Topics
Number System
Rational Numbers
- Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Rational Numbers on a Number Line
- Inserting Rational Numbers Between Two Given Rational Numbers
- Method of Finding a Large Number of Rational Numbers Between Two Given Rational Numbers
Exponents
- Concept of Exponents
- Law of Exponents (For Integral Powers)
- Negative Integral Exponents
- More About Exponents
Squares and Square Root
- Concept of Square Roots
- Finding Square Root by Division Method
- Finding Square Root Through Prime Factorisation
- To Find the Square Root of a Number Which is Not a Perfect Square (Using Division Method)
- Properties of Square Numbers
Cubes and Cube Roots
Playing with Numbers
- Arranging the Objects in Rows and Columns
- Generalized Form of Numbers
- Some Interesting Properties
- Letters for Digits (Cryptarithms)
- Divisibility by 10
- Divisibility by 2
- Divisibility by 5
- Divisibility by 3
- Divisibility by 6
- Divisibility by 11
- Divisibility by 4
Sets
- Concept of Sets
- Representation of a Set
- Cardinality of a Set
- Types of Sets
- Subset
- Proper Subset
- Super Set
- Universal Set
- Complement of a Set
- Difference of Two Sets
- Distributive Laws
- Venn Diagrams
Ratio and Proportion
Percent and Percentage
Profit, Loss and Discount
- Overhead Expenses
- To Find S.P., When C.P. and Gain (Or Loss) Percent Are Given
- To Find C.P., When S.P. and Gain (Or Loss) Percent Are Given
- Concept of Discount
Interest
- Calculation of Interest
- Concept of Compound Interest
- To Find the Principle (P); the Rate Percent (R) and the Time
- Interest Compounded Half Yearly
- Applications of Compound Interest Formula
Direct and Inverse Variations
- Types of Variation
- Unitary Method
- Concept of Arrow Method
- Time and Work
Algebra
Algebraic Expressions
- Algebraic Expressions
- Degree of Polynomial
- Factors and Common Factors
- Classification of Terms in Algebra
- Combining like Terms
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Polynomial
- Multiplying a Polynomial by a Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Simplification of Expressions
Identities
- Algebraic Identities
- Product of Sum and Difference of Two Terms
- Expansion Form of Numbers
- Important Formula of Expansion
- Cubes of Binomials
- Application of Formulae
Factorisation
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out Common Factors
- Factorisation by Grouping
- Factorisation by Difference of Two Squares
- Factorisation of Trinomials
- Factorising a Perfect Square Trinomial
- Factorising Completely
Linear Equations in One Variable
Linear Inequations
- Pair of Linear Equations in Two Variables
- Replacement Set and Solution Set
- Operation of Whole Numbers on Number Line
- Concept of Properties
Geometry
Understanding Shapes
Special Types of Quadrilaterals
Constructions
- Introduction of Constructions
- Construction of an Angle
- To Construct an Angle Equal to Given Angle
- To Draw the Bisector of a Given Angle
- Construction of an Angle Bisector: 30°, 45°, 60°, 90°
- Construction of Bisector of a Line
- The Perpendicular Bisector
- Construction of Parallel Lines
- Constructing a Quadrilateral
- Construction of Parallelograms
- Construction of a Rectangle When Its Length and Breadth Are Given.
- Construction of Rhombus
- Square: Properties and Construction
- Reflection Symmetry (Mirror Symmetry)
Representing 3-D in 2-D
- 2dimensional Perspective of 3dimensional Objects
- Concept of Polyhedron
- Faces, Edges and Vertices of Polyhedron
- Euler's Formula
- Concept of Polyhedron
- Nets of 3D Figures
Mensuration
Area of a Trapezium and a Polygon
Surface Area, Volume and Capacity
Data Handling (Statistics)
Data Handling
- Mathematical Data Collection and Organisation
- Frequency
- Raw Data, Arrayed Data and Frequency Distribution
- Cumulative Frequency and Cumulative Frequency Table
- Frequency Distribution Table
- Class Intervals and Class Limits
- Frequency Distribution and Its Applications
Probability
- Definition: Regular Polygon
- Formula: Sum of Interior Angles
- Formula: Each Interior Angle
- Formula: Each Exterior Angle
- Formula: Number of Sides
- Formula: Relationship between Interior and Exterior Angles
- Example 1
- Example 2
- Example 3
- Example 4
- Real-Life Applications
Definition: Regular Polygon
A polygon is a closed, flat figure made up of straight-line segments. A polygon is called a regular polygon if it is both:
-
Equiangular: All its interior angles are equal.
