Practice Maths

Line Symmetry

What is Line Symmetry?

A shape has line symmetry (also called reflective symmetry) when a line divides it into two congruent mirror-image halves. The dividing line is called the line of symmetry (or axis of symmetry).

The fold test: if you fold the shape along the line and both halves match exactly, the line is a line of symmetry.

Lines of Symmetry for Common Shapes

  • Equilateral triangle: 3 lines
  • Square: 4 lines
  • Rectangle: 2 lines (horizontal and vertical midlines — NOT the diagonals)
  • Rhombus: 2 lines (the two diagonals)
  • Parallelogram: 0 lines
  • Regular pentagon: 5 lines
  • Regular hexagon: 6 lines
  • Regular n-gon: n lines
  • Isosceles triangle: 1 line
  • Scalene triangle: 0 lines
  • Circle: infinite lines
  • Kite: 1 line (the main diagonal)

Letters of the Alphabet

  • 2 or more lines: H, I, O, X
  • 1 vertical line: A, M, T, U, V, W, Y
  • 1 horizontal line: B, C, D, E, K
  • 0 lines: F, G, J, L, N, P, Q, R, S, Z
Hot Tip
A parallelogram looks like it might have symmetry, but it does NOT — the fold test fails. Neither diagonal is a line of symmetry. Do not confuse a rhombus (which has 2 lines) with a parallelogram (which has 0).

Worked Example 1 — Rectangle

How many lines of symmetry does a rectangle have?

A rectangle has 2 lines of symmetry: one horizontal midline and one vertical midline. The diagonals are NOT lines of symmetry because folding along a diagonal does not produce matching halves (the corners don’t line up).

Worked Example 2 — Regular Hexagon

How many lines of symmetry does a regular hexagon have?

A regular hexagon has 6 lines of symmetry: 3 lines through opposite vertices and 3 lines through midpoints of opposite sides.

Rectangle — 2 lines Equilateral Δ — 3 lines Parallelogram — 0 lines Isosceles Δ — 1 line

Red dashed lines are lines of symmetry.

What Is Line Symmetry?

A shape has line symmetry (also called reflective symmetry or mirror symmetry) if you can draw a line through it so that one half is the mirror image of the other half. If you folded the shape along this line, both halves would match exactly.

The line is called the line of symmetry or axis of symmetry. A shape can have zero, one, or many lines of symmetry.

Lines of Symmetry in Common Shapes

  • Equilateral triangle: 3 lines of symmetry
  • Square: 4 lines of symmetry (2 across opposite vertices, 2 across midpoints of opposite sides)
  • Rectangle (not square): 2 lines of symmetry (both through midpoints of sides — NOT the diagonals)
  • Regular hexagon: 6 lines of symmetry
  • Circle: infinite lines of symmetry (any diameter)
  • Scalene triangle: 0 lines of symmetry
  • Isosceles triangle: 1 line of symmetry (through the apex and midpoint of the base)

For a regular polygon with n sides, there are n lines of symmetry.

Finding Lines of Symmetry

To find lines of symmetry in an irregular shape, imagine folding the shape in different ways and check if the halves match. You can also:

  • Look for matching pairs of points on opposite sides of a possible line
  • Check that each point on one side has a matching point at the same distance from the line on the other side
  • Use a ruler to draw a proposed line and measure to check

Drawing Lines of Symmetry

When asked to draw the line(s) of symmetry on a shape, use a ruler and draw a dashed line. Make sure the line extends to the edges of the shape (or beyond) so it's clearly visible. For shapes on a grid, the line should pass exactly through grid points or midpoints of sides.

Key tip: A common mistake is thinking that the diagonals of a rectangle are lines of symmetry. They are NOT — if you fold a rectangle along its diagonal, the corners don't match up (they only match for a square). The lines of symmetry of a rectangle go through the midpoints of opposite sides.

Key Terms

Line of symmetry
A line that divides a shape into two congruent mirror-image halves; also called an axis of symmetry.
Line symmetry
A shape has line symmetry (reflective symmetry) if at least one line of symmetry exists.
Congruent
Identical in size and shape.
Fold test
A way to check a line of symmetry by imagining folding the shape along the line — if both halves match exactly, it is a line of symmetry.

Mastery Practice

  1. Lines of Symmetry — Shapes Fluency

    Count the lines of symmetry for each shape (shown as red dashed lines).

    (a) Equilateral Δ
    (b) Square
    (c) Rectangle
    no lines
    (d) Parallelogram
    (e) Rhombus
    (f) Regular hexagon
    (g) Isosceles Δ
    ∞ lines
    (h) Circle
    1. Equilateral triangle
    2. Square
    3. Rectangle
    4. Parallelogram
    5. Rhombus
    6. Regular hexagon
    7. Isosceles triangle
    8. Circle

  2. Classifying Letters Fluency

    Study each capital letter below. Red dashed lines show lines of symmetry where they exist. Classify each letter into one of the groups in the sub-questions.

    A
    A
    B
    B
    C
    C
    H
    H
    M
    M
    N no lines
    N
    S no lines
    S
    T
    T
    X
    X
    Y
    Y
    Z no lines
    Z
    O (& more…)
    O
    1. Letters with exactly 1 line of symmetry (list them all)
    2. Letters with 0 lines of symmetry (list them all)
    3. Letters with 2 or more lines of symmetry (list them all)

  3. True or False Fluency

    Write True or False for each statement.

