Important Formulas Symmetry - (Maths) Class 7 (Old NCERT)

(1) If a line divides a figure into two parts such that when the figure is folded about the line the two parts of the figure coincide, then the line is known as the line of symmetry. The line of symmetry is also known as the axis of symmetry.

(2) A figure is said to have rotational symmetry if it fits on to itself more than once during a full turn, i.e. rotation through 360°.

(3) The number of times a figure fits onto itself in one full turn is called the order of rotational symmetry.

(4) Following table provides the details of linear and rotational symmetries of various figures:

Basic Concepts and Definitions

Line (Mirror) Symmetry

Definition: A figure has line symmetry if there exists a straight line (called the axis of symmetry) such that reflecting the figure about that line maps the figure onto itself.

• Vertical axis - the line of symmetry is vertical (example: letter A, if drawn with a straight crossbar appropriately).
• Horizontal axis - the line of symmetry is horizontal (example: letter B does not have a horizontal mirror, but some shapes do).
• Oblique axis - the line of symmetry is slanted (example: some kite shapes).

How to check for line symmetry:

• Fold the figure along a guessed line; if the two halves coincide exactly, that line is an axis of symmetry.
• Alternatively, reflect all points of the figure in the line and check whether the image coincides with the original figure.

Rotational Symmetry

• Definition: A figure has rotational symmetry if there exists an angle of rotation less than 360° about a point (the centre of rotation) that maps the figure onto itself.
• Order of rotational symmetry: If the figure maps onto itself n times during a full turn of 360°, then the figure has rotational symmetry of order n.
• Key formula:The smallest angle of rotation that maps the figure onto itself is θ = 360° ÷ n
  where n is the order of rotational symmetry.

Common Examples and Their Symmetries

• Circle: Infinite lines of symmetry through its centre; rotational symmetry of infinite order (fits on itself for every rotation).
• Square: 4 lines of symmetry; rotational symmetry of order 4; smallest rotation = 90°.
• Rectangle (not a square): 2 lines of symmetry; rotational symmetry of order 2; smallest rotation = 180°.
• Equilateral triangle: 3 lines of symmetry; rotational symmetry of order 3; smallest rotation = 120°.
• Regular pentagon: 5 lines of symmetry; rotational symmetry of order 5; smallest rotation = 72°.
• Regular hexagon: 6 lines of symmetry; rotational symmetry of order 6; smallest rotation = 60°.
• Isosceles triangle: 1 line of symmetry; rotational symmetry of order 1 (no rotation less than 360° maps it onto itself).
• Scalene triangle: 0 lines of symmetry; rotational symmetry of order 1.
• Parallelogram (general): 0 lines of symmetry; rotational symmetry of order 2; smallest rotation = 180°.
• Rhombus (not square): 2 lines of symmetry (along its diagonals); rotational symmetry of order 2; smallest rotation = 180°.

Methods and Short Procedures

• To count lines of symmetry: Try reflecting the figure about possible axes (vertical, horizontal, diagonal). Each axis that produces coincidence is a line of symmetry.
• To find the order of rotational symmetry: Find the smallest angle θ (less than 360°) such that a rotation by θ maps the figure onto itself. Then compute n = 360° ÷ θ.
• For regular polygons: A regular n-gon has n lines of symmetry and rotational symmetry of order n. The smallest rotation is 360° ÷ n.

Symmetry in Letters and Everyday Objects

• Many alphabet letters display line symmetry: for example, A, H, I, M, O, T, U, V, W, X, Y (vertical symmetry) depending on font style.
• Objects like butterflies, leaves, and many manufactured parts often use line symmetry for balance.
• Rotational symmetry appears in star designs, wheel rims, flower petals and architectural patterns.

Applications and Importance

• Symmetry helps in visual design, architecture, and art to create pleasing and balanced forms.
• In geometry, symmetry simplifies proofs and calculations by reducing the number of distinct cases.
• In science and engineering, recognising symmetry reduces complexity in problems involving forces, motion and structures.