Symmetry and Reflection

What is symmetry?

Something has symmetry when it is balanced and matches up neatly. The most common kind is reflective symmetry (also called line symmetry or mirror symmetry). A shape has reflective symmetry if you can draw a straight line through it that splits it into two halves that match exactly — so that if you folded the shape along that line, the two halves would land perfectly on top of each other.

That special line is called a line of symmetry or a mirror line. If you held a small mirror standing up along the line, the half you can see plus its reflection would look exactly like the whole original shape.

You can spot symmetry everywhere: a butterfly's wings, your own face, a love-heart, a snowflake, the letter "A", and many flowers. Nature loves symmetry because balanced shapes are often stronger and help animals move straight.

Lines of symmetry: how many?

Different shapes have different numbers of lines of symmetry.

• No lines of symmetry — a scalene triangle, or the letter "F". No fold makes the halves match.
• One line of symmetry — an isosceles triangle, a love-heart, the letter "A".
• Two lines of symmetry — a rectangle (one vertical, one horizontal) and the letter "H".
• Many lines of symmetry — a square has four, an equilateral triangle has three, a regular hexagon has six.
• Infinitely many — a circle! Any straight line through its centre splits it into two matching halves.

Notice that the more "regular" a shape is (the more its sides and angles are equal), the more lines of symmetry it tends to have. You can revisit these regular shapes in our lesson on types of quadrilaterals.

What is a reflection?

A reflection is an action in geometry. You take a shape and flip it across a mirror line to make a mirror image on the other side. Three things are always true about a reflection:

1. The image is the same size as the original — reflection never stretches or shrinks.
2. The image is flipped, so it faces the opposite way (your right hand reflects to look like a left hand).
3. Every point of the image is the same distance from the mirror line as the matching point on the original, just on the opposite side.

That third rule is the key to reflecting accurately. If a corner is 3 squares to the left of the mirror line, its reflection lands 3 squares to the right.

A diagram in words

Imagine a vertical mirror line drawn down the middle of a piece of squared paper.

On the left of the line, draw a small flag: a dot 2 squares from the line for the bottom of the pole, a vertical line up for the pole, and a little triangle sticking out to the left at the top.

Now picture its reflection on the right. The bottom of the pole appears 2 squares to the right of the line (the same distance across). The pole stands up just the same. But the triangle now points to the right instead of the left — it has been flipped. The two flags lean away from each other like a butterfly's wings, perfectly mirrored across the centre line.

Worked example 1

How many lines of symmetry does a rectangle (that is not a square) have? Draw them in words.

A rectangle has two lines of symmetry. One runs vertically down the middle (splitting it into a left and right half), and one runs horizontally across the middle (splitting it into a top and bottom half). The diagonals do not work, because folding along a diagonal would not make the halves match.

Worked example 2

A triangle has a corner at point A, which is 4 squares to the left of a vertical mirror line. Where does the reflection of A land?

Reflection keeps the same distance on the other side. So the image of A lands 4 squares to the right of the mirror line, at the same height. It is exactly 8 squares from the original in total.

Worked example 3

Is the letter "B" symmetrical? If so, where is the line of symmetry?

Yes. The letter B has one line of symmetry, and it is horizontal — running through the middle from left to right, because the top loop matches the bottom loop. It does not have a vertical line of symmetry, because the flat side is on the left and the curves are on the right.

Why this matters

Symmetry is one of the first big ideas that connects maths to art, science and the real world. Designers use it to make logos, buildings and patterns that feel balanced and pleasing. Scientists find symmetry in crystals, leaves and even the laws of physics. Reflection is also your first transformation — a way of moving a shape around a grid without changing its size. Soon you will meet two more transformations, rotation and translation, and together they let you describe exactly how any shape can move. Learning to reflect carefully now builds the careful, point-by-point thinking you will use in coordinate geometry later.

Activity: fold, blot and mirror

1. The fold test: cut out paper shapes — a square, a rectangle, a heart and a random "blob". Try folding each one different ways. Every fold that makes the two halves match exactly is a line of symmetry. Count and record how many each shape has.
2. Paint blots: fold a sheet of paper in half, dab some paint on one side only, then press the halves together and open it up. The blot is now perfectly symmetrical — the fold was your mirror line!
3. Letter detective: write out the alphabet in capital letters. Sort each letter into three groups: vertical line of symmetry (like A), horizontal line of symmetry (like B), or no symmetry (like F).
4. Challenge: use a small mirror. Stand it on a half-drawing and move it until the reflection completes a butterfly. What did you learn about where the mirror line must go?