Lines of symmetry: folding and mirrors

Geometry · level 2, lesson 2 of 6

A question to start

Fold a sheet of paper in half. Draw half a heart against the fold, with the middle of the heart on the fold line. Cut along your line, through both layers, and open it up.

You only drew half a heart, but you got a whole one, and the two halves match perfectly. How did the paper know how to draw the other half?

Here is a second puzzle. A square can be folded in half so the halves match in 4 different ways. A rectangle (like a sheet of paper) can only be folded in 2 ways. And a circle can be folded in half in more ways than you could ever count. Why?

The idea

What a line of symmetry is

Fold a shape along a straight line. If the two halves land exactly on top of each other, with nothing sticking out, then that line is a line of symmetry (say "SIM-e-tree"). It is also called a mirror line, and you will soon see why.

A shape with a line of symmetry is called symmetric. One half is the mirror image of the other: the same, but facing the opposite way, like your left hand and your right hand.

The heart puzzle is now easy to explain. When you cut through two layers of folded paper, each cut goes through both layers at once, at the same distance from the fold. When you open it, one layer's cut is on one side and the other layer's cut is on the other side. The fold becomes a line of symmetry automatically. The paper "drew" the other half by copying your cut, flipped over.

Testing with a mirror

You do not always have to fold. Stand a small mirror upright on the line you want to test. Look into the mirror from one side. If the half you see on the paper, together with its reflection in the mirror, looks exactly like the whole shape, then the mirror is standing on a line of symmetry.

That is why it is called a mirror line. A mirror shows you a flipped copy of what is in front of it. Every point in the reflection is the same distance behind the mirror as the real point is in front of it.

Lines of symmetry of some shapes

• A square has 4 lines of symmetry: up the middle, across the middle, and both corner-to-corner lines.
• A rectangle that is not a square has only 2: up the middle and across the middle.
• An equilateral triangle (all sides equal) has 3: each from a corner to the middle of the opposite side.
• A circle has endless lines of symmetry. Any straight line through the centre works.

Why does the circle have so many? Every point of a circle is the same distance from the centre. So whichever line you fold along through the centre, each point on one half lands on a point of the other half that is the same distance from the centre: a point of the circle. Every direction works, and there are endless directions.

Why the rectangle's corner-to-corner line fails

The corner-to-corner line of a rectangle cuts it into two equal triangles. So it seems like it should be a line of symmetry. But try folding a sheet of paper that way!

The folded half does not land on the other half. It sticks out over the edge, and leaves parts of the rectangle uncovered. Cutting a shape into two equal pieces is not the same as folding it onto itself. The two triangles are the same size and shape, but one is turned round compared with the other, not flipped. A line of symmetry needs the halves to be mirror images, landing on top of each other when you fold.

For a square, the corner-to-corner fold does work. That is because a square's sides are all equal, so when a side folds over, it lands exactly on the next side.

Regular polygons

Look at the regular pentagon and the regular hexagon.

• The regular pentagon has 5 lines of symmetry. Each goes from a corner to the middle of the opposite side.
• The regular hexagon has 6 lines of symmetry. 3 go from a corner to the opposite corner, and 3 go from the middle of a side to the middle of the opposite side.

Put it all together:

Regular polygon	Sides	Lines of symmetry
equilateral triangle	3	3
square	4	4
regular pentagon	5	5
regular hexagon	6	6
regular octagon	8	8

A regular polygon has as many lines of symmetry as it has sides. Why? Each line goes through the very middle of the shape. A regular polygon is the same all the way round, so whatever works at one corner works at every corner. Every corner gets a line through it, and every side gets a line through its middle. When the number of sides is odd, each line goes through one corner and one side-middle, so there is one line for each corner. When the number is even, corners pair up with opposite corners and side-middles with opposite side-middles: half as many corner lines as corners, plus half as many side lines as sides, which adds up to the number of sides again.

Shapes with no line of symmetry

Some shapes have no line of symmetry at all. The slanted four-sided shape (a parallelogram) is a surprise: it looks very even, but no fold makes its halves match. Try it with a paper one! A triangle with three different sides has none. The letter S has none either: fold it up-down or left-right and the curves go the wrong way.

Symmetry in letters and in nature

Many capital letters (written plainly, without curly bits) have symmetry.

• Up-and-down line (a line from top to bottom): A, H, I, M, O, T, U, V, W, X, Y.
• Side-to-side line (a line across the middle): B, C, D, E, H, I, K, O, X.
• Both: H, I, O, X.
• None: F, G, J, L, N, P, Q, R, S, Z.

The word MOM has an up-and-down line of symmetry. The word BOX has a side-to-side one (cover the bottom half and look).