    1. A square has exactly 2 lines of symmetry.
    2. A circle has infinitely many lines of symmetry.
    3. A parallelogram has 2 lines of symmetry along its diagonals.
    4. A rhombus has 2 lines of symmetry along its diagonals.
    5. Every line of symmetry divides a shape into two congruent halves.
    6. A scalene triangle has 1 line of symmetry.
    7. A regular polygon with n sides has n lines of symmetry.
    8. The diagonals of a rectangle are lines of symmetry.

  4. Rectangles vs Squares Understanding

    1. A rectangle has diagonals. Explain clearly why the diagonal of a rectangle is NOT a line of symmetry. What happens to the corners when you fold along a diagonal?
    2. A square also has diagonals. Are the diagonals of a square lines of symmetry? Explain why or why not.
    3. A rectangle has 2 lines of symmetry. A square has 4 lines of symmetry. What is the extra pair of lines in a square that the rectangle does not have?
    4. True or False: every square is also a rectangle. Does a square always have at least as many lines of symmetry as a rectangle?

  5. Finding Reflected Vertices Understanding

    The left half of a shape has vertices at (0, 0), (0, 3), (2, 3), and (2, 0). The mirror line is x = 2.

    1 2 3 4 1 2 3 x y (0,3) (2,3) (0,0) (2,0) x = 2 ?
    Parts (a)–(c): left half shown. Find the reflected right half.
    1 2 3 4 5 1 2 3 4 x y (3,0) (3,4) (5,2) x = 3 ?
    Part (d): right half shown. Find the left-half vertices.
    1. Find the coordinates of the four right-half vertices after reflecting in the line x = 2.
    2. What shape is formed by the combined left and right halves?
    3. How many lines of symmetry does the final shape have?
    4. A different shape has its right half with vertices at (3, 0), (3, 4), (5, 2). The mirror line is x = 3. Find the left-half vertices.

  6. Symmetry in the Real World Problem Solving

    1. The Australian flag has a particular design. Does it have line symmetry? Explain your reasoning.
    2. Design (describe in words) a shape that has exactly 3 lines of symmetry. What shape did you choose? Where are the lines?
    3. A butterfly’s wings are symmetric. If the left wing has a spot at coordinates (2, 5) from the body centre, where is the corresponding spot on the right wing? Assume the line of symmetry is the vertical axis (x = 0).
    4. Name two real-world objects or logos that have line symmetry and state the number of lines each has.
    5. Can a shape have exactly 2 lines of symmetry? Give an example. Can a shape have exactly 3 lines? Give an example. Can a shape have exactly 5 lines? Give an example.

  7. Lines of Symmetry — Regular Polygons Pattern Fluency

    Use the rule “a regular n-gon has n lines of symmetry” to complete the table. For each polygon, state the number of sides and the number of lines of symmetry.

    1. Regular triangle (equilateral triangle)
    2. Regular quadrilateral (square)
    3. Regular pentagon
    4. Regular hexagon
    5. Regular heptagon (7 sides)
    6. Regular octagon
    7. Regular nonagon (9 sides)
    8. Regular decagon (10 sides)

  8. Completing Symmetric Shapes Understanding

    In each case, describe the completed shape you would get, and state how many lines of symmetry the final shape has. Blue = given half-shape; green dashed = completed reflection.

    y-axis (−3,0) (0,4) (0,0)
    (a) Reflected in the y-axis
    x-axis Reflected lower half
    (b) Reflected in the x-axis
    ? → rectangle
    (c) Reflected in centre line
    hyp. A′ → kite
    (d) Reflected in hypotenuse
    1. Half a shape is drawn to the left of a vertical mirror line. It is a right-angled triangle with vertices at (0, 0), (0, 4), (−3, 0). After reflecting in the y-axis, describe the full shape formed and its lines of symmetry.
    2. Half a shape is drawn above a horizontal mirror line. It is a semi-circle. After reflecting in the x-axis, what shape is formed? How many lines of symmetry does a circle have?
    3. An “L” shape is reflected in a vertical line through its centre. The combined shape is a rectangle. How many lines of symmetry does a rectangle have?
    4. A right-angled triangle has only one right angle. If you reflect it in its longest side (hypotenuse), what shape is formed? Does the result have line symmetry?

  9. Lines of Symmetry — Sorting and Reasoning Understanding

    1. Sort these shapes into groups: “0 lines”, “1 line”, “2 lines”, “more than 2 lines”: scalene triangle, isosceles triangle, square, parallelogram, rhombus, regular hexagon, kite, circle, rectangle, equilateral triangle.
    2. Can a quadrilateral have exactly 3 lines of symmetry? Explain why or why not.
    3. A shape has 8 lines of symmetry. Name the shape and describe where the lines are.
    4. Explain why the diagonal of a rectangle is NOT a line of symmetry, but the diagonal of a square IS.

  10. Symmetry Design Challenge Problem Solving

    1. A company wants a logo with exactly 4 lines of symmetry. What shape (or shapes) could they use as a basis? Describe the lines of symmetry.
    2. A tiling pattern uses equilateral triangles. Each tile has 3 lines of symmetry. If you join two equilateral triangles along one edge to form a rhombus, how many lines of symmetry does the rhombus have?
    3. A word is printed in capitals: “MOM”. Does the word have line symmetry? If so, describe the line of symmetry.
    4. Design a simple 5-sided polygon (pentagon) that has exactly 1 line of symmetry. Describe the shape and the position of the line of symmetry.
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