Learn Class 6 Mathematics Chapter 9 Symmetry with clear notes on lines of symmetry, rotational symmetry, angles of symmetry, reflection and real-life examples.
Symmetry is all around us — in flowers, butterflies, rangolis, pinwheels, buildings and even in the wheels of a bicycle. This chapter helps you discover the hidden patterns that make a figure look balanced and beautiful. You will learn what symmetry means and how to check whether a figure is symmetrical. You will explore lines of symmetry by folding paper and finding mirror halves. You will also learn about rotational symmetry, where a figure looks the same after being turned about a fixed point. You will find angles of symmetry for different shapes, including squares, circles and figures with radial arms. Along the way, you will create your own symmetric designs using ink blots, paper folding and cutting. By the end, you will be able to identify lines of symmetry, angles of symmetry and understand how reflection and rotation work together to create symmetry.
What you'll learn
1Define symmetry and identify symmetrical and asymmetrical figures.
2Find lines of symmetry by folding figures and checking for mirror halves.
3Recognise figures with multiple lines of symmetry, such as squares and regular polygons.
4Explain reflection symmetry and how points move when reflected across a line.
5Identify rotational symmetry and find angles of symmetry for various figures.
6Determine the centre of rotation and the smallest angle of symmetry.
7Create symmetric shapes using ink blots, paper folding and cutting.
8Analyse real-life objects like fans, flowers and wheels for rotational symmetry.
Chapter at a glance
01Chapter Overview
02Lines of Symmetry
03Rotational Symmetry
04Symmetry in Shapes and Figures
05Reflection and Mirror Images
Detailed chapter notes
01
What is Symmetry?
A figure has symmetry when its parts repeat in a definite pattern. When you look at a flower, a butterfly or a rangoli, you may notice that one part of the figure is repeated in some way. If you fold the figure along a certain line, the two halves may overlap exactly. Such a figure is called symmetrical. If the parts do not repeat in any pattern, the figure is asymmetrical. For example, a cloud does not have a repeating pattern, so it is not symmetrical. Symmetry makes objects look balanced and attractive. In this chapter, you will learn about two main types of symmetry: line symmetry (also called reflection symmetry) and rotational symmetry.
Symmetrya figure has symmetry if its parts repeat in a definite pattern.
Symmetrical figurea figure that has at least one line of symmetry or rotational symmetry.
Asymmetrical figurea figure that has no symmetry.
02
Line of Symmetry
A line of symmetry is a line that divides a figure into two parts that exactly overlap when folded along that line. These two parts are called mirror halves. For example, if you fold a blue triangle along a dotted line and one half covers the other half completely, that line is a line of symmetry. Not every line through a figure is a line of symmetry. If you fold a figure and the halves do not overlap exactly, then that line is not a line of symmetry. A figure can have more than one line of symmetry. A square has four lines of symmetry: one vertical, one horizontal and two diagonal. A rectangle that is not a square has only two lines of symmetry: vertical and horizontal. Its diagonal is not a line of symmetry. Regular polygons have as many lines of symmetry as the number of sides. For example, an equilateral triangle has 3 lines of symmetry, a regular pentagon has 5, and a regular hexagon has 6.
Line of symmetrya line that cuts a figure into two mirror halves.
Mirror halvesthe two parts of a figure that overlap exactly when folded along the line of symmetry.
A square has 4 lines of symmetry.
A rectangle (not a square) has 2 lines of symmetry.
An equilateral triangle has 3 lines of symmetry.
A regular hexagon has 6 lines of symmetry.
03
Reflection Symmetry
When a figure is folded along a line of symmetry, one half is reflected onto the other half. This is called reflection symmetry. Imagine a square with corners labelled A, B, C and D in order. If you reflect it along the vertical line of symmetry, point B moves to the position of A, and point C moves to the position of D. Points A and D move to the positions of B and C respectively. If you reflect along the diagonal from A to C, then point D moves to the position of B, while A and C stay in their places. Reflection symmetry helps us understand how points move when a figure is mirrored. Many objects around us, like the Taj Mahal and gopurams, show reflection symmetry.
Reflection symmetrya figure has reflection symmetry if it has at least one line of symmetry.
When reflecting along a line, points on one side move to matching positions on the other side.