-
Equilateral: All its sides are equal in length.
Formula: Sum of Interior Angles
Sum of interior angles = (n − 2) × 180∘
Formula: Each Interior Angle
Each interior angle = `"(n − 2) × 180° " /"n"`
Formula: Each Exterior Angle
Each exterior angle = `"360°"/"n"`
Formula: Number of Sides
Number of sides (n) of a regular polygon = `"360°"/"exterior angle "`
Formula: Relationship between Interior and Exterior Angles
Relationship between interior and exterior angles = Exterior angle + Interior angle = 180°
Example 1
Find each interior angle of the regular polygon with the number of sides.
(i) 20 (ii) 15
Solution:
(i) Given number of sides n = 20
Each interior angle of the 20-sided regular polygon
= `"(n − 2) × 180° " /"n"`
= `"(20 − 2) × 180° " /"20"`
= `"18 × 180° " /"20"`
= 162°
(ii) Given number of sides, n = 15
Each interior angle of 15-sided regular polygon
= `"(n − 2) × 180° " /"n"`
= `"(15 − 2) × 180° " /"15"`
= `"13 × 180° " /"15"`
= 156°
Example 2
Find each exterior angle of a regular polygon with sides:
(i) 12 (ii) 18
Solution:
If the number of sides of a regular polygon = n,
its each exterior angle = `"360°" /"n"`
(i) Number of sides, n = 12
∴ Each exterior angle = `"360°" /"n"`
= `"360°" /"12"`
= 30°
(ii) Number of sides, n = 18
∴ Each exterior angle = `"360°" /"n"`
= `"360°" /"18"`
= 20°
Example 3
Find the number of sides of a regular polygon with each exterior angle:
(i) 18° (ii) 72°
Solution:
When exterior angle of a regular polygon is given,
the number of its sides = `"360°"/"exterior angle "`
(i) Exterior angle of the regular polygon = 18°
⇒ Number of sides in it = `"360°"/"exterior angle "`
= `"360°"/"18"`
= 20
(ii) Exterior angle of the regular polygon = 72°
⇒ Number of sides in it = `"360°"/"exterior angle "`
= `"360°"/"72"`
= 5
Example 4
Is it possible to have a regular polygon with each interior angle 135°?
Solution:
Method 1:
If n be the number of sides of the regular polygon,
∴ `"(n − 2) × 180° " /"n"` = 135° => 180n − 360 = 135n
=> 180n − 135n = 360
=> 45n = 360
and, n = `360/5` = 8
Since n = 8 is a whole number, a regular polygon with each interior angle 135° is possible.
Method 2:
Given each interior angle = 135°
and we know, at each vertex of a polygon,
interior angle + exterior angle = 180°
=> 135° + exterior angle = 180°
=> Exterior angle = 180° − 135°
= 45°
Now, number of sides = `"360°"/"exterior angle"`
= `"360°"/"45° "` = 8
Hence, a regular polygon with each interior angle of 135° is possible (an octagon).
Real-Life Applications
The Natural Engineer: Bees and Hexagons
Honeybees construct their honeycombs using regular hexagonal cells. This is one of nature's most efficient designs.
Why hexagons?
-
Each interior angle of a regular hexagon = 120°
-
Hexagons tessellate (fit together) perfectly with no gaps
-
This structure uses the least amount of wax while providing maximum storage space
-
The 120° angles create strong structural integrity
Mathematical calculation:
-
For a hexagon (n = 6):
-
Sum of interior angles = (6 − 2) × 180° = 720°
-
Each interior angle = 720° ÷ 6 = 120°
-
Each exterior angle = 360° ÷ 6 = 60°
Bees instinctively create this mathematically optimal shape without any formal training in geometry!
