Class 6 Maths Chapter 9 Symmetry – easy notes, concepts, NCERT book solutions, important questions, worksheet and answers. Learn about symmetry, lines of symmetry, symmetrical shapes, mirror images and patterns with simple explanations and examples.
🌈 Complete Concept Notes + Book Questions + Easy Solutions
🌟 Chapter 9 – Symmetry
Pages 217–218
📌 What is Symmetry?
Symmetry means a definite pattern in which parts of a figure repeat or match in a particular way.
Many objects around us show symmetry. Examples include flowers,
butterflies, rangoli designs, the Taj Mahal and many other structures.
A figure is called symmetrical when its parts repeat in a definite pattern.
Remember: Symmetry helps us identify balance and repeated patterns in shapes and objects.
Examples
Flower – symmetrical
Butterfly – symmetrical
Rangoli – symmetrical
Some structures such as the Taj Mahal – symmetrical
Cloud shown in the chapter – not symmetrical
9.1 Line of Symmetry
Page 219
📘 Concept Notes
A line that cuts a figure into two parts that exactly overlap when folded along that line is called a
line of symmetry.
The two matching parts are called mirror halves.
Imagine folding a figure along a line. If both halves fit exactly on each other, the fold line is a line of symmetry.
Q1. Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
Solution:
Yes. The flower, rangoli and butterfly have lines of symmetry.
The pinwheel and the cloud shown in the chapter do not have a line of symmetry.
Q2. For each of the following figures, identify the line(s) of symmetry if it exists.
Solution:
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📘 Figures with More Than One Line of Symmetry
A figure can have more than one line of symmetry.
A square is an important example.
A square has 4 lines of symmetry:
one vertical line
one horizontal line
one diagonal
the other diagonal
Page 221
Q. Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have?
Solution:
No, there is no other way to fold the square so that the two halves exactly overlap.
A square has 4 lines of symmetry.
Q. We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry?
Solution:
No. The diagonal of a rectangle that is not a square is not a line of symmetry because the two parts do not exactly overlap when folded along the diagonal.
Page 222
🔄 Reflection
Reflection means that one side of a figure is reflected across a line of symmetry to the other side.
Q. What if we reflect along the diagonal from A to C? Where do points A, B, C and D go? What if we reflect along the horizontal line of symmetry?
Solution:
When reflected along diagonal AC, points A and C remain in their positions, while B and D exchange their positions.
When reflected along the horizontal line of symmetry, the upper and lower points exchange their positions.
Page 224
Q1. In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded. Figure (d) was created by punching a single hole. How was the paper folded?
Solution:
Q2. Given the line(s) of symmetry, find the other hole(s):
Solution:
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Q4. After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer.
Solution:
Page 226
Q5. Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it?
Solution:
Q6. How many lines of symmetry do these shapes have?
a.
b. A triangle with equal sides and equal angles.
c. A hexagon with equal sides and equal angles.
Solution:
Page 227
Q7. Trace each figure and draw the lines of symmetry, if any:
Solution: p>
Page 228
Q8. Find the lines of symmetry for the kolam below.
Solution:
Q9. Draw the following.
a. A triangle with exactly one line of symmetry.
b. A triangle with exactly three lines of symmetry.
c. A triangle with no line of symmetry.
Is it possible to draw a triangle with exactly two lines of symmetry?
Solution:
Q10. Draw the following. In each case, the figure should contain at least one curved boundary.
a. A figure with exactly one line of symmetry.
b. A figure with exactly two lines of symmetry.
c. A figure with exactly four lines of symmetry.
Solution:
Page 228–229
Q11. Copy the following on squared paper. Complete them so that the blue line is a line of symmetry. Problem (a) has been done for you.
Solution:
Q12. Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry.
Solution:
Page 229
Q13. Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Solution:
9.2 Rotational Symmetry
Page 230
🔄 Concept Notes
A figure has rotational symmetry if it looks exactly the same after being rotated through an angle less than 360° about a fixed point.
The fixed point about which the figure is rotated is called the centre of rotation.
An angle through which a figure can be rotated to look exactly the same is called an
angle of rotational symmetry or simply an angle of symmetry.
Important:
For a figure with n equal rotational positions,
Smallest angle = 360° ÷ n
Q. Will the windmill above look exactly the same when rotated through an angle of less than 90°?
Solution:
No. It will first look exactly the same after a rotation of 90°.
Page 231
Q. Do you know of any other shape that has exactly four angles of symmetry? How many angles of symmetry does a square have? How much rotation does it require to get the initial square?
Solution:
A square has exactly four angles of symmetry between 0° and 360°:
90°, 180°, 270° and 360°.
A square comes back to its original position after a rotation of 90°.
Q. What are the other angles of symmetry?
Solution:
For a square, the other angles of symmetry are
180°, 270° and 360°, along with 90°.
Page 232
Q. Example: Find the angles of symmetry of the following strip.
Solution:
The strip does not overlap itself after 180° rotation.
It returns to its original position only after one complete rotation of 360°.
Therefore, it does not have rotational symmetry.
Q. Consider this figure with 4 radial arms. How many angles of symmetry does it have? What are they?
Solution:
The four radial arms are equally spaced.
Therefore, the smallest angle is
360° ÷ 4 = 90°.
The angles of symmetry are:
90°, 180°, 270° and 360°.
Q. Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.
Solution:
The four arms must remain equally spaced around the centre.
Therefore, the angle between adjacent arms must be 90°.
Q. How will you modify the figure above so that it has only two angles of symmetry?
Solution:
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Q. We have seen figures having 4 and 2 angles of symmetry. Can we get a figure having exactly 3 angles of symmetry? Can you use radial arms for this?
