The Letter S Has No Mirror Line, Yet It Is Still Symmetric
Learn to find every line of symmetry in a figure, identify point symmetry and its centre, and work out the order of rotational symmetry and the angle of rotation.
Can a figure be symmetric without having a mirror line?
Yes — and the letter S proves it.
Fold a printed S along a vertical line and the halves do not match. Fold it horizontally and they do not match either. No line of symmetry exists, in any direction.
But turn it upside down — rotate it through — and it lands exactly on itself. The S is unchanged by that half turn, which is a genuine symmetry even though no mirror is involved.
So symmetry comes in more than one kind. Line symmetry is about reflection, rotational symmetry is about turning, and point symmetry is the particular case of turning through half a revolution. A figure can have any combination of the three, including one without the others. This page covers the first part of the ICSE Class 8 Mathematics chapter on symmetry.
Fold a printed S along a vertical line and the halves do not match. Fold it horizontally and they do not match either. No line of symmetry exists, in any direction.
But turn it upside down — rotate it through — and it lands exactly on itself. The S is unchanged by that half turn, which is a genuine symmetry even though no mirror is involved.
So symmetry comes in more than one kind. Line symmetry is about reflection, rotational symmetry is about turning, and point symmetry is the particular case of turning through half a revolution. A figure can have any combination of the three, including one without the others. This page covers the first part of the ICSE Class 8 Mathematics chapter on symmetry.
How many lines of symmetry does a figure have?
As many as the folds that leave the two halves matching exactly.
A line of symmetry is a line along which the figure could be folded so that one half lands precisely on the other. The test is that strict: roughly matching is not symmetry.
Regular polygons follow a simple rule — a regular polygon with sides has exactly lines of symmetry:
- Equilateral triangle:
- Square:
- Regular pentagon:
- Regular hexagon:
- Regular decagon:
- Circle: infinitely many, since every diameter works
**Where those lines run depends on whether is odd or even. In a regular pentagon each line goes from a vertex to the midpoint of the opposite side — there is no opposite vertex to aim at. In a regular hexagon**, three lines join opposite vertices and three join the midpoints of opposite sides. So for odd all the lines are of one type, and for even they split evenly into two types.
Now the quadrilaterals, where marks are lost:
- Square:
- Rectangle: — the lines through the midpoints of opposite sides
- Rhombus: — its two diagonals
- Parallelogram:
- Kite: — the diagonal joining the unequal angles
- Isosceles trapezium:
The rectangle is the trap. Its diagonals are not lines of symmetry. Fold a non-square rectangle along a diagonal and the halves are triangles that do not coincide — the shorter side folds onto the longer one. It is worth actually folding a piece of paper once, because the drawing looks convincing and the fold settles it.
Triangles: equilateral , isosceles (the line through the apex), scalene .
Capital letters, worth memorising for the standard question:
- Vertical line only: A, M, T, U, V, W, Y
- Horizontal line only: B, C, D, E, K
- Both lines: H, I, O, X
- No line at all: F, G, J, L, N, P, Q, R, S, Z
Notice that N, S and Z are in the last group, yet each has the half-turn symmetry described in the opening — which is exactly why lines alone do not describe symmetry completely.
A line of symmetry is a line along which the figure could be folded so that one half lands precisely on the other. The test is that strict: roughly matching is not symmetry.
Regular polygons follow a simple rule — a regular polygon with sides has exactly lines of symmetry:
- Equilateral triangle:
- Square:
- Regular pentagon:
- Regular hexagon:
- Regular decagon:
- Circle: infinitely many, since every diameter works
**Where those lines run depends on whether is odd or even. In a regular pentagon each line goes from a vertex to the midpoint of the opposite side — there is no opposite vertex to aim at. In a regular hexagon**, three lines join opposite vertices and three join the midpoints of opposite sides. So for odd all the lines are of one type, and for even they split evenly into two types.
Now the quadrilaterals, where marks are lost:
- Square:
- Rectangle: — the lines through the midpoints of opposite sides
- Rhombus: — its two diagonals
- Parallelogram:
- Kite: — the diagonal joining the unequal angles
- Isosceles trapezium:
The rectangle is the trap. Its diagonals are not lines of symmetry. Fold a non-square rectangle along a diagonal and the halves are triangles that do not coincide — the shorter side folds onto the longer one. It is worth actually folding a piece of paper once, because the drawing looks convincing and the fold settles it.
Triangles: equilateral , isosceles (the line through the apex), scalene .
Capital letters, worth memorising for the standard question:
- Vertical line only: A, M, T, U, V, W, Y
- Horizontal line only: B, C, D, E, K
- Both lines: H, I, O, X
- No line at all: F, G, J, L, N, P, Q, R, S, Z
Notice that N, S and Z are in the last group, yet each has the half-turn symmetry described in the opening — which is exactly why lines alone do not describe symmetry completely.
