Symmetry
Introduced| English | Meaning |
|---|---|
| line symmetry/laɪn ˈsɪmətri/ | A property of a shape that maps onto itself when reflected in a mirror line. |
| mirror line/ˈmɪrə laɪn/ | The line in which a reflection flips a shape. |
| rotational symmetry/rəʊˈteɪʃənl ˈsɪmətri/ | The property of a shape that matches itself after a rotation of less than one full turn. |
| order/ˈɔːdə/ | The number of times a shape matches its original appearance during one complete turn, when describing rotational symmetry. |
| plane of symmetry/pleɪn ɒv ˈsɪmətri/ | A flat plane dividing a solid into two mirror-image halves. |
The snowflake test
- A snowflake has 6 lines of symmetry — fold it along any one and the halves match perfectly.
- Symmetry is nature's way of saying "this shape is balanced" — and mathematicians have a precise vocabulary for it.
Line symmetry 轴对称
- A shape has line symmetry (reflection symmetry) if a mirror line 对称轴 splits it into two matching halves.
- A square has 4 lines of symmetry: two diagonals and two through the midpoints of opposite sides.
- A nonsquare rectangle has 2 (only through midpoints); an equilateral triangle has 3.
Letter test. The letter "A" has 1 vertical line of symmetry. "H" has 2 (vertical and horizontal). "S" has 0 lines but has rotational symmetry 旋转对称.
Angles around a point add up to 360 degrees
Symmetry as a reflection
A shape has line symmetry if reflecting it leaves it unchanged. Reflect the shape and watch what is preserved.
How many lines of symmetry does a square have?
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Two diagonals and two through the side midpoints: 4 lines.
Line symmetry means a mirror line splits the shape into:
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A line of symmetry reflects one half exactly onto the other.
An equilateral triangle has ______ lines of symmetry.
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Each line goes from a vertex to the midpoint of the opposite side: 3 lines.
Rotational symmetry
- A shape has rotational symmetry if it fits onto itself as you turn it around its centre.
- The order 阶 is how many times it fits in one full $360^{\circ}$ turn.
- A square has order $4$ (fits at $0^{\circ}$, $90^{\circ}$, $180^{\circ}$, $270^{\circ}$).
- A nonsquare rectangle has order $2$ (fits at $0^{\circ}$ and $180^{\circ}$).
Order 1 means no symmetry. Every shape fits itself after a full turn, so order 1 is not counted as rotational symmetry.
What is the order of rotational symmetry of a square?
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A square fits onto itself 4 times in one full turn.
What is the order of rotational symmetry of a rectangle (not a square)?
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A rectangle fits itself at 0° and 180° — order 2.
A shape with rotational symmetry of order 1 has rotational symmetry.
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Order 1 means it only fits itself after a full turn — that is no rotational symmetry.
Symmetry in 3D (Extended)
- A plane of symmetry 对称面 slices a solid into two mirror halves (like cutting an apple).
- An axis of symmetry is a line you can rotate the solid around so it fits itself.
- A cube has 9 planes of symmetry and 13 axes of symmetry.
Reflection in a mirror line produces a symmetric image — the basis of line symmetry.
You've got it
- line symmetry: a mirror line gives two matching halves
- rotational symmetry order = times it fits itself in one full turn
- a square: 4 lines of symmetry, rotational order 4
Write from 3 lines of symmetry
- An equilateral triangle.
- Count the lines of symmetry.
How many lines of symmetry does an equilateral triangle have?
Three identical corners give three mirror lines.
How many lines of symmetry?
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Each vertex starts a line of symmetry.