Symmetry – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash, tailored for Class 6 in Mathematics.
This one-pager compiles key formulas and equations from the Symmetry chapter of Ganita Prakash. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Line of Symmetry: Folded halves overlap
A line of symmetry divides a figure into two mirror-image halves. This concept is used in art, nature, and design to create balanced and harmonious structures.
Rotational Symmetry: 360°/n
The angle of rotation for a figure with n identical segments to match its original position. This applies to objects like wheels and stars, ensuring uniform appearance upon rotation.
Angle of Symmetry: 360°/n
For a figure with n angles of symmetry. Each angle can be calculated using this relation, useful in understanding regular polygons and their properties.
Symmetrical Shapes: Mirror Image
Figures such as squares and circles exhibit symmetry. The identification of symmetrical properties aids in designing aesthetically pleasing shapes.
Square: 4 lines of symmetry
A square has lines of symmetry along its vertical, horizontal, and both diagonal axes, making it a perfect example of symmetry in two dimensions.
Rectangle: 2 lines of symmetry
Defines that a rectangle can be divided into two identical parts along its vertical and horizontal midlines, useful in architectural designs.
Triangle: 3 lines of symmetry
An equilateral triangle has three lines of symmetry. This can be demonstrated through folding, highlighting the balance in triangular designs.
Circle: Infinite lines of symmetry
A circle can be divided into symmetric sections from any diameter, demonstrating perfect rotational and reflectional symmetry.
Regular Polygon: n lines of symmetry
A regular polygon has the same number of lines of symmetry as it has sides (n), showcasing uniformity in geometric shapes.
Symmetric Patterns in Nature: Repeating Units
Symmetry in nature, such as flowers and animals, follows mathematical principles to create visually appealing patterns.
Equations
Rotation of 90°: 4 angles of symmetry
A figure with 4 angles of symmetry will look the same when rotated by 90°, 180°, 270°, and 360°, useful in identifying symmetrical properties of shapes.
Angle of Reflection: θ = 180° - φ
φ is the angle of incidence, demonstrating the relationship of angles when light is reflected on a symmetrical surface.
Reflection Symmetry: Line of Symmetry
If a figure can be divided into two identical parts, it can be represented mathematically as having a line of symmetry.
Symmetry in art/mathematics: Rigid Motions
Symmetrical properties involve transformations like reflection or rotation while maintaining the original figure, crucial in design and construction.
Angles of Polygons: Sum = (n - 2) x 180°
This formula helps derive the interior angles of polygons, linking symmetry to geometry.
Cyclic Nature of Symmetry: S_n = 360°/n
Defines periodic symmetry where n stands for number of identical sections in a figure, applicable in various geometric contexts.
Folded Symmetry: F = 1/n
Where n refers to the number of equivalent sections when a shape is folded symmetrically, illustrating balance.
Mirror Symmetry: M = 180°/k
Here, k indicates the number of segments reflecting across a center line; this highlights design principles in aesthetics.
Artistic Symmetry in Design: Area = L x W
Each section's area can be determined, combining symmetric principles in practical applications like tile patterns or fabric designs.
Reflective Surface: R = 2S
R is the reflective edge ratio of a symmetric shape to its base area, linking geometrical and artistic perspectives.
