Page 258 - Ganit Kaushal

Symmetries of a Circle

The circle is an interesting shape.

What happens if you rotate a circle clockwise around its centre?
It always aligns perfectly with itself, regardless of the angle of rotation. This means
that every angle is an angle of symmetry for a circle.
Therefore, a circle has an infinite number of angles
of symmetry and infinite lines of symmetry. Remember
1. A figure may have multiple lines of symmetry.
Every line passing through the centre of a circle is 2. A figure may have multiple angles of symmetry.
its line of symmetry. 3. Some figures may have a line of symmetry, but

Now, we present a table that shows the reflection no angles of symmetry, while others may have
and rotational symmetry of some standard plane angles of symmetry but no lines of symmetry.
shapes by giving the number of lines of symmetry Some figures may have both lines of symmetry
and the number of angles of symmetry. as well as angles of symmetry.

Sr. No. Standard plane Number of lines Number of angles Least angle of Order of
shapes of symmetry of symmetry symmetry Symmetry
1. Rectangle 2 2-(180°, 360°) 180° 2
2. Rhombus 2 2-(180°, 360°) 180° 2
3. Square 4 4-(90°, 180°, 270°, 360°) 90° 4
4. Parallelogram 0 2-(180°, 360°) 180° 2
5. Trapezoid 0 1-(No rotational symmetry) 360° 1
6. Trapezoid (Equal 1 1-(No rotational symmetry) 360° 1
Base Angles)
7. Isosceles triangle 1 1-(No rotational symmetry) 360° 1
8. Equilateral triangle 3 3-(120°, 240°, 360°) 120° 3
9. Scalene triangle 0 1-(No rotational symmetry) 360° 1
10. Regular Pentagon 5 5-(72°, 144°, 216°, 288°, 360°) 72° 5
11. Regular Hexagon 6 6-(60°, 120°, 180°, 240°, 300°, 360°) 60° 6
12. Circle Infinite Infinite 0° Infinite

Solved Examples

Ex 1. Find the order and angles of symmetry for the given figures about the point
marked ‘•’. Also find the least angle of symmetry and the number of lines of
symmetry, and draw them if any.
Sol. (a) Angles of symmetry = 90°, 180°, 270°, and 360°. (a) (b)
Least angle of symmetry = 90°.
Order of symmetry = 360° ÷ 90° = 4
Number of lines of symmetry = 4. (shown in the figure by dotted lines)
(b) There is no rotational symmetry, as the only angle of symmetry is 360°. 90°
\ Least angle of symmetry = Angle of symmetry = 360° (a) (b)

Number of lines of symmetry = 1 (shown in the figure by dotted lines)
Order of symmetry = 360 ÷ 360 = 1.

256 Mathematics-6