Symmetry 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°.
In Spanish
simetría
Also called
line symmetry, reflection symmetry, mirror symmetry, rotational symmetry
A line of symmetry works like a mirror. A square has 4, a rectangle 2, an isosceles triangle 1 and a scalene triangle none.
Formal definition and notation
A figure has symmetry when some rigid motion other than the identity carries it onto itself. If a reflection across a line ℓ maps the figure onto itself, ℓ is a line of symmetry. If a rotation about a point by an angle between 0° and 360° maps it onto itself, the figure has rotational symmetry; the number of such positions in a full turn is its order.
Symbol
Read it as
Meaning
ℓ
line l
a line of symmetry
order n
order n
the shape matches itself n times in one full turn; the smallest turn is 360° ÷ n
Key facts
Line symmetry: fold along the line and the two halves land exactly on each other.
Every point and its match are the same distance from the line of symmetry, on a segment perpendicular to it.
A regular polygon with nsides has n lines of symmetry and rotational symmetry of order n: a square has 4 of each.
A rectangle (not a square) has 2 lines of symmetry; its diagonals are not lines of symmetry.
A parallelogram that is not a rectangle or rhombus has no line of symmetry but does have rotational symmetry of order 2 (a half turn).
Try it: test for symmetry
Interactive
This house shape is symmetric about the y-axis. Pick the y-axis as the line of reflection: the image lands exactly on the original. Pick the x-axis: it does not, so the x-axis is not a line of symmetry.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Symmetry examples, step by step
Example 1: Lines of symmetry of a rectangle
Warm-up
How many lines of symmetry does a 6 cm by 3 cm rectangle have?
A vertical fold through the middle matches the left and right halves: 1 line.
A horizontal fold through the middle matches the top and bottom: 2 lines.
A diagonal fold does not work: the corners miss each other because the sides are unequal.
Answer: The rectangle has 2 lines of symmetry.
Example 2: Complete a symmetric figure
Core
The line x = 4 is a line of symmetry of a figure. One vertex is at (1, 5). Where is its matching vertex?
The vertex is 4 − 1 = 3 units left of the line.
Its match is 3 units to the right, at the same height.
x = 4 + 3 = 7, and y stays 5.
Answer: The matching vertex is at (7, 5).
Example 3: Symmetry of a regular hexagon
Test-style
How many lines of symmetry does a regular hexagon have, and what is the smallest turn that maps it onto itself?
3 lines join opposite vertices and 3 lines join midpoints of opposite sides: 6 lines.
It matches itself 6 times in a full turn, so the smallest turn is 360° ÷ 6 = 60°.
Answer: A regular hexagon has 6 lines of symmetry and matches itself after a 60° turn.
Common mistakes
✗ MistakeThe diagonal of a 6 by 3 rectangle is a line of symmetry.
✓ InsteadFold along the diagonal and the corners do not meet. Only the vertical and horizontal middle lines work.
A line of symmetry must send every point to a matching point. A diagonal works only when all sides are equal, as in a square.
✗ MistakeA parallelogram has a line of symmetry down the middle.
✓ InsteadA slanted parallelogram has no line of symmetry. It does match itself after a half turn (180°).
Folding a slanted parallelogram leaves the halves leaning in opposite directions. Rotational symmetry and line symmetry are different tests.
✗ MistakeEvery shape has rotational symmetry because it matches itself after a full 360° turn.
✓ InsteadRotational symmetry needs a match after a turn of less than 360°.
Any shape returns to its start after a full turn, so that turn tells you nothing.
A reflection is the flip itself, a move that can be applied to any shape. Symmetry is a property: the shape lands on itself after that flip.
Symmetryvs.Axis of symmetry (parabola)
For a parabola, the axis of symmetry is the vertical line through its vertex, such as x = 2 for y = (x − 2)2. It is a line of symmetry of that graph.
Quick check: 3 questions
Question 1 of 3
How many lines of symmetry does a square have?
Show answer and explanation
Answer: D. A square has 2 lines through the midpoints of opposite sides and 2 diagonals: 4 in all. Choice B counts only a rectangle’s lines, and 8 counts each line twice.
Question 2 of 3
Which capital letter has a vertical line of symmetry?
Show answer and explanation
Answer: A. A vertical line through the point of A splits it into matching halves. F, G and J have parts on one side only, so their halves do not match.
Question 3 of 3
What is the smallest turn that maps an equilateral triangle onto itself?
Show answer and explanation
Answer: B. It matches itself 3 times per turn: 360 ÷ 3 = 120°. 60° is its interior angle, 180° is for a parallelogram, and 360° does not count.
Symmetry: frequently asked questions
What is the difference between line symmetry and rotational symmetry?
Line symmetry means you can fold a shape along a line so the two halves match, like a heart or a butterfly. Rotational symmetry means you can turn the shape less than a full turn around its center and it looks the same, like a pinwheel. Some shapes have both (a square), some only one (a parallelogram has only rotational).
Does a circle have symmetry?
Yes, endless amounts. Every line through the center of a circle is a line of symmetry, so a circle has infinitely many. It also matches itself after a turn of any angle about its center, so its rotational symmetry has no smallest turn. That is why a circle looks the same however you flip or spin it.
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, GeometryLine of SymmetryA 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.Geometry · Grade 4, Grade 8, GeometryRotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryNumberA 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 2Regular 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, 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 2SquareA 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 3ParallelogramA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Geometry · Grade 3, Grade 5, Grade 6RhombusA 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 5HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6MidpointA 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, 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 2CircleA 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 7TransformationA 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 1CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometryScalene 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 7Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7