Symmetry | FIO

Question 12

Give the order of rotational symmetry for each figure:

Question diagram 1
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Solution
Understand the Question
  • Rotational symmetry is when an object looks identical to its original position after being rotated about its centre by an angle less than or equal to 360∘360^\circ.
  • The order of rotational symmetry is the number of times a figure matches its original appearance during one complete turn of 360∘360^\circ. Order of Rotational Symmetry=360∘Angle of Rotation\text{Order of Rotational Symmetry} = \dfrac{360^\circ}{\text{Angle of Rotation}}

(i) Give the order of rotational symmetry for the first figure (circle with cross).

Step 1 · Find the Order of Rotational Symmetry

Diagram 5

The figure has four identical arms and coincides with itself at rotations of: 90∘,180∘,270∘, and 360∘90^\circ, 180^\circ, 270^\circ, \text{ and } 360^\circ

Order of rotational symmetry=360∘90∘=4\text{Order of rotational symmetry} = \dfrac{360^\circ}{90^\circ} = 4

Answer

(i) 4

(ii) Give the order of rotational symmetry for the second figure (triangle / three-armed figure).

Step 1 · Find the Order of Rotational Symmetry

Diagram 3

The figure has three identical arms and coincides with itself at rotations of: 120∘,240∘, and 360∘120^\circ, 240^\circ, \text{ and } 360^\circ

Order of rotational symmetry=360∘120∘=3\text{Order of rotational symmetry} = \dfrac{360^\circ}{120^\circ} = 3

Answer

(ii) 3

(iii) Give the order of rotational symmetry for the third figure (circle with three spokes / triskelion).

Step 1 · Find the Order of Rotational Symmetry

The figure has three identical curved spokes and coincides with itself at rotations of: 120∘,240∘, and 360∘120^\circ, 240^\circ, \text{ and } 360^\circ

Order of rotational symmetry=360∘120∘=3\text{Order of rotational symmetry} = \dfrac{360^\circ}{120^\circ} = 3

Answer

(iii) 3

(iv) Give the order of rotational symmetry for the fourth figure (four-blade fan).

Step 1 · Find the Order of Rotational Symmetry

The figure has four identical blades and coincides with itself at rotations of: 90∘,180∘,270∘, and 360∘90^\circ, 180^\circ, 270^\circ, \text{ and } 360^\circ

Order of rotational symmetry=360∘90∘=4\text{Order of rotational symmetry} = \dfrac{360^\circ}{90^\circ} = 4

Answer

(iv) 4

(v) Give the order of rotational symmetry for the fifth figure (cross).

Step 1 · Find the Order of Rotational Symmetry

The cross shape coincides with itself at rotations of: 90∘,180∘,270∘, and 360∘90^\circ, 180^\circ, 270^\circ, \text{ and } 360^\circ

Order of rotational symmetry=360∘90∘=4\text{Order of rotational symmetry} = \dfrac{360^\circ}{90^\circ} = 4

Answer

(v) 4

Common Mistakes
  • Angle vs. Order Confusion: Confusing the angle of rotation (e.g. 90∘90^\circ or 120∘120^\circ) with the order of rotational symmetry (44 or 33).
  • Order Zero Misconception: Believing a figure can have order 00. Every figure has at least order 11 because it looks identical after a full rotation of 360∘360^\circ.

More questions in FIO

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?

Q2

For each of the following figures, identify the line(s) of symmetry if it exists.

Q3

Punching Game

The fold is a line of symmetry. Punch holes at different locations of a folded square sheet of paper using a punching machine and create different symmetric patterns.

Q4

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?

Q5

Given the line(s) of symmetry, find the other hole(s):

Q6

Here are some questions on paper cutting. Consider a vertical fold. We represent it this way:

Vertical Fold

Similarly, a horizontal fold is represented as follows:

Horizontal Fold

Q7

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.

Q8

Trace each figure and draw the lines of symmetry, if any:

Q9

Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.

Q10

Find the angles of symmetry for the given figures about the point marked •.

(a) (b) (c)

Q11

Which of the following figures have more than one angle of symmetry?

Q12

Give the order of rotational symmetry for each figure:

Q13

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?

Q14

Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.

Q15

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.

Q16

In a figure, 60∘60^\circ is the smallest angle of symmetry. What are the other angles of symmetry of this figure?

Q17

In a figure, 60∘60^\circ is an angle of symmetry. The figure has two angles of symmetry less than 60∘60^\circ. What is its smallest angle of symmetry?

Q18

Can we have a figure with rotational symmetry whose smallest angle of symmetry is:

a. 45∘45^\circ?

b. 17∘17^\circ?

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