A shape has line symmetry if it folds onto itself, and rotational symmetry if it fits onto itself more than once in a full turn. Solids have planes and axes of symmetry too (Extended).
Count lines of symmetry by asking where a mirror could go; count the order of rotational symmetry by asking how many times the shape fits onto itself in one full turn. Every shape has order at least 1, and a regular \(n\)-sided polygon has \(n\) lines and order \(n\).
Fig. 1: lines of symmetry (dashed) and order of rotational symmetry. A parallelogram has no lines of symmetry but fits onto itself after a half turn about its centre
Section 01
Line symmetry.
In the lab, place a mirror line on each shape and reflect it. If the reflection lands exactly on the original, the line is a line of symmetry. Try the diagonal of a rectangle: it looks promising, but the reflection does not fit.
Shape
Lines of symmetry
Order of rotational symmetry
Equilateral triangle
3
3
Isosceles triangle
1
1
Square
4
4
Rectangle
2
2
Rhombus
2 (the diagonals)
2
Parallelogram
0
2
Kite
1
1
Isosceles trapezium
1
1
Regular \(n\)-gon
\(n\)
\(n\)
Worked example Symmetry tells you about the shape
Triangle ABC has exactly one line of symmetry, through A. Angle BAC = \(40^\circ\). Find angle ABC.
One line of symmetry through A means the triangle is isosceles with AB = AC.
The mirror swaps B and C, so angle ABC = angle ACB.
Angles in a triangle add to \(180^\circ\): \(180^\circ - 40^\circ = 140^\circ\) for the two base angles.
Angle ABC \(= 140^\circ \div 2 = 70^\circ\).
Answer: \(70^\circ\)
Examiner trap
A rectangle's diagonals are NOT lines of symmetry: folding along a diagonal does not match the corners. A rhombus is the shape whose diagonals are lines of symmetry.
Section 02
Rotational symmetry.
Imagine tracing the shape and spinning the tracing about its centre. Count how many times it fits exactly during one full turn, including the starting position. That count is the order.
Order 1 means the shape only fits back at the full turn: it has no rotational symmetry.
Smallest angle of turn that fits \(= 360^\circ \div \text{order}\). A regular hexagon has order 6, so it fits after every \(60^\circ\).
Lines and order are separate questions: a parallelogram has order 2 but no lines (Fig. 1); a kite has one line but order 1.
Worked example Letters
For the capital letters H, S and A, write down the number of lines of symmetry and the order of rotational symmetry.
H: a vertical and a horizontal mirror line both work, and a half turn fits: 2 lines, order 2.
S: no mirror line works, but a half turn fits: 0 lines, order 2.
A: one vertical mirror line, and it only fits after a full turn: 1 line, order 1.
Answer: H: 2, order 2. S: 0, order 2. A: 1, order 1.
Worked example Regular polygons
A regular polygon fits onto itself after a turn of \(40^\circ\), and after no smaller turn. How many sides does it have, and how many lines of symmetry?
Order \(= 360^\circ \div 40^\circ = 9\).
A regular polygon has order equal to its number of sides, so it has 9 sides.
A regular \(n\)-gon has \(n\) lines of symmetry: 9 lines.
Answer: 9 sides (a regular nonagon), 9 lines of symmetry
Your turn
Write down the number of lines of symmetry and the order of rotational symmetry of a regular octagon. What is the smallest angle of turn that fits?
Order 0 does not exist. A shape with no rotational symmetry has order 1, because it always fits once, at the full turn.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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How many lines of symmetry does a square have?
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What is the order of rotational symmetry of an equilateral triangle?
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How many lines of symmetry does a rectangle (not a square) have?
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A shape has no rotational symmetry. What is its order of rotational symmetry?
Section 03
Symmetry of solids (Extended).
In the solids tab, slice each solid with a plane and see whether the two halves are mirror images. Then spin it about an axis and count how often it looks the same.
Solid
Planes of symmetry
Axes of rotational symmetry
Cuboid (all edges different)
3
3 axes, each order 2
Cube
9
13 axes (beyond IGCSE)
Right prism, regular \(n\)-gon cross-section
\(n + 1\)
main axis of order \(n\)
Right pyramid, square base
4
1 axis, order 4
Cylinder
infinitely many, plus 1 across the middle
main axis, infinite order
Cone
infinitely many (all through the axis)
1 axis, infinite order
Worked example Planes of a prism (Extended)
A cuboid measures 2 cm by 2 cm by 5 cm. How many planes of symmetry does it have?
The cross-section along the 5 cm length is a 2 cm by 2 cm square, which has 4 lines of symmetry.
Each line of the square, extended along the length, gives a plane: 4 planes.
One more plane cuts the 5 cm length in half, parallel to the square ends.
Total \(4 + 1 = 5\).
Answer: 5 planes of symmetry
Your turn
(Extended) How many planes of symmetry does a right prism with an equilateral triangle cross-section have?
The triangle has 3 lines of symmetry, giving 3 planes along the length, plus 1 plane across the middle: 4 planes.
Examiner trap
For prisms, remember the extra plane that cuts across the middle of the length. For a cube, the six diagonal planes (through opposite edges) are easy to miss: \(3 + 6 = 9\).
Stage 2 · Exam
Exam-style questions
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Which shape has rotational symmetry of order 2 but no lines of symmetry?
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Describe the symmetry of a regular octagon.
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(Extended) How many planes of symmetry does a cuboid measuring 3 cm by 4 cm by 6 cm have?
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(Extended) How many planes of symmetry does a right prism with a regular pentagon as its cross-section have?
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