A line of symmetry is a line that divides a figure into two halves that are mirror images of each other. If you fold the figure along that line, the two halves match exactly. A square has 4 lines of symmetry, a rectangle has 2, and a parallelogram that is not a rectangle has none.
In Spanish
eje de simetría
Also called
axis of symmetry, mirror line, line symmetry, reflection symmetry
Test a line by folding: if every point lands on a matching point, it is a line of symmetry. Diagonals work for a square but not for a rectangle.
Formal definition and notation
A line ℓ is a line of symmetry of a plane figure if the reflection across ℓ maps the figure onto itself: every point of the figure has its mirror image, the same distance from ℓ on the other side, also in the figure. A figure with at least one such line has line (reflection) symmetry. For the graph of a parabola, the same idea is called the axis of symmetry.
Symbol
Read it as
Meaning
dashed line
line of symmetry
diagrams usually draw lines of symmetry dashed
x = h
x equals h
a vertical line of symmetry on a coordinate graph, such as the axis of a parabola
A regular polygon with n sides has exactly n lines of symmetry.
A circle has infinitely many lines of symmetry: every line through its center.
A rectangle’s diagonal is not a line of symmetry: folding along it makes the corners overshoot.
Capital letters A, M, T, U, V, W, Y have a vertical line of symmetry; B, C, D, E, K have a horizontal one; H, I, O, X have both.
The graph of y = ax2 + bx + c is symmetric about the vertical line x = −b2a.
Try it: test a mirror line
Interactive
Switch to Reflect and choose the y-axis: the reflected house lands exactly on the original, so the y-axis is a line of symmetry. Try the x-axis and the line y = x to see lines that are not.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Line of Symmetry examples, step by step
Example 1: Count the lines
Warm-up
How many lines of symmetry does a regular hexagon have?
A regular polygon with n sides has n lines of symmetry.
A hexagon has 6 sides.
3 lines go through opposite vertices and 3 through midpoints of opposite sides.
Answer: A regular hexagon has 6 lines of symmetry.
Example 2: Complete the figure
Core
The line x = 2 is a line of symmetry of a shape. The point (5, 3) is on the shape. Which other point must be on it?
(5, 3) is 5 − 2 = 3 units to the right of the line x = 2.
Its mirror image is 3 units to the left: x = 2 − 3 = −1.
Reflection across a vertical line keeps y the same.
Answer: The point (−1, 3) must be on the shape.
Example 3: Axis of a parabola
Test-style
Find the line of symmetry of the graph of y = x2 − 8x + 7.
For y = ax2 + bx + c, the axis is x = −b2a.
Here a = 1, b = −8: x = −−82 = 4.
Check: x = 3 and x = 5 both give y = −8.
Answer: The line of symmetry is x = 4.
Common mistakes
✗ MistakeDrawing the diagonals of a rectangle as lines of symmetry: rectangle has 4 lines.
✓ InsteadA non-square rectangle has 2 lines: the vertical and horizontal lines through its center.
Folding a rectangle along a diagonal does not match the halves; the corners stick out. Only a square has diagonal lines of symmetry.
✗ MistakeAny line through the middle of a parallelogram is a line of symmetry.
✓ InsteadA slanted parallelogram has 0 lines of symmetry. It does have rotational symmetry: a half turn maps it onto itself.
Splitting a shape into two equal areas is not enough. The halves must be mirror images.
✗ MistakeReflecting (7, 1) across x = 3 to get (−7, 1).
✓ Instead(7, 1) is 4 units right of x = 3, so its image is 4 units left: (−1, 1).
Changing the sign of x reflects across the y-axis, x = 0, not across x = 3.
Not to be confused with
Line of Symmetryvs.Rotational symmetry
Rotational symmetry means a turn of less than 360° maps the shape onto itself, as with the letter S or a parallelogram. Line symmetry needs a fold line. A shape can have one kind without the other.
