Class 6 Mathematics

Chapter 9 Solutions — Symmetry

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Solutions

Overview

Step-by-step NCERT solutions for Symmetry (Chapter 9, NCERT Class 6 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Symmetry textbook chapter.

Solved

What these solutions cover

All 28 questions in Symmetry are solved in the PDF. Here's what's inside, exercise by exercise:

Line of Symmetry

Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
For each of the following figures, identify the line(s) of symmetry if it exists. (Five figures are shown: an irregular zigzag polygon, a kite/diamond shape, a stepped rectangle, an L-shape, and an arrow/chevron.)

Generating Shapes Having Lines of Symmetry

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? (a) Two holes symmetrically on the left and right sides. (b) Two holes near the top — one slightly left of centre and one to the right, both near top…
2.
Given the line(s) of symmetry, find the other hole(s):
- (a) Square with a diagonal line of symmetry; one hole in the upper-left region.
- (b) Rectangle with a horizontal line of symmetry; one hole in the lower-left region.
- (c) Triangle with a vertical line of symmetry; one hole on the left side.
- (d) Circle with a diagonal line of symmetry; one hole near the top.
- (e) Circle with a diagonal line of…
3.
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.
- (a) Paper folded vertically once; a zigzag/irregular cut made from the folded edge inward.
- (b) Paper folded vertically once; a triangular/arrow notch cut from the folded edge.
- (c) Paper folded vertically and then again (double…
Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it? a. The hole in the centre is a square (the paper shows a square hole with sides parallel to the paper edges). b. The hole in the centre is a square (the paper shows a square hole rotated 45°, i.e., a square standing on a corner).
How many lines of symmetry do these shapes have? a. Two shapes shown: a square rotated 45° (rhombus orientation) and an 8-pointed star. b. A triangle with equal sides and equal angles (equilateral triangle). c. A hexagon with equal sides and equal angles (regular hexagon).
Trace each figure and draw the lines of symmetry, if any. (Eight figures shown: four rhombus/diamond cluster patterns in a row, then two grid figures and two grid figures.)
Find the lines of symmetry for the kolam below. (A kolam is shown: a hexagonal arrangement of star/flower patterns made from rhombuses and dots.)
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?
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.
10.
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. (Six sub-parts:
- (b) horizontal blue line,
- (c) diagonal blue line,
- (d) vertical blue line,
- (e) horizontal blue line,
- (f) diagonal blue line. Each shows a partial figure on squared paper.)
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. (Six sub-parts (a)–(f), each showing a partial figure with two blue lines — either two diagonals, a diagonal and a vertical, a vertical and a horizontal, etc.)
Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry. (Six incomplete figures shown on dot grids, each needing 2 more line segments to form a shape with at least one line of symmetry.)

Rotational Symmetry

1.
Find the angles of symmetry for the given figures about the point marked •.
- (a) A plus/cross shape made of squares — a square in the centre with four square arms, centre marked •.
- (b) An arrow or angular chevron shape pointing left, centre marked •.
- (c) A horizontal rectangle (long and thin), centre marked •.
Which of the following figures have more than one angle of symmetry? (Seven figures shown: a circle with cross inside and centre dot, a downward-pointing triangle with centre dot, a circle divided into 3 equal parts, a 4-blade fan/pinwheel shape, an X-cross made of two lines, a 5-pointed star with centre dot, and a D-shape/half-circle.)
3.
Give the order of rotational symmetry for each figure:
- (a) A line segment with an arrowhead at each end (a double-headed arrow).
- (b) An X-shape (two straight lines crossing at the centre, like an oblique X).
- (c) A 6-pointed star.
- (d) A 3-legged running figure (like the Isle of Man triskelion).
- (e) A plus/cross shape made of rectangles, with a centre dot.
- (f) A regular pentagon with a centre dot.

