Question 18
Can we have a figure with rotational symmetry whose smallest angle of symmetry is:
a. 45°?
b. 17°?
A figure looks the same after turning it. This is rotational symmetry. The smallest turn is the smallest angle of symmetry.
Step 1 — Understanding smallest angle of symmetry
Turning a figure brings it back to its start point. This full turn is 360 degrees. The smallest angle can turn the figure many times. Each turn makes it look the same. It completes a full 360 degree circle. So 360 degrees must be a multiple of the angle. Divide 360 by the smallest angle. We must get a whole number. This number shows how many times it looks the same.
Step 2 — Checking for 45°
Let us see if 45 degrees can be the smallest angle of symmetry. We need to divide 360 degrees by 45 degrees.
The number 8 is a whole number. So 45 degrees can be the smallest angle. For example, a regular octagon has 8 sides. It looks the same after a 45 degree turn.
Step 3 — Checking for 17°
Now, let us see if 17 degrees can be the smallest angle of symmetry. We need to divide 360 degrees by 17 degrees.
The number is not a whole number. So 17 degrees cannot be the smallest angle. Repeated 17 degree turns do not work. It will not match perfectly at 360 degrees.
Answer
(a) Yes, 45 degrees can be the smallest angle of symmetry. (b) No, 17 degrees cannot be the smallest angle of symmetry.
More questions in FIO
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.
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.
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?
Given the line(s) of symmetry, find the other hole(s):
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
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.
Trace each figure and draw the lines of symmetry, if any:
Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Find the angles of symmetry for the given figures about the point marked •.
(a) (b) (c)
Which of the following figures have more than one angle of symmetry?
Give the order of rotational symmetry for each figure:
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?
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°?