Lines of Symmetry of a Rectangle

A rectangle measures $6$ cm by $4$ cm. How many lines of symmetry does it have?

Fold it top to bottom: The fold line goes through the midpoints of the two short sides. Both halves are $6$ cm by $2$ cm and match. = 1 line of symmetry

Fold it left to right: The fold line goes through the midpoints of the two long sides. Both halves are $3$ cm by $4$ cm and match. = 2nd line of symmetry

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

Answer: The rectangle has $2$ lines of symmetry, both through midpoints of opposite sides. The diagonals are not lines of symmetry.

Rotational Symmetry of a Regular Pentagon

Find the order of rotational symmetry and the smallest angle of rotation of a regular pentagon.

Count the positions: Each vertex can move to the place of the next one, and the pentagon looks the same. There are $5$ vertices. = Order $5$

Use the formula: $\text{smallest angle} = \frac{360°}{5}$ = $72°$

Check with the full turn: $5 \times 72° = 360°$ = Five turns of $72°$ make one full turn

Answer: A regular pentagon has rotational symmetry of order $5$ with a smallest angle of rotation of $72°$.

All the Symmetry of a Regular Octagon

A stop sign is a regular octagon. Describe all its lines of symmetry and its rotational symmetry.

Count the lines through opposite vertices: $8$ vertices make $\frac{8}{2} = 4$ pairs of opposite vertices. = $4$ lines

Count the lines through midpoints of opposite sides: $8$ sides make $\frac{8}{2} = 4$ pairs of opposite sides. = $4$ lines

Add them: $4 + 4 = 8$, equal to the number of sides = $8$ lines of symmetry

Find order and smallest angle: Order $8$, smallest angle $\frac{360°}{8} = 45°$ = $45°$

Check for point symmetry: $180° = 4 \times 45°$, so a half turn also fits = Point symmetry: yes

Answer: A regular octagon has $8$ lines of symmetry, rotational symmetry of order $8$ (smallest angle $45°$) and point symmetry.

A Parallelogram: Turning Without Folding

Does a parallelogram (that is not a rectangle or a rhombus) have line symmetry? Does it have rotational symmetry?

Try folding through the midpoints of two opposite sides: The slanted sides lean the same way, so after the fold they cross each other instead of matching. = Not a line of symmetry

Try folding along a diagonal: The two triangles are congruent, but folding flips one of them, so the vertices do not land on each other. = Not a line of symmetry

Turn it $180°$ about the point where the diagonals cross: The diagonals bisect each other, so each vertex lands on the opposite vertex. = It fits after a half turn

State order and point symmetry: It fits after $180°$ and $360°$: $\frac{360°}{180°} = 2$ = Order $2$, point symmetry

Answer: A parallelogram has $0$ lines of symmetry but rotational symmetry of order $2$, so it has point symmetry.

Working Backwards from the Angle

A regular polygon fits onto itself after a turn of $40°$, and $40°$ is the smallest such angle. How many sides and how many lines of symmetry does it have?

Find the order: $\text{order} = \frac{360°}{40°} = 9$ = Order $9$

Find the number of sides: For a regular polygon the order equals the number of sides = $9$ sides (a nonagon)

Find the lines of symmetry: A regular polygon has as many lines as sides = $9$ lines

Describe the lines: $9$ is odd, so every line goes through a vertex and the midpoint of the opposite side = No line joins two vertices

Answer: It is a regular nonagon: $9$ sides, $9$ lines of symmetry (each through a vertex and the midpoint of the opposite side), order $9$.

Mistake: Counting the diagonals of a rectangle as lines of symmetry

Why: 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 $2$ lines of symmetry.

Mistake: Writing the angle instead of the order (saying a square has order $90$)

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

Correct: The order counts the fits in a full turn ($4$ for a square); the angle is $\frac{360°}{\text{order}} = 90°$.

Mistake: Thinking every shape with rotational symmetry also has line symmetry

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

Correct: A parallelogram has order $2$ but $0$ lines of symmetry. Always check folds and turns separately.

Mistake: Counting each line of symmetry of an even polygon twice

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

Correct: Count pairs: a regular hexagon has $3$ lines through opposite vertices and $3$ through opposite midpoints, $6$ in total.

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.

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

Hubcaps and Wheels

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

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

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.

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

Q: Why do we say order 1 instead of order 0?

A: Every shape fits onto itself after a full $360°$ turn, so the count of fits is at least $1$. Order $1$ means that full turn is the only fit, which is the same as saying the shape has no rotational symmetry.

Q: Is point symmetry the same as rotational symmetry?

A: Point symmetry is one special kind of rotational symmetry: a half turn ($180°$) fits. A shape with order $2$, $4$ or $6$ has it. An equilateral triangle (order $3$) has rotational symmetry but no point symmetry.

Q: Does the number of lines always equal the order?

A: For a polygon with at least one line of symmetry, yes: the square has $4$ and $4$, the kite $1$ and $1$. A polygon with no lines can still have rotational symmetry, like the parallelogram with order $2$.

Q: How many lines of symmetry does a circle have?

A: 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.