NCERT Solutions Ganita Prakash Chapter 9 Rotational Symmetry of Figures with Radial Arms — In-text Questions
Book page 232 to 235 Updated on2026-09-05
Q1.
Consider this figure, a picture with 4 radial arms. How many angles of symmetry does it have? What are they? Note that the angle between adjacent central dotted lines is 90°.
Answer
The four arms are identical and the angle between neighbouring arms is 90°, so each arm can take the place of the next one.
Four radial arms, 90° apart. Every quarter turn brings the figure back onto itself.
Why: a turn of 90° carries arm 1 onto arm 2, arm 2 onto arm 3, arm 3 onto arm 4 and arm 4 onto arm 1 — the picture is unchanged. The same is true of 180°, 270° and the full turn of 360°.
Q2.
Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.
Answer
No — not if all four arms are to be kept. With 4 arms the only way to have 4 angles of symmetry is to keep them at 90° to each other.
Smallest angle of symmetry = 360° ÷ 4 = 90°
Why: if the figure looks the same after a turn of 90°, that turn must carry each arm onto the next arm. So the gaps between consecutive arms must all be equal, and four equal gaps in a full turn means 90° each. If you make one gap 80° and another 100°, the arms no longer match after a quarter turn and only the 360° turn is left.
What you can change: the arms themselves. Make them longer, shorter, thicker, curved like a fan blade or shaped like a leaf — as long as all four are identical and 90° apart, the figure still has exactly 4 angles of symmetry. Try drawing a four-blade fan and a four-petal flower and check them with a paper cut-out.
Q3.
How will you modify the figure above so that it has only two angles of symmetry?
Answer
Make the arms unequal in pairs: keep the two opposite arms alike, and make the other two opposite arms different.
Two long arms and two short arms. A quarter turn no longer matches — only a half turn does.
Angles of symmetry = 180°, 360° → only 2 angles
A turn of 90° would try to put a long arm on a short arm — it fails.
A turn of 180° puts each arm on its opposite arm, which is identical — it works.
Another easy way: keep the arms equal but add a small bent flag at the end of each arm, all bending the same way round (the shape shown in the book). A half turn still matches, a quarter turn does not.
Q4.
Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they? Trace and cut out a copy of this figure. By rotating the cutout over this figure determine its angles of rotation.
Answer
Cut out a tracing and turn it slowly over the printed figure. You will find that no partial turn ever matches.
Angle of symmetry = 360° only → 1 angle
So this figure does not have rotational symmetry, even though it has three arms.
Why it fails: in the printed figure the three arms are not equally spaced — the angles between them are all different. For a turn to bring arm 1 onto arm 2, the gap between arm 1 and arm 2 must be the same as the gap between arm 2 and arm 3 and between arm 3 and arm 1. That is not the case here, so only the full turn of 360° works.
Remember: 360° is an angle of symmetry of every figure. A figure is said to have rotational symmetry only when it has some angle of symmetry smaller than 360°.
Q5.
However, can anything in the figure be changed to make it have 3 angles of symmetry? Can it be done by changing the angles between the dotted lines? Observe that ∠A must overlap ∠B, ∠B must overlap ∠C and ∠C must overlap ∠A. So, ∠A = ∠B = ∠C. What must this angle be? Now how many angles of rotation does the figure have and what are they?
Answer
Yes — change the angles between the arms so that all three are equal.
Three equal arms with 120° between neighbouring arms — now the figure has rotational symmetry.
With 120° between the adjacent dotted lines, the figure has 3 angles of symmetry:
120° (one third of a turn), 240° (120° + 120°), 360° (120° + 120° + 120°)
Why exactly these: a turn of 120° sends arm A to arm B, B to C and C to A. Doing it twice (240°) sends A to C, and doing it three times (360°) brings everything home. Notice that all the angles of symmetry are multiples of the smallest one, 120°.
Q6.
Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case. Hint: Use 5 radial arms for the first case. What should the angle between two adjacent radial arms be?
Answer
Use as many equal arms as the number of angles you want, and share the full turn of 360° equally between them.
a) Exactly 5 angles of symmetry — draw 5 equal arms.
Angle between two adjacent arms = 360° ÷ 5 = 72° Angles of symmetry = 72°, 144°, 216°, 288°, 360°
b) Exactly 6 angles of symmetry — draw 6 equal arms.
Angle between two adjacent arms = 360° ÷ 6 = 60° Angles of symmetry = 60°, 120°, 180°, 240°, 300°, 360°
Equal arms spread equally round the centre: 5 arms give 72°, 6 arms give 60°.
The general rule: a figure with n equal arms, equally spaced, has n angles of symmetry, and its smallest angle of symmetry is 360° ÷ n. All the other angles are multiples of it.
Q7.
Consider a figure with radial arms having exactly 7 angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.
Answer
Smallest angle of symmetry = 360° ÷ 7
Divide 360 by 7: 7 × 51 = 357, and 360 − 357 = 3.
360 ÷ 7 = 51 remainder 3 → 513⁄7°
No, it is not a whole number of degrees. The smallest angle of symmetry is 513⁄7° (about 51.43°).
Why it is not a whole number: 7 is not a factor of 360 (360 = 2 × 2 × 2 × 3 × 3 × 5). Whenever the number of arms is not a factor of 360, the smallest angle of symmetry comes out as a fraction. For 3, 4, 5, 6, 8, 9, 10, 12 arms the answer is a whole number; for 7 or 11 arms it is not.