NCERT Solutions Ganita Prakash Chapter 9 & 239Symmetries of a Circle; Regular Polygons; Ashoka Chakra — Figure it Out
Book page 238 Updated on2026-09-05
Q1.
Colour the sectors of the circle below so that the figure has i) 3 angles of symmetry, ii) 4 angles of symmetry, iii) what are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?
Answer
The circle is cut into 12 equal sectors, so each sector is
360° ÷ 12 = 30°
Colour every 4th sector for 3 angles of symmetry; colour every 3rd sector for 4 angles of symmetry.
i) For 3 angles of symmetry — colour every 4th sector (3 coloured sectors, 120° apart).
iii) What numbers are possible? The colouring pattern must repeat after a whole number of sectors, so the order must divide 12.
Possible numbers of angles of symmetry = 1, 2, 3, 4, 6 and 12 — the factors of 12
Colour every … sector
Coloured sectors
Smallest angle
Number of angles of symmetry
1st (all 12)
12
30°
12
2nd
6
60°
6
3rd
4
90°
4
4th
3
120°
3
6th
2
180°
2
any lop-sided colouring
—
360°
1
Why only factors of 12: a turn that matches the colouring must move every coloured sector onto a coloured sector, so it must be a whole number of sectors — 30°, 60°, 90°, … Repeating it must reach 360° exactly, so the number of steps in a full turn (the order) has to divide 12. The numbers 5, 7, 8, 9, 10 and 11 are therefore impossible.
Q2.
Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
Answer
Any regular polygon or any regular star works. Two easy ones:
An equilateral triangle and a regular hexagon: both can be folded and both can be turned.
Equilateral triangle — 3 lines of symmetry and 3 angles of symmetry (120°, 240°, 360°).
Regular hexagon — 6 lines of symmetry and 6 angles of symmetry (60°, 120°, 180°, 240°, 300°, 360°).
More examples: a rectangle (2 lines, order 2), a rhombus (2 lines, order 2), a regular pentagon (5 lines, order 5), a five-pointed star (5 lines, order 5) and most rangoli designs.
Q3.
Draw, wherever possible, a rough sketch of: a. A triangle with at least two lines of symmetry and at least two angles of symmetry. b. A triangle with only one line of symmetry but not having rotational symmetry. c. A quadrilateral with rotational symmetry but no reflection symmetry. d. A quadrilateral with reflection symmetry but not having rotational symmetry.
Answer
Lines of symmetry are dashed; the red dot marks a centre of rotation.
a. Equilateral triangle. It has 3 lines of symmetry (more than two) and 3 angles of symmetry — 120°, 240°, 360°. In fact a triangle can never have exactly two of either, so the equilateral triangle is the only possible answer.
b. Isosceles triangle (two sides equal, the third different). It has exactly 1 line of symmetry, and its only angle of symmetry is 360°, so it has no rotational symmetry.
c. Parallelogram that is neither a rectangle nor a rhombus. It has no line of symmetry, yet a half turn (180°) about the point where its diagonals meet brings it back — order 2. (An “S” or “Z” shape and the paper windmill are other examples of rotation without reflection.)
d. Kite (two pairs of equal adjacent sides) — or an isosceles trapezium. It has exactly 1 line of symmetry (its long diagonal) but no rotational symmetry: a half turn would swap its sharp corner with its blunt corner.
The lesson of this question: lines of symmetry and angles of symmetry are two different things. A figure can have both (a), only lines (b, d), only angles (c) — or neither.
Q4.
In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
Answer
Every angle of symmetry is a multiple of the smallest one, and 360° is always an angle of symmetry. So keep adding 60° until you reach 360°.
60°, 60° + 60° = 120°, 180°, 240°, 300°, 360°
The other angles of symmetry are 120°, 180°, 240°, 300° and 360°.
Order of rotational symmetry = 360° ÷ 60° = 6
Such a figure: a regular hexagon, a six-petal flower, a snowflake or a six-blade fan.
Q5.
In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?
Answer
Let the smallest angle of symmetry be s. All angles of symmetry are the multiples s, 2s, 3s, …
The figure has exactly two angles of symmetry smaller than 60°; these must be s and 2s. And since 60° itself is an angle of symmetry, the next multiple after 2s must be 60°:
3s = 60° s = 60° ÷ 3 = 20°
The smallest angle of symmetry is 20°.
Check: the angles below 60° are 20° and 40° — exactly two ✔ and 60° is the third multiple ✔ Order = 360° ÷ 20° = 18
Q6.
Can we have a figure with rotational symmetry whose smallest angle of symmetry is: a. 45°? b. 17°?
Answer
The test is simple: the smallest angle of symmetry must divide 360° exactly.
a. 45°? Yes.
360° ÷ 45° = 8 (a whole number)
Such a figure has 8 angles of symmetry: 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360°. A regular octagon, an eight-pointed star or an eight-blade fan is an example.
b. 17°? No.
