Symmetry | Mathematics Grade 6 CBSE

Play with it

The six ideas of symmetry

Symmetry comes in two flavours — flipping across a line and turning about a centre. Tap each term to see what it means with a quick example.

Explore · Symmetrytap a term

Learn

The three big ideas

A figure has line symmetry (also called reflective or mirror symmetry) if a line can divide it into two halves that are mirror images of each other.
That line is the line of symmetry (or axis of symmetry). The simplest test: fold along the line — if the two halves land exactly on top of each other, the figure is symmetric.
One side is the reflection of the other: every point on the left has a matching point the same distance away on the right.
Many capital letters have line symmetry — A, M, T, U have a vertical line; B, C, D, E have a horizontal one; H, I, O, X have both.

Worked example. Does the capital letter T have line symmetry? Where is its line of symmetry?

Imagine folding T down a vertical line through its centre. The left arm of the top bar lands exactly on the right arm, and the stem maps onto itself.

So T has line symmetry, with one vertical line of symmetry.

Common mistake: assuming any line through the middle works. The two halves must be true mirror images — the letter R, for instance, has no line of symmetry at all.

A shape can have more than one line of symmetry — or none at all.
Square — 4 lines: two diagonals plus two lines through the midpoints of opposite sides.
Rectangle (not a square) — 2 lines: only the two through the midpoints of opposite sides. Its diagonals are not lines of symmetry.
Equilateral triangle — 3 lines; a regular pentagon — 5; a regular hexagon — 6. A regular figure with n sides has n lines of symmetry.
Circle — infinitely many: every line through the centre is a line of symmetry. A scalene triangle has none.

Worked example. A student claims a rectangle has 4 lines of symmetry, like a square. Is this correct?

Fold a rectangle along a diagonal: the two halves are triangles that do not overlap, because the sides are different lengths.

Only the two lines through the midpoints of opposite sides work. So a rectangle has 2 lines of symmetry, not 4.

Common mistake: treating every diagonal as a line of symmetry. Diagonals are symmetry lines for a square, but not for a non-square rectangle.

A figure has rotational symmetry if, when turned about its centre by less than a full turn, it looks exactly the same as before.
The order of rotational symmetry is how many times the figure matches itself in one full 360° turn.
The smallest turning angle that maps the figure onto itself is the angle of rotation = 360° ÷ order.
A square has order 4 (matches at 90°, 180°, 270°, 360°); an equilateral triangle has order 3 (every 120°); the letter S has order 2 (looks the same after a half-turn) even though it has no line symmetry.

Worked example. A wheel-like design repeats every 60° as it turns. What is its order of rotational symmetry?

Order = 360° ÷ (angle of rotation) = 360 ÷ 60 = 6.

So the design looks identical 6 times during one full turn — order 6.

Common mistake: thinking line symmetry and rotational symmetry always go together. The letter S has rotational symmetry (order 2) but no line of symmetry — they are different ideas.

Where you'll meet it

Symmetry at work

Rangoli & mandalas

Festival rangoli and temple mandalas are built around a centre and repeat at fixed angles — beautiful real examples of rotational symmetry and many lines of symmetry at once.

Nature's designs

A butterfly's wings, a leaf's two halves, and the petals of a flower all show symmetry. Spotting it helps biologists and artists alike describe shapes precisely.

Logos & architecture

Designers use symmetry to make logos feel balanced, and architects use it so that buildings like the Taj Mahal look stable and harmonious from the front.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

Score 0/11