E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Line Symmetry and Rotational Symmetry

E1.1About 40 minutes

Learning goals

I can find all the lines of symmetry of a quadrilateral.
I can find the order of rotational symmetry of a quadrilateral and the smallest angle of turn.
I can add symmetry to my property lists for squares, rectangles, rhombuses, parallelograms, trapezoids and kites.

Words to know

line of symmetry
A line that divides a shape into two halves that match exactly when you fold along it. Each half is a mirror image of the other.

rotational symmetry
A shape has rotational symmetry if it fits exactly onto its own outline after a turn of less than 360° around its centre.

order of rotational symmetry
The number of times a shape fits onto its own outline during one full turn (360°). A shape that only fits after a full turn has order 1, which means no rotational symmetry.

centre of rotation
The point a shape turns around.

isosceles trapezoid
A trapezoid whose two non-parallel sides are equal in length.

Remember from Grade 4

In Grade 4 you found lines of symmetry: if you fold a shape along the line, the two halves match exactly. You found that a rectangle has 2 lines of symmetry and a square has 4. Today you will check every quadrilateral, and learn a second kind of symmetry: rotational symmetry.

Let's start with a window

Look at a square window pane. Close your eyes while someone turns a square tile a quarter turn. When you open your eyes, can you tell it was turned? No! The square fits exactly into its own outline.

A shape that fits onto its own outline after a turn of less than a full turn has rotational symmetry. The number of times it fits during one full turn is its order of rotational symmetry.

Lines of symmetry, drawn as dashed lines. A rectangle has 2 (through the middles of opposite sides). A square has 4. A kite has 1 (the diagonal through the two corners where equal sides meet).

Testing for rotational symmetry

Trace the shape on a piece of paper. This tracing is the outline.
Put the shape on its outline. Hold the centre still with a pencil tip.
Turn the shape slowly through one full turn (360°).
Count how many times it fits exactly onto the outline. Count the full turn at the end as 1.

That count is the order. To find the smallest turn that works, divide: 360° ÷ order. A square has order 4, so it fits every 360° ÷ 4 = 90°.

A parallelogram ABCD turns a half turn (180°) around the point where its diagonals cross. It lands exactly on its own outline, so it has rotational symmetry of order 2.

Shape	Lines of symmetry	Order of rotational symmetry	Smallest turn that fits
Square	4	4	90°
Rectangle (not a square)	2	2	180°
Rhombus (not a square)	2 (the diagonals)	2	180°
Parallelogram (no right angles, sides not all equal)	0	2	180°
Kite (not a rhombus)	1	1 (none)	360° only
Isosceles trapezoid	1	1 (none)	360° only
Other trapezoids	0	1 (none)	360° only

Symmetry of quadrilaterals. "Order 1" means the shape only fits after a full turn, so it has no rotational symmetry.

Worked example

Example 1: Is a rectangle's diagonal a line of symmetry?

A rectangle is 8 cm long and 3 cm wide. Lee thinks the diagonal is a line of symmetry. Is he right?

Call the rectangle ABCD, with AB = 8 cm and AD = 3 cm. Imagine folding along diagonal AC. Corners A and C stay on the fold.
For the two halves to match, side AB (8 cm) would have to land exactly on side AD (3 cm), because both start at corner A.
They can't match, because 8 cm ≠ 3 cm. The folded corner sticks out past the edge. Try it with a paper rectangle to see!

No. A rectangle's lines of symmetry go through the middles of opposite sides, not along the diagonals. (Only in a square do the diagonals work too.)

Worked example

Example 2: Symmetry of a rhombus

Find the lines of symmetry and the order of rotational symmetry of a rhombus that is not a square.

Fold along each diagonal. All 4 sides are equal, so each half matches. That's 2 lines of symmetry.
Fold through the middles of opposite sides. The slanted corners don't match, so these are not lines of symmetry.
Turn it around the centre. It fits after a half turn (180°) and again after the full turn. That's 2 fits.

2 lines of symmetry (the diagonals) and rotational symmetry of order 2.

