Line and Rotational Symmetry

Symmetry is a fundamental concept in geometry that describes how a shape stays unchanged under certain transformations. There are two main types of symmetry you need to know: line symmetry and rotational symmetry.

Line Symmetry

A figure has line symmetry (or reflectional symmetry) if you can draw a straight line through it such that one half is the exact mirror image of the other half. This line is called the line of symmetry.

If you fold the shape along the line of symmetry, both halves will match up perfectly. A shape can have zero, one, or multiple lines of symmetry. For example, a scalene triangle has $0$, an isosceles triangle has $1$, and a rectangle has $2$.

Rotational Symmetry

A figure has rotational symmetry if it can be rotated around its center point by an angle strictly greater than $0^\circ$ and strictly less than $360^\circ$, and still look exactly like the original figure.

The angle of rotational symmetry is the smallest angle for which the figure maps onto itself. The order of rotational symmetry is the number of times the figure maps onto itself during a full $360^\circ$ rotation.

Symmetry in Regular Polygons

A regular polygon is a shape with all sides of equal length and all interior angles equal. Because of their uniformity, regular polygons possess both line and rotational symmetry.

For a regular polygon with $n$ sides:

• Lines of Symmetry: It has exactly $n$ lines of symmetry.
• Angle of Rotational Symmetry: The smallest angle of rotational symmetry is calculated using the formula: $\text{Angle} = \frac{360^\circ}{n}$

Example Problems

Example 1: How many lines of symmetry does a regular hexagon have?

A regular hexagon is a polygon with $6$ equal sides. According to the rule for regular polygons, an $n$-sided regular polygon has exactly $n$ lines of symmetry. Therefore, a regular hexagon has $6$ lines of symmetry.

Example 2: What is the smallest angle of rotational symmetry for a regular pentagon?

A regular pentagon has $5$ equal sides, meaning $n = 5$. To find the smallest angle of rotational symmetry, divide a full circle ($360^\circ$) by the number of sides: $\text{Angle} = \frac{360^\circ}{5} = 72^\circ$ The smallest angle of rotational symmetry for a regular pentagon is $72^\circ$.