04
Rotational Symmetry
Some figures look exactly the same when rotated about a fixed point. This fixed point is called the centre of rotation. If a figure looks the same after a rotation of less than 360°, it is said to have rotational symmetry. The angle through which the figure is rotated to look exactly the same is called an angle of symmetry. For example, a paper windmill looks the same when rotated by 90° about its centre. So, 90° is an angle of symmetry for the windmill. The windmill also looks the same after 180°, 270° and 360°. So, it has four angles of symmetry. Every figure has 360° as an angle of symmetry because a full turn brings it back to its original position. A figure has rotational symmetry if it has at least one angle of symmetry strictly between 0° and 360°.
Centre of rotationthe fixed point about which a figure is rotated.
Angle of symmetryan angle through which a figure can be rotated to look exactly the same.
Rotational symmetrya figure has rotational symmetry if it has an angle of symmetry between 0° and 360°.
360° is always an angle of symmetry for any figure.
05
Angles of Symmetry for Different Figures
The number of angles of symmetry depends on the figure. For a figure with exactly 2 angles of symmetry, the angles are 180° and 360°. For a figure with exactly 3 angles of symmetry, the angles are 120°, 240° and 360°. For a figure with exactly 4 angles of symmetry, the angles are 90°, 180°, 270° and 360°. Notice that in each case, the angles are multiples of the smallest angle. For a figure with n angles of symmetry, the smallest angle is 360° ÷ n. For example, a figure with 5 angles of symmetry has a smallest angle of 72°, and the other angles are 144°, 216°, 288° and 360°. A figure with 7 angles of symmetry has a smallest angle of 360° ÷ 7 = 51 3/7°, which is not a whole number. The smallest angle of symmetry must be a factor of 360° if it is a whole number.
If a figure has n angles of symmetry, the smallest angle is 360° ÷ n.
The angles of symmetry are multiples of the smallest angle.
A circle is a very special shape. When you rotate a circle about its centre, it looks exactly the same no matter what angle you rotate it by. So, every angle is an angle of symmetry for a circle. A circle also has infinitely many lines of symmetry. Every diameter of a circle is a line of symmetry. This is because if you fold a circle along any diameter, the two halves overlap perfectly. The circle's symmetry is the reason wheels can rotate smoothly. Other objects like fans, flowers and wheels also show rotational symmetry.
A circle has infinitely many lines of symmetry (every diameter).
Every angle is an angle of symmetry for a circle.
A circle has rotational symmetry about its centre.
07
Creating Symmetric Figures
You can create beautiful symmetric figures using simple methods. One way is the ink blot method: fold a paper in half, open it, put a few drops of ink on one half, press the halves together, and then open. The resulting pattern is symmetric, and the fold line is a line of symmetry. Another way is paper folding and cutting: fold a paper, make a cut along a line, and then unfold it to see a symmetric shape. You can also use a punching machine to punch holes in a folded paper; when you unfold it, the holes form a symmetric pattern. These activities help you understand how symmetry works and allow you to design your own symmetric rangolis, decorations and patterns.
Ink blotfold, drop ink, press, unfold to get a symmetric figure.
Paper folding and cuttingfold, cut, unfold to create symmetric shapes.
Punching holes in folded paper creates symmetric patterns.
Want the complete chapter resources?Topic notes, quizzes and flashcards for Symmetry.
A square has 4 lines of symmetry: 2 diagonal lines and 2 lines through the middle (horizontal and vertical).
Question 02
What is rotational symmetry?
AWhen a shape looks the same after turning it around a point
BWhen a shape is cut into equal halves
CWhen a shape has straight lines
DWhen a shape is very big
Show answer
Answer: (A) When a shape looks the same after turning it around a point
Rotational symmetry means a shape looks identical when rotated around a central point by less than 360°.
Question 03
What is point symmetry also called?
ARotational symmetry of order 2
BLine symmetry
CRadial symmetry
DBilateral symmetry
Show answer
Answer: (A) Rotational symmetry of order 2
Point symmetry is when a shape looks the same after rotating it 180 degrees around a central point. This is also called rotational symmetry of order 2.
Question 04
What is a line of symmetry?
AA line that divides a shape into two identical halves
BA line that goes around a shape
CA line that makes a shape bigger
DA line that connects two corners
Show answer
Answer: (A) A line that divides a shape into two identical halves
A line of symmetry divides a shape into two equal parts that are mirror images of each other.