Worked example

Example 3: Use the order to find the turns

A shape has rotational symmetry of order 4. Which turns of less than 360° make it fit?

Smallest turn: 360° ÷ 4 = 90°.
Keep adding 90°: 90°, 180°, 270°. (The next one, 360°, is the full turn.)

90°, 180° and 270°.

Just for fun

Write CHOICE in capital letters and fold it along a horizontal line through the middle: every letter matches itself! BOX and DECODE work too. Can you find another word?

Count the full turn

Every shape fits its outline after a full 360° turn, so every shape has order at least 1. When you count, include that last full turn. If the only fit is the full turn, the order is 1 and we say the shape has no rotational symmetry.

Hands-on activity

Fold it, turn it

You need: paper, a ruler, a pencil, scissors (ask a grown-up to help with cutting) and a pushpin or a pencil tip.

With a ruler, draw a square, a rectangle, a rhombus, a parallelogram, a kite and a trapezoid. Make each one about the size of your hand.
Cut them out. Trace each one onto another sheet to make its outline.
Line symmetry: fold each shape in every way you can think of. Draw the fold lines that make the halves match.
Rotational symmetry: put each shape on its outline, hold the centre with the pencil tip (or a pushpin, with a grown-up), and turn it slowly. Count the fits in one full turn.
Make your own table like the one in this lesson. Does yours match?

Math talk: "Which shape surprised you? Why?"

Your turn

How many lines of symmetry does a square have?

What is the order of rotational symmetry of a rectangle that is not a square?

How many lines of symmetry does a parallelogram with no right angles and unequal sides have?

Which line is the kite's line of symmetry?

A shape has rotational symmetry of order 5. What is the smallest turn that makes it fit?

Name all the quadrilaterals in the table with rotational symmetry of order exactly 2.

More practice: warm-up

What happens when you fold a shape along a line of symmetry?

How many degrees is a full turn?

How many degrees is a half turn?

How many lines of symmetry does a rhombus (not a square) have?

Does an isosceles trapezoid have rotational symmetry?

More practice: on your own

A quadrilateral has 4 right angles, 2 lines of symmetry and rotational symmetry of order 2. Name it.

A quadrilateral has rotational symmetry of order 2 but no lines of symmetry. Name it.

Can a quadrilateral have 1 line of symmetry and still have no rotational symmetry? Give an example.

A shape has rotational symmetry of order 3. Find the smallest turn that makes it fit.

List every turn between 0° and 360° that makes a square fit onto its outline.

Challenge

Explain why every parallelogram has rotational symmetry of at least order 2. (Hint: think about its diagonals.)

Draw a quadrilateral that has exactly 1 line of symmetry and is not a kite.

A quadrilateral has rotational symmetry of order 4. Must it be a square? Explain.

Check your understanding

Quiz

How many lines of symmetry does a rectangle (not a square) have?

What is the order of rotational symmetry of a square?

Which quadrilateral has rotational symmetry of order 2 but no lines of symmetry?

A shape has rotational symmetry of order 6. What is the smallest turn that makes it fit?

Where are a rhombus's lines of symmetry?

Which quadrilateral has no rotational symmetry (order 1)?

For parents and teachers

Big idea: E1.1 includes line symmetry and rotational symmetry in the property lists for quadrilaterals. Rotational symmetry is new: a shape has it if it fits onto its own outline after less than a full turn. The order is how many fits there are in one full turn, and the smallest turn is 360° ÷ order.

Common difficulties:

Thinking a rectangle's diagonals are lines of symmetry. Fold a paper rectangle along its diagonal to see that it doesn't work.
Forgetting the full turn. When counting the order, include the fit at 360°.
Mixing up the two kinds of symmetry. Folding tests lines; turning tests rotation. A parallelogram has rotational symmetry but no lines of symmetry.

How to help: Cut paper shapes together (you handle the scissors if needed) and test them. Look for rotational symmetry in tiles, hubcaps, snowflakes and quilt blocks, and ask "How many times does it fit in one turn?"