What is point symmetry and how do you find its centre?
**A figure has point symmetry if turning it through about some point leaves it looking unchanged. That point is the centre of symmetry.
Equivalently: for every part of the figure at a certain distance and direction from the centre, there is a matching part at the same distance in the exactly opposite direction.
Figures that have point symmetry, with the centre named:
- Parallelogram — the intersection of the diagonals
- Rectangle, rhombus, square — the intersection of the diagonals
- Regular hexagon — its centre
- Circle — its centre
- Letters H, I, O, X, N, S, Z
Figures that do not:
- Equilateral triangle
- Regular pentagon
- Kite and isosceles trapezium
- Letters A, B, C, D, E, T, U, V, W, Y
Why the equilateral triangle fails.** Rotate it and it lands point-down instead of point-up — a different orientation, so not a symmetry. It certainly has rotational symmetry, but through and only, and neither of those is a half turn.
The rule that decides every regular polygon. A regular -gon has point symmetry exactly when is even. The square () and the regular hexagon () have it; the equilateral triangle () and the regular pentagon () do not. A half turn must be one of the allowed rotations, and divides evenly into the turning pattern only when is even.
The parallelogram is the instructive case. It has no lines of symmetry at all, yet it has point symmetry about the crossing of its diagonals. Line symmetry and point symmetry are independent properties, and a figure can hold either without the other.
A useful everyday check. Most playing cards are printed with point symmetry about their centre — which is why a court card looks the same whichever way up it is dealt, and why you never need to turn one round. Rotate the card a half turn and the design maps onto itself.
Equivalently: for every part of the figure at a certain distance and direction from the centre, there is a matching part at the same distance in the exactly opposite direction.
Figures that have point symmetry, with the centre named:
- Parallelogram — the intersection of the diagonals
- Rectangle, rhombus, square — the intersection of the diagonals
- Regular hexagon — its centre
- Circle — its centre
- Letters H, I, O, X, N, S, Z
Figures that do not:
- Equilateral triangle
- Regular pentagon
- Kite and isosceles trapezium
- Letters A, B, C, D, E, T, U, V, W, Y
Why the equilateral triangle fails.** Rotate it and it lands point-down instead of point-up — a different orientation, so not a symmetry. It certainly has rotational symmetry, but through and only, and neither of those is a half turn.
The rule that decides every regular polygon. A regular -gon has point symmetry exactly when is even. The square () and the regular hexagon () have it; the equilateral triangle () and the regular pentagon () do not. A half turn must be one of the allowed rotations, and divides evenly into the turning pattern only when is even.
The parallelogram is the instructive case. It has no lines of symmetry at all, yet it has point symmetry about the crossing of its diagonals. Line symmetry and point symmetry are independent properties, and a figure can hold either without the other.
A useful everyday check. Most playing cards are printed with point symmetry about their centre — which is why a court card looks the same whichever way up it is dealt, and why you never need to turn one round. Rotate the card a half turn and the design maps onto itself.
Formula
What is the order of rotational symmetry and the angle of rotation?
The order is how many times a figure fits onto itself in one full turn, and the angle is the smallest turn that works:
Worked example 1 — a square. Turn a square about its centre through and it looks unchanged. So do , and — four positions in a full turn, so the order is 4 and the angle of rotation is
Worked example 2 — regular polygons. A regular -gon has order and angle :
- Equilateral triangle: order , angle
- Square: order , angle
- Regular pentagon: order , angle
- Regular hexagon: order , angle
- Regular octagon: order , angle
- Regular polygon with sides: order , angle
A regular polygon has the same number of lines of symmetry as its order — both equal . That coincidence holds for regular polygons and for almost nothing else.
Worked example 3 — the quadrilaterals.
- Rectangle: order , angle — but only lines of symmetry, so the two counts happen to agree here
- Rhombus: order , angle
- Parallelogram: order , angle , with zero lines of symmetry — the counts now disagree completely
- Kite: order
- Isosceles trapezium: order , despite having a line of symmetry
Order 1 means no real rotational symmetry. Every figure returns to itself after a full turn, so every figure has order at least . Writing order 0 is always wrong.
Worked example 4 — working backwards. *A figure has an angle of rotation of . What is its order?*
And *a figure has lines of symmetry and is a regular polygon* means order and an angle of .
The connection back to point symmetry. A figure has point symmetry exactly when is one of its rotations — that is, when its order is even. Order , , and all include a half turn; order and do not. So the parallelogram (order ) has point symmetry and the regular pentagon (order ) does not, and the one test settles both.
Do not confuse order with the angle. An order of goes with an angle of , and the two numbers multiply to . Quoting or an order of is a slip that a single multiplication would have caught.