Question 05
What is a mirror image?
AAn image that looks exactly like the object but reversed
BAn image that is smaller than the object
CAn image that is upside down
DAn image that is rotated 90 degrees
Show answer
Answer: (A) An image that looks exactly like the object but reversed
A mirror image is the reflection of an object in a mirror, which appears reversed left to right.
Ready for more practice?Unlock the full quiz for this chapter.
Q1. Based on the chapter overview, define symmetry and explain why a cloud is considered asymmetrical compared to a butterfly or a flower.
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Model answer
According to the chapter, symmetry refers to a part or parts of a figure that are repeated in a definite pattern. A flower or a butterfly is considered symmetrical because their parts occur in a repeating, organized pattern that appeals to the eye. In contrast, a picture of clouds is asymmetrical because it lacks any such repetitive pattern or definite structure where parts correspond to one another.
Sample question3 marks
Q2. Explain the difference between the lines of symmetry in a square and a rectangle that is not a square. Why is the diagonal a line of symmetry for one but not the other?
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Model answer
A square has four lines of symmetry: one vertical, one horizontal, and two along its diagonals. This is because folding a square along its diagonal results in two halves that overlap exactly. In contrast, a rectangle that is not a square has only two lines of symmetry (vertical and horizontal). If you fold a rectangle along its diagonal, the two parts do not exactly fit over each other, so the diagonal is not a line of symmetry.
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Q3. What is a 'line of symmetry' and how can the paper-folding method be used to identify 'mirror halves'?
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Model answer
A line of symmetry is a line that cuts a figure into two parts that exactly overlap when folded along that line. These two overlapping parts are called mirror halves. To identify them using the paper-folding method, one folds a shape along a dotted line; if the left half exactly fits over the right half without any gaps or overlaps, the fold line is a line of symmetry. For example, a square has four such lines, while a rectangle's diagonal is not a line of symmetry.
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Q4. If a figure has rotational symmetry and its smallest angle of symmetry is 120 degrees, list all its angles of symmetry. Explain the relationship between these angles as described in the NCERT text.
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Model answer
The angles of symmetry for this figure are 120 degrees, 240 degrees, and 360 degrees. According to the text, the angles of symmetry are always multiples of the smallest angle. Since 120 is the smallest, the next positions where the figure looks identical are 120 + 120 (240 degrees) and 120 + 120 + 120 (360 degrees). Every figure eventually returns to its original position at 360 degrees.
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Q5. Explain the concept of rotational symmetry using the example of a paper windmill. Include the terms 'centre of rotation' and 'angle of symmetry'.
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Model answer
A figure has rotational symmetry if it looks exactly the same after being rotated by an angle less than 360 degrees. The fixed point about which the object rotates is the centre of rotation. For a paper windmill, even though it has no line of symmetry, it looks identical when rotated by 90 degrees. This 90-degree turn is called its smallest angle of symmetry. Other angles of symmetry for the windmill are 180, 270, and 360 degrees.
Want more questions with answers?Get the full practice set for this chapter.
A line of symmetry is a line that cuts a figure into two parts that exactly overlap when folded along that line. These two parts are called mirror halves. For example, a square has four lines of symmetry.
What is the difference between line symmetry and rotational symmetry?
Line symmetry (reflection symmetry) means a figure can be folded along a line to get two identical halves. Rotational symmetry means a figure looks the same after being rotated by an angle less than 360° about a fixed point. Some figures have both, some have only one, and some have neither.
How many lines of symmetry does a square have?
A square has 4 lines of symmetry: one vertical, one horizontal, and two diagonal lines. Each of these lines divides the square into two identical mirror halves.
What is an angle of symmetry?
An angle of symmetry is the angle through which a figure can be rotated about its centre so that it looks exactly the same as the original. For example, a square has angles of symmetry 90°, 180°, 270° and 360°.
How do you find the smallest angle of symmetry?
If a figure has n angles of symmetry, the smallest angle is 360° ÷ n. For example, if a figure has 4 angles of symmetry, the smallest angle is 360° ÷ 4 = 90°.
Does a circle have rotational symmetry?
Yes, a circle has rotational symmetry about its centre. In fact, every angle is an angle of symmetry for a circle because it looks the same no matter what angle you rotate it by.