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.
Showing a random 20 of 50 problems.
Example 1
medium
Which has more lines of symmetry: a regular pentagon or a non-square rhombus?
Example 2
medium
How many lines of symmetry does a regular dodecagon (12-gon) have?
Example 3
easy
Which capital letter has both a horizontal and a vertical line of symmetry: H, A, or P?
Example 4
challenge
A shape has exactly 1 line of symmetry. Can it have rotational symmetry of order greater than 1? Explain.
Example 5
medium
Order of rotational symmetry of a regular octagon?
Example 6
easy
True or false: every shape has at least one line of symmetry.
Example 7
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 8
easy
How many lines of symmetry does a square have?Count the lines of symmetry of this square.
Example 9
medium
A figure has rotational symmetry of order 4 but no line symmetry. Give an example of such a figure.
Example 10
hard
Among all triangles, which type has the largest number of symmetries (rotations and reflections combined)?
Example 11
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 12
easy
How many lines of symmetry does a circle have?
Example 13
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 14
medium
How many lines of symmetry does a regular pentagon have? Explain your reasoning.
Example 15
hard
A regular polygon has 20 axes of symmetry. What is the smallest angle of rotational symmetry?
Example 16
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 17
medium
A regular polygon has each smallest rotational-symmetry angle equal to 30∘. How many sides does it have?
Example 18
medium
Does a rectangle (not a square) have any rotational symmetry? If yes, of what order?
Example 19
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 20
hard
A shape has rotational symmetry of order 3. What is the smallest angle you can rotate it so it looks the same?