Symmetry — Definition, Formula & Examples
Symmetry is when a shape or graph can be divided or transformed so that one part is a mirror image or exact match of another part. A figure has symmetry if you can flip, fold, or rotate it and it looks the same as the original.
A geometric figure possesses symmetry if there exists a rigid transformation — such as a reflection across a line or a rotation about a point — that maps the figure onto itself. The two most common types are line symmetry (reflective) and point symmetry (rotational by 180°).
How It Works
To check for line symmetry, imagine folding the figure along a line. If both halves match perfectly, that line is a line of symmetry. A figure can have zero, one, or many lines of symmetry — a square has four, while a scalene triangle has none. To check for point symmetry, rotate the figure 180° around a central point. If the figure looks identical after the rotation, it has point symmetry. On a coordinate plane, the graph of has point symmetry about the origin, while the graph of has line symmetry about the -axis.
Worked Example
Problem: Determine how many lines of symmetry a regular hexagon has.
Step 1: A regular hexagon has 6 equal sides and 6 equal angles. Draw lines connecting each vertex to the opposite vertex.
Step 2: That gives 3 lines through opposite vertices. Now draw lines connecting the midpoints of each pair of opposite sides.
Step 3: That gives 3 more lines. Each of these 6 lines divides the hexagon into two congruent halves.
Answer: A regular hexagon has 6 lines of symmetry.
Why It Matters
Symmetry helps you sketch and analyze graphs more efficiently — if you know a parabola is symmetric about its axis of symmetry, you only need to plot points on one side. Architects, designers, and engineers rely on symmetry to create balanced structures and patterns.
Common Mistakes
Mistake: Assuming every shape has at least one line of symmetry.
Correction: Many shapes, like scalene triangles or the graph of , have no line of symmetry at all. Always test by checking whether a fold or reflection produces an exact match.