# [[Rotational Symmetry of Regular Polygons]] (Mathematics > Geometry) ## Definition The **rotational symmetry** of a figure refers to the number of times the figure appears unchanged when rotated about its center within a full rotation of $360^\circ$. The **order of rotational symmetry** is the number of distinct positions in which the figure looks the same during a $360^\circ$ rotation. For a **regular polygon** with $n$ sides, the order of rotational symmetry is $n$, meaning the polygon can be rotated $n$ times before looking identical. To calculate the **smallest angle** of rotation that maps the polygon onto itself, divide $360^\circ$ by the number of sides $n$. $ \text{Smallest Angle of Rotation} = \frac{360^\circ}{n} $ ## Key Concepts - **Regular Polygon**: A polygon with all sides and angles equal. - **Order of Rotational Symmetry**: The number of times a figure looks the same when rotated within $360^\circ$. - **Smallest Angle of Rotation**: The angle through which the figure can be rotated to appear identical. ## Important Properties 1. For any regular polygon with $n$ sides, the order of rotational symmetry is $n$. 2. The smallest angle of rotation for a regular polygon is $\frac{360^\circ}{n}$. 3. For an equilateral triangle ($n=3$), the smallest angle of rotation is $120^\circ$. ## Essential Formulas - **Order of Rotational Symmetry** for a regular polygon with $n$ sides: $ n $ - **Smallest Angle of Rotation**: $ \frac{360^\circ}{n} $ ## Core Examples 1. **Equilateral Triangle**: - An equilateral triangle has $3$ sides, so its order of rotational symmetry is $3$. - The smallest angle of rotation is: $ \frac{360^\circ}{3} = 120^\circ $ 2. **Square**: - A square has $4$ sides, so its order of rotational symmetry is $4$. - The smallest angle of rotation is: $ \frac{360^\circ}{4} = 90^\circ $ ## Related Theorems/Rules - **Symmetry in Regular Polygons**: Every regular polygon has both rotational and reflectional symmetry. ## Common Pitfalls - Confusing the **smallest angle of rotation** with the **order of rotational symmetry**. - Assuming non-regular polygons have the same rotational symmetry as regular polygons. ## Related Topics - [[Symmetry (Mathematics > Geometry)]] - [[Regular Polygons (Mathematics > Geometry)]] ## Quick Review Questions 1. What is the order of rotational symmetry of a regular pentagon? 2. How do you calculate the smallest angle of rotation for a regular polygon with $n$ sides? *** ![[Rotational_Symmetry_of_Regular_Polygons_visualization]]