Worked example 1 — a square. Turn a square about its centre through and it looks unchanged. So do , and — four positions in a full turn, so the order is 4 and the angle of rotation is
Worked example 2 — regular polygons. A regular -gon has order and angle :
- Equilateral triangle: order , angle
- Square: order , angle
- Regular pentagon: order , angle
- Regular hexagon: order , angle
- Regular octagon: order , angle
- Regular polygon with sides: order , angle
A regular polygon has the same number of lines of symmetry as its order — both equal . That coincidence holds for regular polygons and for almost nothing else.
Worked example 3 — the quadrilaterals.
- Rectangle: order , angle — but only lines of symmetry, so the two counts happen to agree here
- Rhombus: order , angle
- Parallelogram: order , angle , with zero lines of symmetry — the counts now disagree completely
- Kite: order
- Isosceles trapezium: order , despite having a line of symmetry
Order 1 means no real rotational symmetry. Every figure returns to itself after a full turn, so every figure has order at least . Writing order 0 is always wrong.
Worked example 4 — working backwards. *A figure has an angle of rotation of . What is its order?*
And *a figure has lines of symmetry and is a regular polygon* means order and an angle of .
The connection back to point symmetry. A figure has point symmetry exactly when is one of its rotations — that is, when its order is even. Order , , and all include a half turn; order and do not. So the parallelogram (order ) has point symmetry and the regular pentagon (order ) does not, and the one test settles both.
Do not confuse order with the angle. An order of goes with an angle of , and the two numbers multiply to . Quoting or an order of is a slip that a single multiplication would have caught.
Exam tip
Exam tip: count the lines by folding, the order by turning
Test a line of symmetry by imagining the fold, and be strict — the halves must coincide exactly.
**A rectangle has lines of symmetry, not . Its diagonals are not mirror lines; fold a paper rectangle along one and watch the halves miss.
A regular -gon** has lines of symmetry and order . For odd every line runs vertex to opposite-side midpoint; for even half are vertex to vertex and half are midpoint to midpoint.
Quadrilateral lines of symmetry: square , rectangle , rhombus , kite , isosceles trapezium , **parallelogram .
Angle of rotation **, so order and angle always multiply to — a free check.
**Order is never .** A figure with no rotational symmetry has order .
**Point symmetry means a turn works, which happens exactly when the order is even. A parallelogram has it; an equilateral triangle and a regular pentagon do not.
Line symmetry and point symmetry are independent. The parallelogram has point symmetry with no mirror line; the kite has a mirror line with no point symmetry.
Learn the letters: vertical A M T U V W Y, horizontal B C D E K, both H I O X, and N S Z with point symmetry but no line at all.
And name the centre or the line** in your answer — about the intersection of the diagonals, not just yes it is symmetric.
**A rectangle has lines of symmetry, not . Its diagonals are not mirror lines; fold a paper rectangle along one and watch the halves miss.
A regular -gon** has lines of symmetry and order . For odd every line runs vertex to opposite-side midpoint; for even half are vertex to vertex and half are midpoint to midpoint.
Quadrilateral lines of symmetry: square , rectangle , rhombus , kite , isosceles trapezium , **parallelogram .
Angle of rotation **, so order and angle always multiply to — a free check.
**Order is never .** A figure with no rotational symmetry has order .
**Point symmetry means a turn works, which happens exactly when the order is even. A parallelogram has it; an equilateral triangle and a regular pentagon do not.
Line symmetry and point symmetry are independent. The parallelogram has point symmetry with no mirror line; the kite has a mirror line with no point symmetry.
Learn the letters: vertical A M T U V W Y, horizontal B C D E K, both H I O X, and N S Z with point symmetry but no line at all.
And name the centre or the line** in your answer — about the intersection of the diagonals, not just yes it is symmetric.
Did you know
Where symmetry is chosen on purpose
Symmetry is not only something to be spotted. It is something designers reach for, and usually for a practical reason rather than a decorative one.
A nut on a bolt is a regular hexagon, giving it order . A spanner can therefore be fitted in six different positions, and after each quarter of a turn in a tight space you can lift off, reposition and carry on. Order means a fresh grip every . A square nut, with order , would only offer a new grip every — noticeably worse in a corner.
Rangoli and jali patterns are built on rotational symmetry of a chosen order. A design with order can be drawn once in a wedge and then repeated, which is both faster to lay out and why such patterns look balanced from any side of the courtyard.
A manhole cover is round precisely because a circle has infinite order. Whatever way it is turned it still fits the opening, and — unlike a square cover — it cannot be dropped through its own hole on the diagonal.
Even the point symmetry of playing cards exists to save the player from turning them round, and the same reasoning puts point symmetry into the design of many flags and logos.
