EN हिं

NCERT Solutions Ganita Prakash Chapter 9 & 237Multiples of the smallest angle; True or False; Symmetries of a circle — In-text Questions

Book page 236 Updated on2026-09-05

Q1.
In each case, the angles are the multiples of the smallest angle. You may wonder and ask if this will always happen. What do you think?
Answer

Yes, it always happens. Every angle of symmetry of a figure is a multiple of its smallest angle of symmetry.

Exactly 2 angles: 180°, 360°  → multiples of 180
Exactly 3 angles: 120°, 240°, 360°  → multiples of 120
Exactly 4 angles: 90°, 180°, 270°, 360°  → multiples of 90
Why it must be so: suppose the smallest angle of symmetry is s. Turning by s leaves the figure looking the same, so turning by s again (that is 2s) also leaves it the same, and so does 3s, 4s, … Now suppose some angle a of symmetry was not a multiple of s. Then a would lie between two multiples, say between ks and (k+1)s. Turning first by a and then turning back by ks would leave the figure unchanged too — but that turn is smaller than s, and s was supposed to be the smallest. So no such a can exist.

Since the full turn 360° is always an angle of symmetry, the smallest angle s must divide 360 exactly. That is why the order of rotational symmetry is always a factor of 360 when s is a whole number of degrees.

Q2.
True or False: Every figure will have 360 degrees as an angle of symmetry.
Answer

True.

Why: a rotation of 360° is one complete turn. It brings every single point of the figure back to exactly where it started, so the figure certainly looks the same. This is true of every figure, however irregular — even a cloud or a scribble.
That is why we say: a figure has rotational symmetry only when it has an angle of symmetry strictly between 0° and 360°. Having 360° alone means order 1 — no rotational symmetry.
Q3.
True or False: If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.
Answer

True.

If the smallest angle of symmetry is s, then 360° must be one of
s, 2s, 3s, …  →  360 = n × s for some whole number n
so s is a factor of 360, and n is the order of rotational symmetry.
Why: 360° is always an angle of symmetry (see the previous question), and every angle of symmetry is a multiple of the smallest one. So 360 is a multiple of s — that is exactly what “s is a factor of 360” means.

The possible whole-number smallest angles are therefore the factors of 360: 1°, 2°, 3°, 4°, 5°, 6°, 8°, 9°, 10°, 12°, 15°, 18°, 20°, 24°, 30°, 36°, 40°, 45°, 60°, 72°, 90°, 120°, 180° and 360°. An angle such as 7°, 17° or 50° can never be a smallest angle of symmetry.

Q4.
The circle is a fascinating figure. What happens when you rotate a circle clockwise about its centre? Now take a point on the rim of the circle and join it to the centre. Extend the segment to a diameter of the circle. Is that diameter a line of reflection symmetry?
Answer

On turning: the circle coincides with itself, whatever the angle — 1°, 37°, 90°, 156.5° — it does not matter.

For a circle, every angle is an angle of symmetry.

So a circle has no smallest angle of symmetry — you can always find a smaller one. It is the most symmetric figure of all.

On folding: Yes, every diameter is a line of symmetry. Fold a paper circle along any diameter and the two semicircles match perfectly, because every point of the rim is at the same distance from the centre.

A few of the circle's lines of symmetry. Every one of its infinitely many diameters is a line of symmetry.
Around you: wheels, fans, flowers, plates, bangles and the top of a tawa all turn onto themselves. A ceiling fan with 3 blades has order 3 (120°), a bicycle wheel with 36 spokes has order 36 (10°).
Was this helpful? Report an error