Symmetry is the balance and proportion found when a figure is reflected, rotated or slid and still looks the same. When a figure can be folded along a line so that the two halves match exactly, it has line symmetry, and the line is called the line of symmetry or the axis of symmetry. A butterfly, the letter A, and a rectangle all have line symmetry. When a figure looks the same after being turned by a certain angle, it has rotational symmetry. A windmill, a wheel and a star have rotational symmetry. Symmetry is everywhere in nature, art, architecture and design, from the symmetry of a leaf to the balance of a building facade.
This chapter teaches us to identify lines of symmetry in figures and shapes, to count the number of lines of symmetry, to determine the order of rotational symmetry of a figure, and to understand how symmetry operations like reflection and rotation transform shapes. Studying symmetry develops visual thinking and an appreciation of pattern, and it connects mathematics with art and nature. Symmetry also forms the foundation for reflections and transformations studied in higher geometry and for crystallography and design.
A figure has line symmetry if a line can be drawn through it such that the part on one side of the line is the exact mirror image of the part on the other side. The figure coincides with itself after reflection across the line. Such a line is called a line of symmetry. For example, an isosceles triangle has one line of symmetry, a rectangle has two, an equilateral triangle has three, and a square has four.
Regular polygons have as many lines of symmetry as they have sides: - Equilateral triangle: 3 lines of symmetry. - Square: 4 lines of symmetry. - Regular pentagon: 5 lines of symmetry. - Regular hexagon: 6 lines of symmetry. - Circle: infinitely many lines of symmetry, since every diameter is a line of symmetry.
Some figures have no line of symmetry at all, such as a scalene triangle or the letter F. Letters like A, H, I, M, O, T, U, V, W, X and Y have vertical lines of symmetry; letters like B, C, D, E and K have horizontal lines of symmetry; while H, I, O and X have both.
A figure has rotational symmetry if it looks the same after being rotated through a certain angle less than a full turn of 360 degrees. The centre of rotation is the fixed point about which the figure turns. A full rotation of 360 degrees always brings the figure back to its original position.
The order of rotational symmetry is the number of times the figure looks exactly like its original position during one complete turn of 360 degrees. For example, a square looks the same after rotations of 90, 180, 270 and 360 degrees, so its order of rotational symmetry is 4. An equilateral triangle has order 3, a regular pentagon order 5, and a windmill with 3 blades has order 3.
The smallest angle through which a figure must be rotated to look the same as its original is called the angle of rotation. For a figure with order n, the angle of rotation is 360/n degrees. For a square, the angle of rotation is 360/4 = 90 degrees.
The centre of rotation is the point around which the figure turns. For regular polygons, the centre of rotation is the centre of the polygon. The position of the centre must be specified to describe rotational symmetry completely.
A reflection across a line produces the mirror image of a figure. The image of a point P across a line is the point P' such that the line is the perpendicular bisector of the segment PP'. A figure and its mirror image are congruent. Many real objects, such as the letters in the word MOM or the shapes of hands and feet, show reflection symmetry.
Observing the alphabet helps us practise identifying symmetry. Letters with vertical symmetry include A, H, I, M, O, T, U, V, W, X and Y. Letters with horizontal symmetry include B, C, D, E, H, I, K, O and X. The letters H, I, O and X have both vertical and horizontal symmetry.
Symmetry is valued in design because balanced forms look pleasing and are structurally stable. Buildings, bridges, vehicles, logos and flags often use symmetry. Natural objects like snowflakes, flowers, shells and many leaves show line or rotational symmetry. Studying symmetry helps us design patterns, understand reflection in mirrors, and appreciate the balance found in the natural world.
| Figure | Number of Lines of Symmetry |
|---|---|
| Isosceles triangle | 1 |
| Rectangle | 2 |
| Equilateral triangle | 3 |
| Square | 4 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | Infinitely many |
| Scalene triangle | 0 |
| Figure | Order of Rotational Symmetry | Angle of Rotation |
|---|---|---|
| Equilateral triangle | 3 | 120 degrees |
| Square | 4 | 90 degrees |
| Rectangle | 2 | 180 degrees |
| Regular pentagon | 5 | 72 degrees |
| Regular hexagon | 6 | 60 degrees |
| Circle | Infinite | Any angle |
Symmetry reveals the harmony in shapes through reflection and rotation. Line symmetry divides a figure into matching mirror halves, while rotational symmetry keeps a figure looking the same after partial turns. Counting lines of symmetry and determining the order of rotational symmetry give us precise ways to describe the balance of figures, from simple letters to regular polygons and the circle. Symmetry is not only a mathematical idea but an artistic and scientific one, appearing in architecture, design, nature and crystallography. Understanding symmetry sharpens spatial reasoning and prepares students for transformations, reflections and coordinate geometry in higher classes.