Page 257 - Ganit Kaushal

The shapes which have 360° as their only angle of symmetry do not have rotational symmetry.

\ There is no rotational symmetry if we have

Angle of symmetry = Least angle of symmetry = 360°

Therefore, for a shape to have rotational symmetry, at least one angle of symmetry should be strictly less than
360°, and the least angle of symmetry can be at most 180°, i.e.,
Angle of symmetry < 360° and Least angle of symmetry ≤ 180°.

Now, what about the other shapes shown above — the Pinwheel, Swastik, and random shape?
Do they have rotational symmetry?

If yes, how many angles of symmetry do they have?
What is the least angle of symmetry?
Students should get ready with the answers to these questions.

Order of Rotational Symmetry

The order of rotational symmetry is defined as the number of times a figure overlaps onto itself in one full
turn (360°). It is also simply called ‘order of symmetry’.

We use the following formula to find the order of rotational symmetry:
360ϒ
Order of rotational symmetry =
Least angle of symmetry

For example: In the rotational symmetry of the windmill as explained above, the least angle of symmetry = 90°.
360ϒ
\ The order of rotational symmetry = = 4,
90 ϒ
which means that the windmill overlaps onto itself four times during a full rotation of 360°.

Rotational Symmetry of Figures with Radial Arms

A figure with ‘n’ radial arms of equal lengths and equal angles between them has ‘n angles of symmetry’, and
the least angle of symmetry is given by the formula:
360ϒ
Least angle of symmetry = , where n = no. of arms = no. of angles of symmetry
n

For example: Consider figure (i), with four radial arms of equal lengths and equal angles between the arms,
as given in the figure. It has four angles of symmetry, and the least angle of symmetry is (360° ÷ 4) = 90°.

A figure with unequal lengths of radial arms or unequal
angles between them will not have rotational symmetry.
For example: consider the adjoining figure (ii). 90°
This figure has unequal angles between its arms. We
see that only a full turn or a rotation of 360° will bring
the figure back to its original position. So, this figure
does not have rotational symmetry, as 360° is its only
angle of symmetry. (i) (ii)

Symmetry 255