Class 6 Maths Chapter 9: Symmetry Notes

What is symmetry?

A flower looks the same from many directions, the two wings of a butterfly match, and the petals of a rangoli come back onto themselves when it is turned a quarter of the way round. In each of these, parts of the figure repeat in a definite pattern. That is symmetry, and such figures are called symmetrical. A cloud has no such pattern, so it is not symmetrical. Buildings like the Taj Mahal and a temple gopuram are symmetrical too: their left and right halves match.

Lines of symmetry

Fold an isosceles triangle along the line through its top corner and the middle of its base, and one half covers the other completely: the two parts are mirror halves. A line that cuts a figure into two parts that exactly overlap when folded along it is a line of symmetry.

A figure can have more than one. Folding a square piece of paper in half vertically, horizontally and along both diagonals shows that a square has 4 lines of symmetry. A rectangle that is not a square has only 2: folding it along a diagonal does not make the halves overlap. More in line of symmetry.

Reflection

We can also say that a line of symmetry reflects the part of a figure on one side onto the other side. Label the corners of a square A, B, C and D. Reflecting in the vertical line of symmetry sends A to B’s place and D to C’s place, and the other way round. A figure that has a line of symmetry is also said to have reflection symmetry.

Making symmetric shapes

• Ink blot devils: drop ink on one half of a folded paper, press the halves together and open it. The pattern is symmetric about the fold.
• Folding and cutting: cut a shape from a folded paper and unfold it. The fold is a line of symmetry of the cut-out. Folding a strip several times gives a repeating design with many lines of symmetry.
• Punching holes: a hole punched in a folded square becomes two holes, one on each side of the fold, at equal distances from it.

Rotational symmetry

A paper windmill has no line of symmetry, yet it looks exactly the same after a turn of 90° about the point at its centre. It has rotational symmetry. The fixed point is the centre of rotation, and an angle through which a figure can be turned to look exactly the same is an angle of symmetry. The windmill’s angles of symmetry are 90°, 180°, 270° and 360°. Every figure comes back to its starting position after 360°, so 360° is always an angle of symmetry. A figure has rotational symmetry only if it also has an angle of symmetry strictly between 0° and 360°. More in rotational symmetry.

Angles of symmetry

A figure made of radial arms has rotational symmetry when the angles between neighbouring arms are equal. With 3 arms the angle must be 360° ÷ 3 = 120°, giving the angles 120°, 240° and 360°; with 4 arms it is 90°, and with 5 arms 72°. With 7 arms the smallest angle is 360° ÷ 7 = 51 3/7°, not a whole number. In every case the angles of symmetry are multiples of the smallest one, and if the smallest angle is a whole number of degrees, it is a factor of 360. The number of angles of symmetry is called the order of rotational symmetry.

Symmetries of a circle

A circle is the most symmetric shape of all. Turn it about its centre through any angle and it fits onto itself, so every angle is an angle of symmetry. And every diameter is a line of symmetry. Wheels, fans and many flowers have rotational symmetry too. Answers to every question of the chapter are in the NCERT solutions for Chapter 9.