Nature loves symmetry. A butterfly, a leaf, a face and most animals have an up-and-down mirror line, more or less. Look at your own face in a mirror: it is nearly symmetric, but not perfectly. Nature is close, and math is exact.

Worked examples

Example 1. Finish the picture. Here is half a picture on squared paper. The dashed line is a mirror line. Finish the picture.

1. Work one square at a time. Pick a coloured square on the left. Count how many squares it is away from the mirror line.
2. Go the same number of squares on the other side of the line, in the same row. Colour that square.
3. A square touching the line on the left gets a partner touching the line on the right. A square 2 squares away on the left gets a partner 2 squares away on the right.
4. Do this for every square. The finished picture is on the right. Check: fold along the line in your head, and every blue square lands on an orange one.

Example 2. How many lines? How many lines of symmetry does the capital letter H have?

1. Up-and-down through the middle: the left post matches the right post. Yes.
2. Side-to-side through the middle: the top half matches the bottom half (the crossbar is in the middle). Yes.
3. Corner-to-corner slanted lines: no, a slanted fold would put the posts in the wrong place.
4. So H has 2 lines of symmetry.

Example 3. Rectangle or square? A shape is a rectangle. Someone tells you that one of its corner-to-corner lines is a line of symmetry. What can you say about it?

1. For a rectangle that is not a square, the corner-to-corner fold does not work: one side folds onto a side of a different length and sticks out.
2. The fold only works when the side folding over is the same length as the side it lands on.
3. So the long sides and short sides must be the same length. It must be a square.

Try it

1. How many lines of symmetry does a rectangle (not a square) have?
2. Which of these letters have a side-to-side line of symmetry: A, B, D, M, E?
3. How many lines of symmetry does a regular octagon have?
4. Does the number 8 have a line of symmetry? How many? What about the number 3?
5. A shape has only 1 line of symmetry, up the middle. Draw one. (Hint: a house shape.)
6. Fold a paper in half, then in half again the other way, making 4 layers. Cut a shape across the corner where the folds meet. How many lines of symmetry will the shape have when you open it?
7. Does a parallelogram (a slanted rectangle) have any lines of symmetry?
8. Find a word that has an up-and-down line of symmetry when written in capitals, one letter under another (like a tower). (Hint: every letter must have an up-and-down line.)

Answers

1. 2: up the middle and across the middle. The corner-to-corner lines do not work.
2. B, D and E. Their top halves match their bottom halves. A and M have an up-and-down line instead.
3. 8, because a regular polygon has as many lines of symmetry as sides: 4 from corner to opposite corner and 4 from side-middle to opposite side-middle.
4. 8 has 2: up-and-down and side-to-side (written as two equal circles one on top of the other). 3 has 1: a side-to-side line across its middle, because its top curve matches its bottom curve.
5. A house shape (a square with a triangle roof, the point in the middle) has one up-and-down line through the point of the roof. It has no side-to-side line, because the roof is not like the floor. A heart, a vase or the letter A also work.
6. At least 2: both folds become lines of symmetry, because each fold copied your cut as a mirror image. (If your cut is very even, like a straight cut at the same slant to both folds, the shape can get more lines, like the diamond in level 1.)
7. No. Fold it up the middle or across: the slanted ends go the wrong way. Fold it corner to corner: it sticks out, like the rectangle. It is one of the surprising shapes with none. (It does have a different kind of symmetry: turn it upside down, half a turn, and it looks the same. More about turning in Slides, flips and turns.)
8. Any word made only of A, H, I, M, O, T, U, V, W, X, Y. For example MOM, TOY, HAT, WAX or TAXI, written in a column. (Written in a row, as normal, only some of them stay symmetric, like MOM, because the order of the letters flips too!)

Common mistake

Thinking every line that cuts a shape into two equal halves is a line of symmetry. The rectangle's corner-to-corner line cuts it into two equal triangles, but folding along it does not make them match. Always imagine the fold, or test with a mirror.

Mirroring by sliding. When you finish a mirror picture, a square near the line must stay near the line on the other side. Students sometimes just copy the half and slide it across, which puts the near squares far away. In a mirror image, the order is reversed: what is closest to the line on one side is closest on the other side too.

Where it leads

• Previous lesson: Polygons and regular shapes.
• Next lesson: Faces, edges and corners of solids.
• This level: Level 2: Shapes all around. The whole topic: Geometry.
• A flip in a mirror line is one of the three moves in Slides, flips and turns.
• Shapes that look the same after a turn have rotational symmetry, which you will count in level 4.
• Symmetric patterns in rows: Patterns.