Question

at which angle will the hexagon rotate so that it maps onto itself? 60° 90° 120° 180°

Explanation:

Step1: Calculate the central angle of a regular hexagon

The formula for the central angle of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For a hexagon, \(n = 6\), so \(\frac{360^{\circ}}{6}=60^{\circ}\).

Step2: Determine the rotation angles that map the hexagon onto itself

A regular hexagon has rotational symmetry. If we rotate it by \(k\times\frac{360^{\circ}}{n}\) (\(k = 1,2,\cdots,n\)), it maps onto itself.

• When \(k = 1\), the rotation angle is \(60^{\circ}\).
• When \(k = 2\), the rotation angle is \(2\times60^{\circ}=120^{\circ}\).
• When \(k = 3\), the rotation angle is \(3\times60^{\circ}=180^{\circ}\).

Since \(90^{\circ}\div60^{\circ}=1.5\) (not an integer), a \(90^{\circ}\) rotation does not map the regular hexagon onto itself.

Answer:

A. \(60^{\circ}\), C. \(120^{\circ}\), D. \(180^{\circ}\)