Line Symmetry — Definition, Formula & Examples
Line symmetry is a property of a shape where a straight line can divide it into two halves that are exact mirror images of each other. The dividing line is called the line of symmetry.
A figure possesses line symmetry (also called reflective symmetry or bilateral symmetry) if there exists a line such that every point on the figure, when reflected across , maps to another point on the same figure. The line is called the axis of symmetry, and the figure is said to be invariant under reflection in .
How It Works
To check for line symmetry, imagine folding the shape along a line. If both halves land perfectly on top of each other, that fold is a line of symmetry. Some shapes have more than one line of symmetry — a square has four, while a circle has infinitely many. Other shapes, like a scalene triangle, have none at all. You can test a possible line of symmetry by picking any point on one side of the line and checking that a matching point exists at the same distance on the other side.
Example
Problem: How many lines of symmetry does a regular hexagon have?
Step 1: A regular hexagon has 6 equal sides and 6 equal angles. Draw a line from each vertex through the center to the opposite vertex.
Step 2: There are 3 such vertex-to-vertex lines. Each one divides the hexagon into two matching halves, so each is a line of symmetry.
Step 3: Now draw a line from the midpoint of each side through the center to the midpoint of the opposite side. There are 3 such midpoint-to-midpoint lines, and each is also a line of symmetry.
Step 4: Count the total: 3 vertex lines + 3 midpoint lines.
Answer: A regular hexagon has 6 lines of symmetry.
Another Example
Problem: Does the letter "B" have a horizontal line of symmetry, a vertical line of symmetry, or neither?
Step 1: Imagine folding the letter B along a vertical line down its center. The left side is a straight edge, but the right side has two bumps. The halves do not match, so there is no vertical line of symmetry.
Step 2: Now imagine folding B along a horizontal line through its middle. The top bump mirrors the bottom bump. Both halves match.
Answer: The letter B has a horizontal line of symmetry but not a vertical one.
Why It Matters
Line symmetry appears throughout elementary and middle-school geometry when classifying shapes, analyzing patterns, and understanding reflections. In art and design, symmetry guides the creation of logos, buildings, and decorative patterns. It also lays the groundwork for the study of geometric transformations — especially reflections — in courses like pre-algebra and high school geometry.
Common Mistakes
Mistake: Assuming every shape has at least one line of symmetry.
Correction: Many shapes, such as scalene triangles and parallelograms (that are not rectangles), have no lines of symmetry at all. Always test by checking whether both halves truly mirror each other.
Mistake: Confusing line symmetry with rotational (point) symmetry.
Correction: Line symmetry involves a mirror reflection across a line. Rotational symmetry means a shape looks the same after being turned around a center point by some angle less than 360°. A shape can have one type without the other.