Question

a regular octagon is shown below. suppose that the octagon is rotated counterclockwise about its center so that the vertex at w is mov degrees does the octagon rotate?

Explanation:

Step1: Calculate the central angle of a regular octagon

The formula for the central angle of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For an octagon, \(n = 8\), so the central angle \(\theta=\frac{360^{\circ}}{8}=45^{\circ}\).

Step2: Determine the number of vertices moved

Assume the vertex at \(W\) moves to \(V\) (a common adjacent - vertex movement in rotation problems for simplicity, since the problem is about the rotation of a regular octagon around its center). The number of vertices between \(W\) and \(V\) (counting counter - clockwise) is \(1\).

Step3: Calculate the rotation angle

Since each vertex corresponds to a central angle of \(45^{\circ}\), and if the vertex moves \(1\) position counter - clockwise, the rotation angle \(\alpha = 45^{\circ}\times1=45^{\circ}\). In general, if the vertex moves \(k\) positions counter - clockwise around the octagon, the rotation angle \(\alpha=45k^{\circ}\). When \(k = 1\) (a minimal non - zero rotation that maps a vertex to another vertex in a non - trivial way for an octagon), the rotation angle is \(45^{\circ}\).

Answer:

\(45\)