Symmetry in Polygons

Find the lines of symmetry and the rotational symmetry of polygons: fold test, order of rotation, smallest angle and point symmetry.

Intermediate25 min18 problems
Practice Problems

Definition

A polygon has symmetry when a move (a fold or a turn) puts it exactly back onto itself.
1. Line (reflection) symmetry
A line of symmetry cuts a shape into two halves that are mirror images. Fold test: if you fold the shape along the line and the two halves match perfectly, the line is a line of symmetry.
2. Rotational symmetry
A shape has rotational symmetry when it looks exactly the same after a turn of less than 360°360° around its center.
  • The order of rotational symmetry is how many times the shape fits onto itself during one full turn (the full turn counts).
  • Order 1 means the shape only fits after the full 360°360° turn: it has no rotational symmetry.
  • The smallest angle of rotation is
smallest angle=360°order\text{smallest angle} = \frac{360°}{\text{order}}
3. Point symmetry
A shape has point symmetry when a half turn (180°180°) about its center puts it back onto itself. This is rotational symmetry with an even order (2, 4, 6, ...).
Regular polygons: a regular polygon with nn sides has exactly nn lines of symmetry and rotational symmetry of order nn, so its smallest angle of rotation is 360°n\frac{360°}{n}.
  • nn odd (triangle, pentagon): every line goes through a vertex and the midpoint of the opposite side.
  • nn even (square, hexagon): n2\frac{n}{2} lines go through two opposite vertices and n2\frac{n}{2} lines through the midpoints of two opposite sides.
ShapeLines of symmetryOrder of rotation
Square44
Rectangle22
Rhombus22
Parallelogram02
Kite11
Isosceles trapezoid11
Equilateral triangle33
Isosceles triangle11
Scalene triangle01

Try it now

How many lines of symmetry does a regular hexagon have?

Worked Examples

A rectangle measures 66 cm by 44 cm. How many lines of symmetry does it have?
1

Fold it top to bottom

The fold line goes through the midpoints of the two short sides. Both halves are 66 cm by 22 cm and match. → 1 line of symmetry
2

Fold it left to right

The fold line goes through the midpoints of the two long sides. Both halves are 33 cm by 44 cm and match. → 2nd line of symmetry
3

Test a diagonal

Folding along a diagonal flips one triangle over, but the corners do not land on each other because 6≠46 \neq 4. → A diagonal is NOT a line of symmetry

Common Mistakes

Counting the diagonals of a rectangle as lines of symmetry

Why it's wrong: A diagonal cuts the rectangle into two congruent triangles, but congruent halves are not enough: they must be mirror images across the line.

Correct: Use the fold test. Folded along a diagonal, the corners of a rectangle do not meet, so it has only 22 lines of symmetry.

Writing the angle instead of the order (saying a square has order 9090)

Why it's wrong: The order and the smallest angle are two different numbers that belong together.

Correct: The order counts the fits in a full turn (44 for a square); the angle is 360°order=90°\frac{360°}{\text{order}} = 90°.

Thinking every shape with rotational symmetry also has line symmetry

Why it's wrong: Many familiar symmetric shapes have both, so students assume they always come together.

Correct: A parallelogram has order 22 but 00 lines of symmetry. Always check folds and turns separately.

Counting each line of symmetry of an even polygon twice

Why it's wrong: A line through two opposite vertices touches two vertices, so it gets counted at both ends.

Correct: Count pairs: a regular hexagon has 33 lines through opposite vertices and 33 through opposite midpoints, 66 in total.

Interactive Visual

Symmetry

Lines of symmetry

5

Order of rotational symmetry

5

Smallest rotation angle

360° ÷ 5 = 72°

A line of symmetry folds the shape exactly onto itself. The order of rotational symmetry counts how many times the shape fits onto its outline during one full turn.

Symmetry

Count first: how many lines of symmetry, and what is the order of rotational symmetry?

A line of symmetry folds the shape exactly onto itself. The order of rotational symmetry counts how many times the shape fits onto its outline during one full turn.

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which shape has exactly 22 lines of symmetry and rotational symmetry of order 22?

Why It Matters

Symmetry is one of the first things our eyes notice, and designers use it on purpose:
  • Road signs: a stop sign is a regular octagon with 8 lines of symmetry and order 8, so it looks right from every side.
  • Logos: many brand logos have rotational symmetry of order 3 or 4, which makes them look balanced.
  • Nature: snowflakes have 6 lines of symmetry and order 6, flowers often have 5.
  • Wheels and hubcaps: identical spokes spaced evenly give rotational symmetry, so the wheel is balanced when it spins.
Symmetry is also a shortcut in geometry: if you know a shape is symmetric, you know which sides and angles are equal without measuring them.

