Lines of symmetry

Here you will learn about lines of symmetry, including symmetry properties within polygons, angle properties, and symmetry of different line graphs.

Students first learn about line symmetry in grade 4 with their work with 2D shapes in geometry.

What are lines of symmetry?

Lines of symmetry are straight lines that divide a shape into two equal parts, where one part is an exact reflection or mirror image of the other.

In geometry, many 2D shapes have line symmetry.

For example,

Equilateral triangle	Square	Rectangle	Regular hexagon
3 lines of symmetry	4 lines of symmetry	2 lines of symmetry	6 lines of symmetry

Line symmetry can be seen in nature.

For example,

Line symmetry can be seen in art.

For example,

Lines of symmetry in regular shapes

Regular polygons are polygons that have equal side lengths and equal angle measures. 

For example,

Equilateral Triangle
3 lines of symmetry Notice how each line of symmetry can be a fold line.
Square
4 lines of symmetry Notice how each line of symmetry can be a fold line.
Regular pentagon
5 lines of symmetry Notice how each line of symmetry can be a fold line.
Regular hexagon
6 lines of symmetry Notice how each line of symmetry can be a fold line.

Notice how the number of lines of symmetry in these regular polygons is equal to the number of sides and angles the polygons have.

All regular polygons share the property that the number of sides is equal to the number of lines of symmetry.

Lines of symmetry in irregular shapes

Irregular polygons are polygons that do not have all equal angles and all equal side lengths. However, they can have line symmetry.

For example, these irregular polygons have line symmetry.

Let’s look at different shapes (specifically quadrilaterals) and their number of lines of symmetry.

Quadrilateral	Image	Number of Lines of Symmetry
Square		4
Rectangle		2
Parallelogram		0
Rhombus		2
Kite		1
Trapezoid		0
Isosceles Trapezoid		1
Irregular		0

Lines of symmetry in circles

The diameter of a circle splits the circle into two equal sized half circles, or semicircles. Think of the diameter as the line of reflection.

A circle can be folded in-half many ways, meaning that there are an infinite number of lines of symmetry. 

What are lines of symmetry?

Common Core State Standards

How does this apply to 4th grade math?

• Grade 4: Geometry (4.G.A.3)
  Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

How to identify lines of symmetry

In order to draw/identify lines of symmetry:

1. Locate the center of the 2D shape.
2. Draw a vertical line through the center and check for line symmetry.
3. Draw a horizontal line through the marked center and check for line symmetry.
4. Draw a line from each vertex through the center and check for symmetry.
5. State the number of lines of symmetry.

Lines of symmetry examples

Example 1: regular shapes

How many lines of symmetry does the equilateral triangle have?

1. Locate the center of the 2D shape.

2Draw a vertical line through the center and check for line symmetry.

The vertical line through the center is a line of symmetry dividing the triangle into two equal halves that are mirror images of each other. Notice that it can be a fold line.

3Draw a horizontal line through the marked center and check for line symmetry.

4Draw a line from each vertex through the center and check for line symmetry.

5State the number of lines of symmetry.

The equilateral triangle has 3 lines of symmetry. An equilateral triangle has 3 equal sides and 3 equal angles, so it will have 3 lines of symmetry.

Example 2: the regular pentagon

How many lines of symmetry does the regular pentagon have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

The vertical line is a line of symmetry because it divides the regular pentagon into two equal halves that are mirror images of each other.

Draw a horizontal line through the marked center and check for line symmetry.

The horizontal line is NOT a line of symmetry because it does not divide the regular pentagon into two equal halves that are mirror images of each other.

Draw a line from each vertex through the center and check for line symmetry.  

State the number of lines of symmetry.

A regular pentagon has 5 lines of symmetry. A regular pentagon has 5 equal sides and 5 equal angles, so it will have 5 lines of symmetry.

Example 3: irregular shapes

How many lines of symmetry does the trapezoid have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

The vertical line is a line of symmetry because it divides the trapezoid into two equal halves that are mirror images of each other.

Draw a horizontal line through the marked center and check for line symmetry.

The horizontal line is NOT a line of symmetry because it does not divide the trapezoid into two equal halves that are mirror images of each other.

Draw a line from each vertex through the center and check for line symmetry.  

State the number of lines of symmetry.

The trapezoid has one line of symmetry.

Example 4: irregular shapes

How many lines of symmetry does the irregular hexagon have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

The vertical line is a line of symmetry because it divides the irregular hexagon into two equal halves that are mirror images.

Draw a horizontal line through the marked center and check for line symmetry.

The horizontal line is a line of symmetry because it divides the irregular hexagon into two equal halves that are mirror images.

Draw a line from each vertex through the center and check for line symmetry.  

State the number of lines of symmetry.

The irregular hexagon has two lines of symmetry.

Example 5: irregular shapes

How many lines of symmetry does the parallelogram have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

The vertical line is NOT a line of symmetry because it does not divide the parallelogram into two equal halves that are mirror images of each other.

Draw a horizontal line through the marked center and check for line symmetry.

The horizontal line is NOT a line of symmetry because it does not divide the parallelogram into two equal halves that are mirror images of each other.

Draw a line from each vertex through the center and check for line symmetry.  

Example 6: irregular shapes

How many lines of symmetry does the shape have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

The vertical line is a line of symmetry because it divides the shape into two equal halves that are mirror images of each other.

Draw a horizontal line through the marked center and check for line symmetry.

The horizontal line is a line of symmetry because it divides the shape into two equal halves that are mirror images of each other.

Draw a line from each vertex through the center and check for line symmetry.  

Teaching tips for lines of symmetry

• Use hands on manipulatives to help students gain understanding of lines of symmetry.

• Infuse creative activities such as origami so that students can build understanding through fun learning tasks.

Easy mistakes to make

• Not visualizing that a line of symmetry is like a fold line
  For example, thinking that this is a line of symmetry. If you were to fold the parallelogram along the line of symmetry, the two halves would not be mirror images of each other.
  
  

  

• Thinking that all quadrilaterals have \bf{4} lines of symmetry
  The only quadrilateral with 4 lines of symmetry is a square because it is a regular polygon. The other quadrilaterals are not regular polygons so do not have 4 lines of symmetry.

• The number of sides always \textbf{=} the number of lines of symmetry
  Although this is true for regular shapes, it is not true for all shapes.

• Thinking all triangles have \bf{3} lines of symmetry
  For example, this right scalene triangle does not have any lines of symmetry.
  
  

  

  
  Equilateral triangles are the only triangles that have 3 lines of symmetry because it is a regular triangle.

Related symmetry lessons

• Symmetry
• Rational symmetry

The next lessons are

• Angles in polygons
• Congruence and similarity
• Transformations

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