So the order of a figure is not simply a number to count. It tells you how many different positions an object can occupy while still doing its job — which, for anything that has to be gripped, fitted or read from more than one direction, is the whole design problem.
A nut on a bolt is a regular hexagon, giving it order . A spanner can therefore be fitted in six different positions, and after each quarter of a turn in a tight space you can lift off, reposition and carry on. Order means a fresh grip every . A square nut, with order , would only offer a new grip every — noticeably worse in a corner.
Rangoli and jali patterns are built on rotational symmetry of a chosen order. A design with order can be drawn once in a wedge and then repeated, which is both faster to lay out and why such patterns look balanced from any side of the courtyard.
A manhole cover is round precisely because a circle has infinite order. Whatever way it is turned it still fits the opening, and — unlike a square cover — it cannot be dropped through its own hole on the diagonal.
Even the point symmetry of playing cards exists to save the player from turning them round, and the same reasoning puts point symmetry into the design of many flags and logos.
So the order of a figure is not simply a number to count. It tells you how many different positions an object can occupy while still doing its job — which, for anything that has to be gripped, fitted or read from more than one direction, is the whole design problem.
Key takeaways
Lines, points and rotational symmetry: quick revision
- Line symmetry is reflection, rotational symmetry is turning, and point symmetry is the special case of a turn. A figure can have any one without the others.
- A **regular -gon** has lines of symmetry and order : triangle , square , pentagon , hexagon , decagon . A circle has infinitely many.
- For **odd every line runs from a vertex to the midpoint of the opposite side; for even half join opposite vertices and half join opposite side midpoints.
- Quadrilateral lines of symmetry**: square , rectangle , rhombus (its diagonals), kite , isosceles trapezium , parallelogram .
- A rectangle's diagonals are not lines of symmetry — folding along one makes the shorter side land on the longer.
- Triangles: equilateral , isosceles , scalene .
- Letters — vertical only: A M T U V W Y. Horizontal only: B C D E K. Both: H I O X. None: F G J L N P Q R S Z.
- Point symmetry means a turn about a centre leaves the figure unchanged. Parallelogram, rectangle, rhombus, square, regular hexagon and circle have it, with the centre at the diagonal intersection or the centre of the figure.
- Letters N, S and Z have point symmetry but no line of symmetry; the equilateral triangle, regular pentagon, kite and isosceles trapezium have no point symmetry.
- A **regular -gon has point symmetry exactly when is even.
- Order** is the number of positions in one full turn, and .
- Orders and angles: triangle , square , pentagon , hexagon , octagon , twelve-sided .
- **Rectangle, rhombus and parallelogram all have order ** and angle — but the parallelogram has no lines of symmetry, so the two counts are independent.
- **Kite and isosceles trapezium have order , and order is never ** since every figure returns after .
- Backwards: an angle of means order ; nine lines of symmetry on a regular polygon means order and angle .
- Point symmetry holds exactly when the order is even, because must be one of the rotations.
Take the letters of your own name in capitals and sort them into the four symmetry groups, then check which of them have point symmetry — it is the fastest way to make the distinction stick.
- A **regular -gon** has lines of symmetry and order : triangle , square , pentagon , hexagon , decagon . A circle has infinitely many.
- For **odd every line runs from a vertex to the midpoint of the opposite side; for even half join opposite vertices and half join opposite side midpoints.
- Quadrilateral lines of symmetry**: square , rectangle , rhombus (its diagonals), kite , isosceles trapezium , parallelogram .
- A rectangle's diagonals are not lines of symmetry — folding along one makes the shorter side land on the longer.
- Triangles: equilateral , isosceles , scalene .
- Letters — vertical only: A M T U V W Y. Horizontal only: B C D E K. Both: H I O X. None: F G J L N P Q R S Z.
- Point symmetry means a turn about a centre leaves the figure unchanged. Parallelogram, rectangle, rhombus, square, regular hexagon and circle have it, with the centre at the diagonal intersection or the centre of the figure.
- Letters N, S and Z have point symmetry but no line of symmetry; the equilateral triangle, regular pentagon, kite and isosceles trapezium have no point symmetry.
- A **regular -gon has point symmetry exactly when is even.
- Order** is the number of positions in one full turn, and .
- Orders and angles: triangle , square , pentagon , hexagon , octagon , twelve-sided .
- **Rectangle, rhombus and parallelogram all have order ** and angle — but the parallelogram has no lines of symmetry, so the two counts are independent.
- **Kite and isosceles trapezium have order , and order is never ** since every figure returns after .
- Backwards: an angle of means order ; nine lines of symmetry on a regular polygon means order and angle .
- Point symmetry holds exactly when the order is even, because must be one of the rotations.
Take the letters of your own name in capitals and sort them into the four symmetry groups, then check which of them have point symmetry — it is the fastest way to make the distinction stick.