Question

a regular octagon is rotated about it center. which angle measure will carry the octagon onto itself? ①36° ②144° ③90° ④160°

Explanation:

Step1: Recall rotation - symmetry of regular polygons

The formula to find the angle of rotation that maps a regular polygon onto itself is $\frac{360^{\circ}}{n}$, 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: $\frac{360^{\circ}}{8}=45^{\circ}$. Also, multiples of this angle will also map the octagon onto itself. Among the given options, $90^{\circ}=2\times45^{\circ}$.

Answer:

③ $90^{\circ}$