Question 27
In each case, the angles are the multiples of the smallest angle. You may wonder and ask if this will always happen. What do you think?
Yes, I think the angles of symmetry will always be multiples of the smallest angle of symmetry.
Step 1 — What is Rotational Symmetry?
Some shapes look the same after we turn them. We call this rotational symmetry. The amount we turn a shape is an angle.
Step 2 — The Smallest Angle
Every shape with rotational symmetry has a smallest angle. This is the smallest turn. It makes the shape look exactly the same. Let us call this smallest angle .
Step 3 — How Other Angles Appear
Let us turn a shape by its smallest angle . It looks the same. Turn it again by angle . Now it has turned . This is . It will still look the same. We can turn it by many times. Each time, the shape will look exactly the same. The angles are , , , and more. These are all multiples of . We do this until we turn a full circle. A full circle is 360 degrees.
Step 4 — Why it Always Happens
If a shape looks the same after turning by angle . It will also look the same after turning by . And it will look the same after and more. These are all multiples of the smallest angle . This will always happen for any shape with rotational symmetry.
Answer
I think this will always happen. In every case, the figure looks the same after turning it by its smallest angle of symmetry, and again after turning it by multiples of that angle — until we reach 360°. For example:
- If the smallest angle is 90°, then it repeats at 180°, 270°, and 360°.
- If the smallest angle is 120°, then it repeats at 240° and 360°. So yes, the angles of symmetry are always multiples of the smallest angle of symmetry.
More questions in IT
What about the butterfly? No doubt, the colours are very attractive. But what else about the butterfly appeals to you?
Can you see what repeats in the beautiful rangoli figure?
What about the pinwheel? Can you spot which pattern is repeating?
Hint: Look at the hexagon first.
Now, can you say what figure repeats along each side of the hexagon? What is the shape of the figure that is stuck to each side? Do you recognise it? How do these shapes move as you move along the boundary of the hexagon? What about the other pictures—what is it about those structures that appeals to you and what are the patterns in those structures that repeat?
What are the symmetries that you see in these beautiful structures?
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?
Thus, figures can have multiple lines of symmetry. The figures below also have multiple lines of symmetry. Can you find them all?
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?
First, see the rectangle and answer this question. Then, take a rectangular piece of paper and check if the two parts overlap by folding it along its diagonal. What do you observe?
Context: Consider a square with its corners labeled A, B, C and D, and its vertical line of symmetry as shown in the figure.
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?
In these two figures, a sheet of paper is folded and a cut is made along the dotted line shown. Draw a sketch of how the paper will look when unfolded.
Do you see a line of symmetry in this figure? What is it?
5 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.
b. The hole in the centre is a square.
Note: For the above two questions, check if the 4-sided figures in the centre satisfy both the properties of a square.
- How many lines of symmetry do these shapes have?
Find the lines of symmetry for the kolam below.
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.
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.
Hint: For (c) and (f), see if rotating the book helps!
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.
Do you know of any other shape that has exactly four angles of symmetry?
Consider this figure, a picture with 4 radial arms. How many angles of symmetry does it have? What are they? Note that the angle between adjacent central dotted lines is 90°.
Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.
How will you modify the figure above so that it has only two angles of symmetry?
Can we get a figure having exactly 3 angles of symmetry? Can you use radial arms for this?
Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they?
However, can anything in the figure be changed to make it have 3 angles of symmetry?
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.
Hint: Use 5 radial arms for the first case. What should the angle between two adjacent radial arms be?
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.
In each case, the angles are the multiples of the smallest angle. You may wonder and ask if this will always happen. What do you think?
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.
Like wheels, we can find other objects around us having rotational symmetry. Find them.