Theory:

When an object shows rotational symmetry, it rotates around its center in an angle. The angle by which the object is turned once is called its angle of rotation. For one complete rotation, an object has to turn \(360°\).
As we know, a square has an order of rotation \(4\), and the angle of rotation of a square is \(90°\). 

91_2 (1)used.svg

Figure-1

Figure 1(i) is the original position of the square. If it is rotated by \(90°\) about the centre \(O\), we get figure 1(ii). Now observe the position of the shape. Rotate the square again by \(90°\), and you get figure 1(iii). When the square completes four quarter turns, the square reaches Figure 1(v), which is similar to its original position, Figure 1(i). You can check the coordinates of the square after each turn and note the number of turns it is taking to come to its original position.

Thus, We can see the square (EFHG)  has rotational symmetry of order \(4\) about its centre \(O\).

Important!

(i) The centre of rotation is the centre of the shape marked as \(O\).

(ii) The angle of rotation is 90°.

(iii) The direction of rotation is clockwise.

(iv) The order of rotational symmetry is \(4\). 

51_2.png

Figure-2

Figure 2(i) is the original position of the star. If it is rotated by \(72°\) about the centre \(O\), we get figure 2(ii). Now observe the position of the star. Rotate the star again by \(72°\), and you get figure 2(iii). When the star completes five turns, the star reaches figure 2(vi) which is similar to its original position figure 2(i). You can check the coordinates of the star after each turn and note the number of turns it is taking to come to its original position.

Thus, We can see the star (PQRST)  has rotational symmetry of order \(5\) about its centre \(O\).

Important!

(i) The centre of rotation is the centre of the shape marked as \(O\).

(ii) The angle of rotation is 72°.

(iii) The direction of rotation is clockwise.

(iv) The order of rotational symmetry is \(5\). 

Know the angle of rotation:
 
We can calculate the angle of rotation when we know the order of rotation of an object.
Example:
A square has an order of rotation 4.
 
We know that one complete rotation of any object is 360\(°\).
 
Then, 3604 \(=\) 90\(°\).
 
Therefore the angle of rotation of a square for one turn is 90\(°\). A square has to complete four turns to reach its original position attaining \(360°\).