 20\), \(\frac{360}{20}=18\). The angles of rotational symmetry would be factors of \(18\), but \(20,40,80\) are not factors of \(18\).

Answer:

18