Question

the regular polygon below is to be rotated about its center. which angle of rotation would carry the figure onto itself?

Explanation:

Step1: Recall rotation - symmetry formula

For a regular polygon with \(n\) sides, the angle of rotation \(\theta\) that maps the polygon onto itself is given by \(\theta=\frac{360^{\circ}}{k}\), where \(k = 1,2,\cdots,n\).

Step2: Identify the number of sides of the polygon

The given polygon is an octagon, so \(n = 8\).

Step3: Calculate the angles of rotation

When \(k = 1\), \(\theta=\frac{360^{\circ}}{1}=360^{\circ}\); when \(k = 2\), \(\theta=\frac{360^{\circ}}{2}=180^{\circ}\); when \(k = 3\), \(\theta=\frac{360^{\circ}}{3}=120^{\circ}\); when \(k = 4\), \(\theta=\frac{360^{\circ}}{4}=90^{\circ}\); when \(k = 5\), \(\theta=\frac{360^{\circ}}{5}=72^{\circ}\); when \(k = 6\), \(\theta=\frac{360^{\circ}}{6}=60^{\circ}\); when \(k = 7\), \(\theta=\frac{360^{\circ}}{7}\); when \(k = 8\), \(\theta=\frac{360^{\circ}}{8}=45^{\circ}\). The angles of rotation that map the octagon onto itself are \(45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ},360^{\circ}\) and their multiples.

Answer:

The angles of rotation that carry the octagon onto itself are \(45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ},360^{\circ}\) and their multiples.