ACT® · Math: Geometry — Transformations, Proof, and Constructions
Counting Lines of Symmetry
A line of symmetry is a fold line that lands a figure exactly on top of itself, vertex on vertex. Count them by testing each candidate fold, not by guessing from the figure's general shape.
A line of symmetry splits a figure into two halves that are mirror images. To test a candidate line, fold the figure over it in your head: every vertex must land exactly on another vertex (or on itself), and every edge must land on an edge of the same length. A rectangle looks like it should have four fold lines because it has four sides, but only two actually work: the horizontal and vertical lines through the center. The diagonals fail because folding along one sends a short side swinging onto a long side, which never lines up.
Regular polygons are the clean case: a regular polygon with n sides always has exactly n lines of symmetry, one through each pair of opposite vertices or midpoints of opposite sides, or a mix of both when n is odd. A regular hexagon has 6 lines of symmetry: three run through pairs of opposite vertices, and three run through midpoints of opposite sides. Letters are trickier because they're irregular: H has 2 lines of symmetry (vertical and horizontal), E has 1 (horizontal only), and S has 0, since any fold sends its curves to empty space.
This topic sits right next to rotational symmetry, and the two don't always travel together. A parallelogram that isn't a rectangle or rhombus has 180-degree rotational symmetry about its center but zero lines of symmetry, because its diagonals are not mirror lines. As a regular polygon gains more sides, its lines of symmetry pile up one per side, and in the limit the shape approaches a circle, which has infinitely many lines of symmetry through its center.
Key terms
- Line of symmetry
- A line where folding the figure over it makes both halves match up exactly, vertex on vertex.
- Regular polygon
- A polygon with all sides equal and all angles equal, like an equilateral triangle or a square.
- Rotational symmetry
- The property of landing back on itself after a rotation of less than 360 degrees about its center.
- Mirror image
- A reflection of a shape across a line, flipped left-right or top-bottom relative to the original.
Watch it explained
- How it worksHow do you test a candidate fold line by folding the figure over it and checking that every vertex lands on a vertex?
- Worked exampleFind how many lines of symmetry a regular hexagon has and where each one runs.
- Worked exampleFind the number of lines of symmetry in each of the capital letters H, S and E.
- Common mistakeWhy doesn't a parallelogram have a line of symmetry along either one of its diagonals?
- Going furtherWhat happens to the number of lines of symmetry as a regular polygon picks up more and more sides?
Check yourself
End of topic.