Symmetry

IMG_4330.jpgThere are two types of symmetry: line symmetry, which involves reflection , and rotational symmetry, which involves rotation.

Line Symmetry

A figure has a line of symmetry if it maps or folds onto itself under reflection in the line.

e.g. Y8_Symmetry_02.gif

Line m is called the axis of symmetry.

Different shapes can have various numbers of axes of symmetry. e.g.

Name of shape
Diagram
Number of axes of symmetry
Rectangle
Y8_Symmetry_03.gif
2
Hexagon
Y8_Symmetry_04.gif
6
Scalene triangle
Y8_Symmetry_05.gif
0

 

 

Rotational Symmetry

A figure has rotational symmetry if it maps onto itself as it is rotated about a point at its centre.
All figures have an order of rotational symmetry of at least 1.

The order of rotational symmetry is the number of times the shape maps onto itself during a rotation of 360°. e.g.

Name of shape
Diagram
Order of rotational symmetry
Rectangle
Y8_Symmetry_06.gif

2

(180° and 360°)
Hexagon
Y8_Symmetry_07.gif

6

(60°, 120°, 180°, 240°, 300°and 360°)
Scalene triangle
Y8_Symmetry_08.gif
1

 

 

Total Order of Symmetry

The total order of symmetry = number of axes of symmetry + order of rotational symmetry.

The table shows the symmetry properties of some common shapes.

 

Shape
Axes of symmetry
Order of rotational symmetry
Total order of symmetry

Scalene triangle
0
1
1

Isosceles triangle
1
1
2

Equilateral triangle
3
3
6

Kite
1
1
2

Trapezium
0
1
1

Isosceles trapezium
1
1
2

Parallelogram
0
2
2

Rhombus
2
2
4

Rectangle
2
2
4

Square
4
4
8

Regular pentagon
5
5
10

Regular hexagon
6
6
12

Regular octagon
8
8
16

 

A figure has point symmetry if it maps onto itself under a rotation of 180° (a half turn). e.g. A parallelogram.