Hinglish

Rotational Symmetry

Class 6 Maths · Chapter 9: Symmetry · NCERT Ganita Prakash section 9.2

Learn

Tenzin makes a paper phirki (pinwheel) with 4 blades. He pins it at the centre and turns it slowly. After a quarter turn (90°) it looks exactly as it did at the start! It looks the same again at 180°, 270° and 360°.

A shape that looks the same after less than a full turn has rotational symmetry. The number of times it looks the same in one full turn is its order of rotational symmetry. The phirki has order 4. The smallest such turn, here 90°, is its angle of symmetry.

360° ÷ 4 = 90°

An equilateral triangle looks the same after 120°, 240° and 360°: order 3. A regular hexagon: every 60°, order 6. A rectangle: only after half turns, order 2. For any shape, order × angle of symmetry = 360°.

Every shape looks the same after a full turn of 360°. If that is the only time, the shape has no rotational symmetry. An arrow or a kite has a line of symmetry but no rotational symmetry.

And the opposite happens too: the phirki and the parallelogram have rotational symmetry but no line of symmetry. The letters S, N and Z look the same upside down (order 2), but no mirror line works.

Explore: turn each shape and watch for the moment it fits its outline again.

  1. Turn the square until it looks the same as at the start.
  2. Now the 3-bladed phirki.
  3. Find the smallest turn that works for the regular hexagon.
  4. Turn the rectangle until it looks the same.

Remember

  • Rotational symmetry: the shape looks the same after less than a full turn.
  • Order = how many times it looks the same in a full turn; angle of symmetry = 360° ÷ order.

Quick check

A shape looks the same after turns of 72°, 144°, 216°, 288° and 360°. What is its order of rotational symmetry?

  1. 5
  2. 72
  3. 4
  4. 360
Answer

5. It looks the same 5 times in one full turn: 360 ÷ 72 = 5.

Think about it

Can a shape have an angle of symmetry of 50°? What about 45°?

One answer

45° works: a regular octagon looks the same 8 times in a full turn, and 8 × 45° = 360°. 50° does not, because 360 is not a multiple of 50, so the shape would not get back to its start after whole steps.

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.