Solution:
360° ÷ 3 = 120°.
Q. Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they?
Solution:
If the three arms are not equally spaced, the figure may have only 360° as an angle of symmetry.
If the three arms are equally spaced by 120°, it has:
120°, 240° and 360°.
Page 234
Q. If these two figures must overlap, what can you tell about the angles?
Solution:
The three angles must be equal:
∠A = ∠B = ∠C
Since their sum is 360°:
Each angle = 360° ÷ 3 = 120°.
Q. Now how many angles of rotation does the figure have and what are they?
Solution:
The angles are:
120°, 240° and 360°.
Page 235
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.
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.
Solution:
Smallest angle:
360° ÷ 7 = 51 3/7°.
Therefore, the smallest angle is 51 3/7°.
It is not a whole number.
Q1. Find the angles of symmetry for the given figures about the point marked •.
Q2. Which of the following figures have more than one angle of symmetry?
Solution:
These all figures have more then 1 angle of symmetry.
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Q3. Give the order of rotational symmetry for each figure:
Solution:
a) 2
b) 4
c) 6
d) 3
e) 4
f) 5
True or False
• Every figure will have 360 degrees as an angle of symmetry.
• If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.
Solution: 1. True. A complete rotation is always 360°. 2. True. If the smallest angle is a natural number and the figure repeats perfectly, it divides 360°.
Page 237
⭕ Symmetries of a Circle
A circle has rotational symmetry for every angle because rotating a circle about its centre does not change its appearance.
Every diameter of a circle is a line of reflection symmetry.
📚 Figure it Out – Page 238
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?
Solution:
Q2. Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
Solution:
Two suitable examples are a regular hexagon and an equilateral triangle.
Both have reflection symmetry and rotational symmetry.
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.
Solution:
Q4. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
Solution:
The other angles are:
120°, 180°, 240°, 300° and 360°.
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?
Solution:
The smallest angle is 20°.
The angles can then be 20°, 40°, 60°, and so on.
Q6. Can we have a figure with rotational symmetry whose smallest angle of symmetry is:
a. 45°?
b. 17°?
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.
Solution: (a) Yes. The outer boundary has 3 lines of symmetry.
(b) Yes. Its rotational symmetry angles are:
120°, 240° and 360°.
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?
Solution:
For a regular polygon, the number of lines of symmetry is equal to the number of its sides.
Thus the sequence is:
3, 4, 5, 6, 7, 8, 9, 10, …
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?
Solution:
A regular polygon with n sides has rotational symmetry of order n.
Therefore, the number of angles of symmetry including 360° follows the same count:
3, 4, 5, 6, 7, 8, 9, 10, …
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?
Solution:
For the Koch Snowflake sequence, the number of lines of symmetry shown is:
3, 6, 6, 6, 6.
The corresponding number of angles of symmetry is:
3, 6, 6, 6, 6.
Q11. How many lines of symmetry and angles of symmetry does Ashoka Chakra have?
Solution:
Number of lines of symmetry = 24.
Number of angles of symmetry = 24.
Playing with Tiles
a. Use the colour tiles given at the end of the book to complete the following figure so that it has exactly 2 lines of symmetry.
b. Use 16 such tiles to make figures that have exactly:
1 line of symmetry
2 lines of symmetry
c. Use these tiles in making creative symmetric designs.
Solution: (a) Arrange the tiles as mirror images about two lines.
(b) Arrange the 16 tiles so that the complete design has either one or two lines of symmetry.
(c) Any creative design is acceptable if its parts follow a definite symmetric pattern.
📖 Chapter Summary
When a figure is made up of parts that repeat in a definite pattern, it has symmetry.
A line that divides a figure into two parts that exactly overlap when folded is called a line of symmetry.
A figure can have more than one line of symmetry.
A square has 4 lines of symmetry.
An equilateral triangle has 3 lines of symmetry.
A regular hexagon has 6 lines of symmetry.
Reflection symmetry means that one part of a figure is reflected to the other side of a line.
A figure may also have rotational symmetry.
The fixed point about which a figure rotates is called the centre of rotation.
An angle through which a figure can rotate and still look the same is called an angle of symmetry.
A complete rotation is 360°.
For a figure with n equally spaced rotational positions:
Smallest angle = 360° ÷ n
A circle has rotational symmetry for every angle.
Every diameter of a circle is a line of reflection symmetry.
A figure may have reflection symmetry, rotational symmetry, both, or neither.
⚡ Quick Revision
Concept
Remember
Symmetry
Repeated definite pattern
Line of symmetry
Divides a figure into matching mirror halves
Reflection
Mirror image across a line
Rotational symmetry
Figure looks same after rotation
Centre of rotation
Fixed point about which the figure rotates
Complete rotation
360°
Square
4 lines of symmetry; rotational order 4
Equilateral triangle
3 lines of symmetry; rotational order 3
Regular hexagon
6 lines of symmetry; rotational order 6
Circle
Every diameter is a line of symmetry; every rotation about centre works
⭐ Exam Tip:
Whenever you are asked whether a figure has symmetry, imagine folding it or rotating it.
If the figure exactly matches itself, symmetry is present.
Practice Class 6 Maths Chapter 8 with this easy and engaging worksheet. This worksheet is useful for practice, revision and exam preparation.
Class 6 Maths Chapter 9, Symmetry, Class 6 Maths notes, Class 6 Maths solutions, NCERT Class 6 Maths Chapter 9, Class 6 Maths worksheet, lines of symmetry, symmetrical shapes, mirror images, symmetry patterns, Class 6 Maths questions and answers.
360° ÷ 3 = 120°.
(a) 5 angles of symmetry:
(b) Yes. Its rotational symmetry angles are:
120°, 240° and 360°.