Symmetries of a Circle

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? (The circle is divided into 12 equal sectors.)
Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
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.
In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
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?
Can we have a figure with rotational symmetry whose smallest angle of symmetry is: a. 45°? b. 17°?
This is a picture of the new Parliament Building in Delhi. (The outer boundary is shown as a regular triangle with concave or flat sides — essentially a triangle with 3 lines of symmetry.) 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…
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?
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?
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?
How many lines of symmetry and angles of symmetry does Ashoka Chakra have?

Sample

Worked examples

3 questions from Symmetry, solved step by step.

Generating Shapes Having Lines of Symmetry

How many lines of symmetry do these shapes have? a. Two shapes shown: a square rotated 45° (rhombus orientation) and an 8-pointed star. b. A triangle with equal sides and equal angles (equilateral triangle). c. A hexagon with equal sides and equal angles (regular hexagon).

Part a:

Square rotated 45° (rhombus orientation): It is still a square — a square always has 4 lines of symmetry (2 through opposite vertices and 2 through midpoints of opposite sides). Lines of symmetry = 4.

8-pointed star (regular octagram): It has 8 lines of symmetry — one through each point and one through each indentation. Lines of symmetry = 8.

Part b — Equilateral triangle:

Each line of symmetry goes from a vertex to the midpoint of the opposite side. There are 3 vertices, so there are 3 lines of symmetry.

Part c — Regular hexagon:

A regular hexagon has 6 lines of symmetry — 3 lines connecting opposite vertices and 3 lines connecting midpoints of opposite sides.

Answer: a. Square (rotated 45°): 4 lines of symmetry; 8-pointed star: 8 lines of symmetry. b. Equilateral triangle: 3 lines of symmetry. c. Regular hexagon: 6 lines of symmetry.

Symmetries of a Circle

How many lines of symmetry and angles of symmetry does Ashoka Chakra have?

The Ashoka Chakra is the 24-spoked wheel appearing on the Indian national flag.

Lines of symmetry: Each spoke is a diameter of the wheel. Folding along any spoke creates two halves that overlap. Since there are 24 spokes, and each diameter is a line of symmetry, but each line of symmetry passes through 2 opposite spokes.

Number of distinct lines of symmetry = 24 ÷ 2 × 2 = 24.

The Ashoka Chakra has 24 lines of symmetry.

Angles of symmetry: The wheel looks the same after rotating by 360°/24 = 15°. So the angles of symmetry are all multiples of 15° from 15° to 360°.

Number of angles of symmetry = 360° ÷ 15° = 24.

The Ashoka Chakra has 24 angles of symmetry.

Answer: The Ashoka Chakra has 24 lines of symmetry and 24 angles of symmetry.

Generating Shapes Having Lines of Symmetry

Find the lines of symmetry for the kolam below. (A kolam is shown: a hexagonal arrangement of star/flower patterns made from rhombuses and dots.)

The kolam has an overall hexagonal shape with a repeating pattern of 6-fold symmetry.

Lines through opposite vertices of the hexagonal outline: There are 3 such lines.

Lines through midpoints of opposite sides of the hexagonal outline: There are 3 such lines.

Total lines of symmetry = 3 + 3 = 6.

Answer: The kolam has 6 lines of symmetry.

All 28 questions in Symmetry are solved step by step in the Solutions PDF above.

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Chapter 9 Solutions — Symmetry

Overview

What these solutions cover

Line of Symmetry

Generating Shapes Having Lines of Symmetry

Rotational Symmetry

Symmetries of a Circle

Worked examples

How many lines of symmetry do these shapes have? a. Two shapes shown: a square rotated 45° (rhombus orientation) and an 8-pointed star. b. A triangle with equal sides and equal angles (equilateral triangle). c. A hexagon with equal sides and equal angles (regular hexagon).

How many lines of symmetry and angles of symmetry does Ashoka Chakra have?

Find the lines of symmetry for the kolam below. (A kolam is shown: a hexagonal arrangement of star/flower patterns made from rhombuses and dots.)

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