The least angle measure by which a figure can be rotated so that it maps onto itself depends on the figure's rotational symmetry, specifically the order of symmetry. For a regular polygon with nnn sides, this minimum angle is the measure of its exterior angle, calculated as:
Minimum angle=360∘n\text{Minimum angle}=\frac{360^\circ}{n}Minimum angle=n360∘
For example:
- A regular hexagon (6 sides) has a minimum rotation of 360∘/6=60∘360^\circ /6=60^\circ 360∘/6=60∘.
- A regular pentagon (5 sides) has a minimum rotation of 360∘/5=72∘360^\circ /5=72^\circ 360∘/5=72∘.
- A square (4 sides) has a minimum rotation of 360∘/4=90∘360^\circ /4=90^\circ 360∘/4=90∘
If the figure is not a regular polygon but has some symmetry, the least angle is the smallest rotation that maps the figure onto itself. For instance, some figures have an order 2 rotational symmetry, meaning the least angle is 180∘180^\circ 180∘
. In summary:
- For regular polygons, use 360∘/n360^\circ /n360∘/n.
- For other figures, identify the order of rotational symmetry; the least angle is 360∘360^\circ 360∘ divided by that order.
- Common least angles are 60∘60^\circ 60∘ for hexagons, 72∘72^\circ 72∘ for pentagons, 90∘90^\circ 90∘ for squares, and 180∘180^\circ 180∘ for figures with order 2 symmetry.
Thus, without a specific figure given, the least angle measure is the smallest positive angle of rotation that maps the figure onto itself, often found by dividing 360° by the figure's symmetry order
