Question

a regular pentagon is a five - sided polygon with all five sides equal in length and all five interior angles. 7. give a list of all rotations of symmetry between 0° and 360°. separate your answers with commas and no spaces.

Explanation:

Step1: Recall rotation - symmetry formula

For a regular \(n\) - sided polygon, the angles of rotation of symmetry are given by \(\theta=\frac{360^{\circ}}{n}\times k\), where \(n\) is the number of sides and \(k = 0,1,\cdots,n - 1\). For a pentagon, \(n = 5\).

Step2: Calculate the angles of rotation

When \(k = 1\), \(\theta_1=\frac{360^{\circ}}{5}\times1 = 72^{\circ}\); when \(k = 2\), \(\theta_2=\frac{360^{\circ}}{5}\times2=144^{\circ}\); when \(k = 3\), \(\theta_3=\frac{360^{\circ}}{5}\times3 = 216^{\circ}\); when \(k = 4\), \(\theta_4=\frac{360^{\circ}}{5}\times4=288^{\circ}\); when \(k = 0\), \(\theta_0 = 0^{\circ}\) (but we want non - zero angles in the range \(0^{\circ}<\theta<360^{\circ}\)).

Answer:

72,144,216,288