360° ÷ 17° = 21 remainder 3 → not a whole number
If 17° were an angle of symmetry, then 17° × 21 = 357° would be one too, and one more 17° turn would give 374° — that is 14° past the full turn. So 14° would also be an angle of symmetry, which is smaller than 17°. That contradicts “17° is the smallest”. Hence 17° is impossible.
Rule to remember: the smallest angle of symmetry (when it is a whole number of degrees) is always a factor of 360.
Q7.
This is a picture of the new Parliament Building in Delhi. a. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many are they? b. Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.
Answer
The outer boundary of the new Parliament Building is a triangle with its three corners cut off — the three long sides are equal and the three cut corners are equal.
The outline of the new Parliament Building: 3 lines of symmetry meeting at the centre.
a. Yes, it has reflection symmetry — there are 3 lines of symmetry. Each one passes through the middle of a side and the opposite cut corner, and all three meet at the centre of the building.
b. Yes, it has rotational symmetry of order 3 about that centre.
Smallest angle of rotation = 360° ÷ 3 = 120° Angles of rotational symmetry = 120°, 240°, 360°
Why 3: the boundary is made of three identical pieces (one long side + one cut corner) placed around the centre. A one-third turn moves each piece exactly onto the next.
Q8.
How many lines of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
Answer
A regular polygon with n sides has exactly n lines of symmetry.
Regular polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Sides
3
4
5
6
7
8
9
10
Lines of symmetry
3
4
5
6
7
8
9
10
The number sequence is 3, 4, 5, 6, 7, 8, 9, 10, … — the counting numbers starting from 3. It is the very same sequence as the number of sides.
Why lines = sides: for an odd number of sides, each line of symmetry joins one corner to the midpoint of the opposite side — one line per corner, so n lines. For an even number of sides, n/2 lines join opposite corners and n/2 lines join midpoints of opposite sides — again n lines in all.
Q9.
How many angles of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
Answer
A regular polygon of n sides has exactly n angles of symmetry, because turning it by one “side” brings each corner onto the next corner.
Regular polygon
Sides
Angles of symmetry
Smallest angle = 360° ÷ n
Triangle
3
3
120°, 240°, 360°
Quadrilateral (square)
4
4
90°, 180°, 270°, 360°
Pentagon
5
5
72°, 144°, …, 360°
Hexagon
6
6
60°, 120°, …, 360°
Heptagon
7
7
513⁄7°, …, 360°
Octagon
8
8
45°, 90°, …, 360°
Nonagon
9
9
40°, 80°, …, 360°
Decagon
10
10
36°, 72°, …, 360°
The number sequence is again 3, 4, 5, 6, 7, 8, 9, 10, … — the counting numbers from 3, exactly the same sequence as in question 8.
A neat conclusion: for a regular polygon, number of sides = number of lines of symmetry = order of rotational symmetry, and the smallest angle of rotation is 360° ÷ n.
Q10.
How many lines of symmetry do the shapes in the last shape sequence in Chapter 1, Table 3, the Koch Snowflake sequence, have? How many angles of symmetry?
Answer
The Koch Snowflake sequence begins with an equilateral triangle; at every step each straight edge is replaced by a “speed bump” of four smaller edges.
The first Koch shape is a triangle (3 lines of symmetry); from the second shape on it is six-pointed, with 6 lines of symmetry.
Shape in the sequence
1st
2nd
3rd
4th
5th
Lines of symmetry
3
6
6
6
6
Angles of symmetry
3
6
6
6
6
So both sequences are 3, 6, 6, 6, 6, …
Why it settles at 6: the first shape is an equilateral triangle — 3 lines, angles 120°, 240°, 360°. Adding a bump to each of its three sides produces the familiar six-pointed star, in which the three old corners and the three new bump-tips are all at the same distance from the centre and 60° apart. From then on every step adds identical bumps to every edge, so the six-fold symmetry is never lost: 6 lines of symmetry, and angles of symmetry 60°, 120°, 180°, 240°, 300°, 360°.
Q11.
How many lines of symmetry and angles of symmetry does Ashoka Chakra have?
Answer
The Ashoka Chakra at the centre of our national flag has 24 spokes, equally spaced around the hub.
The Ashoka Chakra — 24 equally spaced spokes, so 24 lines of symmetry and 24 angles of symmetry.
Number of spokes = 24 Lines of symmetry = 24 Angles of symmetry = 24 Smallest angle of rotation = 360° ÷ 24 = 15°
The angles of symmetry are 15°, 30°, 45°, 60°, …, 345°, 360°.
Why 24 lines: 24 is an even number, so 12 of the lines pass along a pair of opposite spokes and the other 12 pass exactly midway between two neighbouring spokes.
Did you know? The Ashoka Chakra is taken from the Lion Capital of Ashoka at Sarnath. Its 24 spokes are said to stand for the 24 hours of a day — and, being equally spaced, they make the wheel look the same after every 15° turn.