Real World Applications

Road Signs

A stop sign is a regular octagon. Its symmetry makes it easy to recognize, even when it is seen at an angle or is slightly turned.

Example:

A yield sign is an equilateral triangle: 33 lines of symmetry and order 33, so it fits onto itself after 360°3=120°\frac{360°}{3} = 120°.

1Try It Yourself

A worker mounts a stop sign (a regular octagon) on a post that can turn. She wants to turn it as little as possible so that it still looks exactly the same.

By how many degrees should she turn it?

Step 1: Write the mathematical expression

Smallest angle of rotation of a regular octagon:

Hubcaps and Wheels

Wheel designers space identical spokes evenly. The wheel then has rotational symmetry, which keeps it balanced when it spins.

Example:

A wheel with 44 identical spokes has order 44: it looks the same after every 360°4=90°\frac{360°}{4} = 90°.

2Try It Yourself

A hubcap looks exactly the same after a turn of 72°72°, and 72°72° is the smallest such turn.

How many identical spokes does the hubcap have?

Step 1: Write the mathematical expression

Order of rotational symmetry from the smallest angle:

Snowflakes

Ice crystals grow in six directions, so a snowflake has $6$ lines of symmetry and rotational symmetry of order $6$, just like a regular hexagon.

Example:

A snowflake fits onto itself after 360°6=60°\frac{360°}{6} = 60°, and because 66 is even it also has point symmetry.

3Try It Yourself

All 66 lines of symmetry of a snowflake pass through its center and are spaced evenly. Each line passes straight through the center, so it covers a half turn (180°180°) of directions.

What is the angle between two neighboring lines of symmetry?

Step 1: Write the mathematical expression

Split the half turn among the 6 lines:

Key Takeaways

  • 1A line of symmetry splits a shape into two mirror-image halves (fold test)
  • 2The order of rotational symmetry counts how many times a shape fits onto itself in one full turn; order 1 means no rotational symmetry
  • 3Smallest angle of rotation =360°order= \frac{360°}{\text{order}}, and order =360°smallest angle= \frac{360°}{\text{smallest angle}}
  • 4Point symmetry means a half turn (180°180°) fits: it happens exactly when the order is even
  • 5A regular polygon with nn sides has nn lines of symmetry and order nn
  • 6Quadrilaterals: square 4 and 4, rectangle and rhombus 2 and 2, parallelogram 0 and 2, kite and isosceles trapezoid 1 and 1

Frequently Asked Questions

Every shape fits onto itself after a full 360°360° turn, so the count of fits is at least 11. Order 11 means that full turn is the only fit, which is the same as saying the shape has no rotational symmetry.
Every shape fits onto itself after a full 360°360° turn, so the count of fits is at least 11. Order 11 means that full turn is the only fit, which is the same as saying the shape has no rotational symmetry.
Point symmetry is one special kind of rotational symmetry: a half turn (180°180°) fits. A shape with order 22, 44 or 66 has it. An equilateral triangle (order 33) has rotational symmetry but no point symmetry.
For a polygon with at least one line of symmetry, yes: the square has 44 and 44, the kite 11 and 11. A polygon with no lines can still have rotational symmetry, like the parallelogram with order 22.
Infinitely many: every line through the center is a line of symmetry, and a circle fits onto itself after a turn by any angle. A circle is not a polygon, so it does not appear in the table.

Glossary

Line of symmetry
A line that cuts a shape into two halves that are mirror images of each other.
Line (reflection) symmetry
A shape has line symmetry if it has at least one line of symmetry.
Rotational symmetry
A shape has rotational symmetry if it looks the same after a turn of less than 360°360° around its center.
Order of rotational symmetry
The number of times a shape fits onto itself during one full turn. Order 11 means no rotational symmetry.
Angle of rotation
The angle a shape is turned through. The smallest one that fits is 360°order\frac{360°}{\text{order}}.
Point symmetry
A shape has point symmetry if a half turn (180°180°) about its center puts it back onto itself.

Formula Card

Smallest angle of rotation

angle=360°order\text{angle} = \frac{360°}{\text{order}}

Divide the full turn by the order of rotational symmetry

Order from the angle

order=360°angle\text{order} = \frac{360°}{\text{angle}}

Use it when you know the smallest turn that fits

Regular polygon with n sides

nn lines of symmetry, order nn

Odd $n$: each line through a vertex and the opposite midpoint; even $n$: half through vertices, half through midpoints

Point symmetry

order is even\text{order is even}

A half turn of $180°$ fits exactly when the order is 2, 4, 6, ...

Today's challengeOne problem, three tries

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