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 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:A figure is symmetric when a flip, turn, or slide maps it exactly onto itself so it looks the same as before.
Common stuck point:The procedure for symmetry is the easy part; the trap is calling any balanced-looking shape symmetric. Asking "Is there a flip or turn that lands this figure exactly back onto itself?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Is there a flip or turn that lands this figure exactly back onto itself?
Worked Examples
Example 1
easy
Does a square have a line of symmetry? Draw and describe one.How many lines of symmetry does this square have?
Answer
Yes. A square has 4 lines of symmetry.
First step
1
Step 1: A line of symmetry divides a shape into two mirror-image halves.
Full solution
2
Step 2: Draw a vertical line through the midpoints of the top and bottom sides of the square.
3
Step 3: The left half is a mirror image of the right half, so this is a valid line of symmetry.
4
Step 4: A square actually has 4 lines of symmetry (2 through midpoints of opposite sides, 2 through opposite corners).
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.
Example 3
easy
List the capital letters of the English alphabet that have a vertical line of symmetry.
Example 4
medium
A regular polygon has rotational symmetry of order 8. What is its smallest angle of rotational symmetry, and how many lines of symmetry does it have?
Example 5
medium
A regular polygon has each smallest rotational-symmetry angle equal to 30∘. How many sides does it have?
Example 6
medium
Does a rectangle (not a square) have any rotational symmetry? If yes, of what order?
Example 7
medium
A shape has reflection symmetry across the y-axis. The point (3,7) lies on the shape. Give the coordinates of the mirror partner.Point A(3, 7) reflected across the y-axis. What are the coordinates of A'?
Example 8
hard
A regular polygon has 20 axes of symmetry. What is the smallest angle of rotational symmetry?
Example 9
hard
A shape has rotational symmetry of order n>1 but no reflection symmetry. Give the smallest n for which such a shape exists, and describe an example.
Example 10
hard
Two figures have rotational symmetries of orders 4 and 6 respectively. They are joined concentrically so that both rotational symmetries must be preserved. What is the order of the rotational symmetry of the combined figure?
Example 11
challenge
A figure consists of a regular hexagon with one of its 6 corners marked. How many symmetries (rotations + reflections) does the marked figure retain?
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?
Example 3
easy
How many lines of symmetry does a square have?Count the lines of symmetry of this square.
Example 4
easy
Does the letter A have a line of symmetry? If so, describe it.
Example 5
easy
How many lines of symmetry does an equilateral triangle have?How many lines of symmetry does an equilateral triangle have?
Example 6
easy
How many lines of symmetry does a circle have?
Example 7
easy
True or false: every shape has at least one line of symmetry.
Example 8
easy
How many lines of symmetry does a non-square rectangle have?How many lines of symmetry does this rectangle have?
Example 9
easy
Which capital letter has both a horizontal and a vertical line of symmetry: H, A, or P?
Example 10
easy
If you fold a butterfly down its middle and the wings match, what type of symmetry is that?
Example 11
medium
A regular hexagon has how many lines of symmetry?How many lines of symmetry does a regular hexagon have?
Example 12
medium
A shape looks the same after turning it 90∘ about its center. What is its order of rotational symmetry?
Example 13
medium
Which has more lines of symmetry: a regular pentagon or a non-square rhombus?
Example 14
medium
The letter S has no lines of symmetry. Does it have any rotational symmetry?
Example 15
medium
A shape has rotational symmetry of order 3. What is the smallest angle of rotation that maps it onto itself?
Example 16
medium
Half of a symmetric figure is drawn to the left of a vertical mirror line. The left half has area 14 cm². What is the area of the whole figure?
Example 17
medium
Which has rotational symmetry of higher order: a regular octagon or a regular pentagon?
Example 18
medium
A pattern repeats by sliding the same motif a fixed distance over and over. What type of symmetry is this?
Example 19
challenge
A shape has exactly 1 line of symmetry. Can it have rotational symmetry of order greater than 1? Explain.
Example 20
challenge
On a 3×3 grid, you shade some unit squares so the pattern has 4 lines of symmetry (like a square). What is the smallest number of squares you must shade to make a nonempty symmetric pattern, and which?
Example 21
challenge
A figure has rotational symmetry of order 6 and at least one line of symmetry. How many lines of symmetry must it have?
Example 22
challenge
Explain why a shape's number of lines of symmetry, if greater than zero, can never be exactly 2 for a regular polygon but can be 2 for a rectangle.
Example 23
easy
How many lines of symmetry does a regular hexagon have?How many lines of symmetry does a regular hexagon have?
Example 24
easy
Which letter has rotational symmetry of order 2: 'S', 'A', or 'B'?
Example 25
easy
How many lines of symmetry does an isosceles (non-equilateral) triangle have?How many lines of symmetry does this isosceles triangle have?
Example 26
easy
Order of rotational symmetry for an equilateral triangle?
Example 27
easy
A rhombus that is not a square has how many lines of symmetry?How many lines of symmetry does a rhombus (non-square) have?
Example 28
medium
A figure has rotational symmetry of order 4 but no line symmetry. Give an example of such a figure.
Example 29
medium
Order of rotational symmetry of a regular octagon?
Example 30
medium
How many lines of symmetry does the letter X have?
Example 31
medium
Order of rotational symmetry of the letter Z?
Example 32
hard
Among all triangles, which type has the largest number of symmetries (rotations and reflections combined)?
Example 33
hard
A regular polygon has interior angles of 156∘. How many lines of symmetry does it have?