Symmetry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Symmetry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A geometric property where a figure remains unchanged under a specific transformation such as reflection, rotation, or translation. A shape has reflection symmetry when a line divides it into two mirror-image halves, and rotational symmetry when it looks the same after turning by a certain angle.

A butterfly's wings are symmetric—fold it down the middle and both sides match.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Symmetry is about self-similarity under transformation (reflection).

Common stuck point: Students think symmetry only means left-right mirror symmetry. But shapes can have rotational symmetry, and 3D objects can have planes of symmetry.

Sense of Study hint: Try folding the shape along different lines. If both halves match perfectly, that fold line is a line of symmetry.

Worked Examples

Example 1

easy
Does a square have a line of symmetry? Draw and describe one.

Solution

  1. 1
    Step 1: A line of symmetry divides a shape into two mirror-image halves.
  2. 2
    Step 2: Draw a vertical line through the midpoints of the top and bottom sides of the square.
  3. 3
    Step 3: The left half is a mirror image of the right half, so this is a valid line of symmetry.
  4. 4
    Step 4: A square actually has 4 lines of symmetry (2 through midpoints of opposite sides, 2 through opposite corners).

Answer

Yes. A square has 4 lines of symmetry.
Symmetry means one side is a perfect reflection of the other across the line. A square's high regularity gives it 4 such lines — more than any other rectangle.

Example 2

medium
How many lines of symmetry does a regular pentagon have? Explain your reasoning.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Does the letter 'A' have a line of symmetry? If so, describe it.

Example 2

hard
A shape has rotational symmetry of order 3. What is the smallest angle you can rotate it so it looks the same?
← Back to Symmetry Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

shapes