Question

a regular octagon is shown below. what is the smallest angle of rotation about its center that carries the octagon onto itself? a. 22.5° b. 45° c. 90° d. 135°

Explanation:

Step1: Recall rotation - symmetry formula

For a regular \(n\) - sided polygon, the smallest angle of rotation \(\theta\) about its center that maps the polygon onto itself is given by \(\theta=\frac{360^{\circ}}{n}\), where \(n\) is the number of sides of the polygon.

Step2: Identify the number of sides of an octagon

An octagon has \(n = 8\) sides.

Step3: Calculate the angle of rotation

Substitute \(n = 8\) into the formula \(\theta=\frac{360^{\circ}}{n}\), we get \(\theta=\frac{360^{\circ}}{8}=45^{\circ}\).

Answer:

B. \(45^{\circ}\)