Order of rotational symmetry

The order of rotational symmetry of a geometric figure is the number of times you can rotate the geometric figure so that it looks exactly the same as the original figure. 

Let us start with a shape that has an order of rotational symmetry of 1. A rotational symmetry of order 1 means that the shape will look like its original only once after you rotated the shape 360 degrees. The arrow you see below has a rotational symmetry of order 1.

Notice that you could not get the original arrow until you rotated the arrow 360 degrees.

When the order of rotational symmetry is 2

In our example above, we rotated a rectangle 90 degrees each time. Notice that we were able to get the original shape twice. The first time we got the original image, we got it with a rotation of 180 degrees and the second time, we got it with a rotation of 360 degrees. Since we were able to return the original shape 2 times, the rectangle has rotational symmetry of order 2.

Example of rotational symmetry of order 6.

In this last example above, we rotated a hexagon 60 degrees each time. Each 60 degrees rotation returns the original shape as you can see above. Since we were able to return the original shape 6 times, the hexagon has rotational symmetry of  order 6.

Other examples of order of rotational symmetry.

• Each 90 degrees rotation of a square will return the original square, so a square has an order of rotational symmetry of 4. Notice that 4 times 90 degrees = 360 degrees.

• Each 120 degrees rotation of an equilateral triangle will return the original equilateral triangle, so an equilateral triangle has an order of rotational symmetry of 3. Notice that 3 times 120 degrees = 360 degrees.

• Each 45 degrees rotation of an octagon will return the original octagon, so an octagon has an order of rotational symmetry of 8. Notice that 8 times 45 degrees = 360 degrees.