Line of Symmetryvs.Axis (coordinate axis)
The x-axis and y-axis are number lines that form the coordinate plane. They are lines of symmetry only for shapes that happen to be balanced across them.
Quick check: 3 questions
Question 1 of 3
How many lines of symmetry does an equilateral triangle have?
Show answer and explanation
Answer: C. Each line runs from a vertex to the midpoint of the opposite side, so there are 3. One is the count for an isosceles triangle that is not equilateral, and 6 doubles the count.
Question 2 of 3
Which shape has exactly 2 lines of symmetry?
Show answer and explanation
Answer: D. A rectangle that is not a square has a vertical and a horizontal line of symmetry. A square has 4, a circle has infinitely many, and a scalene triangle has none.
Question 3 of 3
The y-axis is a line of symmetry of a shape that contains (4, −2). Which point must also be on the shape?
Show answer and explanation
Answer: B. Reflecting across the y-axis changes the sign of x only: (−4, −2). (4, 2) reflects across the x-axis, (−4, 2) rotates 180°, and (−2, 4) swaps coordinates.
Line of Symmetry: frequently asked questions
How do you find a line of symmetry?
Look for a line that splits the figure into two halves that would match if folded. On paper, actually fold it. On a grid, check that every vertex has a partner the same distance from the line on the other side. For a regular polygon, draw lines through each vertex and through each side’s midpoint toward the center.
Can a shape have no line of symmetry?
Yes. A scalene triangle, a slanted parallelogram and most irregular shapes have none. The letters F, G, J, L, N, P, R, S and Z have no line of symmetry either. Some of these, such as N, S, Z and the parallelogram, still have rotational symmetry: they look the same after a half turn.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.G.A.3
Cite this entry
Effortless Math Team. “Line of Symmetry.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/line-of-symmetry/
Flashcard
TermLine of Symmetrythe fold line
DefinitionA line that splits a figure into two mirror-image halves. A square has 4; a rectangle has 2; a parallelogram has 0.
ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometrySideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2SymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometryParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryAxis of SymmetryThe axis of symmetry of a parabola is the vertical line through its vertex that splits it into two mirror-image halves. For y = ax2 + bx + c, it is the line x = −b2a. Points at equal distances on either side of this line have the same y-value.Functions · Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Equilateral TriangleAn equilateral triangle is a triangle whose three sides all have the same length. Its three angles are also equal, so each one measures 60 degrees. With side length a, its perimeter is 3a and its area is √34a2. Every equilateral triangle is also isosceles and acute.Geometry · Grade 4, Grade 5, Grade 7Isosceles TriangleAn isosceles triangle is a triangle with at least two sides of equal length. The equal sides are called legs, and the third side is the base. The two angles touching the base, the base angles, are always equal. If the angle between the legs is 40 degrees, each base angle is 70 degrees.Geometry · Grade 4, Grade 5, Grade 7Scalene TriangleA scalene triangle is a triangle whose three sides all have different lengths. Because no two sides match, no two of its angles are equal either. A scalene triangle can be acute, right or obtuse: the 3-4-5 right triangle is scalene, and so is a triangle with sides 4, 6 and 7.Geometry · Grade 4, Grade 5, Grade 7RhombusA rhombus is a quadrilateral with four sides of equal length. It is a special parallelogram, so its opposite sides are parallel and its opposite angles are equal. Its diagonals cross at right angles and cut each other in half, and its area is half the product of the diagonals.Geometry · Grade 3, Grade 4, Grade 5SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Regular PolygonA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is (n − 2) × 180°n, and each exterior angle is 360°n.Geometry · Grade 5, Grade 8, GeometryCircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7MidpointA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is (x1 + x22, y1 + y22).Geometry · Grade 8, Algebra 1, GeometryLineA line is a perfectly straight path of points that goes on forever in both directions. It has no thickness and no endpoints, so it has no length you can measure. Through any two points there is exactly one line, which is named by those two points, as in line AB, or by one lowercase letter.Geometry · Grade 4, Grade 8, GeometryTransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6