Topics
Integers
- Natural Numbers
- Whole Numbers
- Negative and Positive Numbers
- Integers
- Representation of Integers on the Number Line
- Ordering of Integers
- Addition of Integers
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers
Fractions and Decimals
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Concept of Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Using Operator 'Of' with Multiplication and Division
- Multiplication of Fraction
- Division of Fractions
- Concept of Reciprocals or Multiplicative Inverses
- Problems Based on Fraction
- The Decimal Number System
- Comparing Decimal Numbers
- Addition of Decimal Fraction
- Subtraction of Decimal Numbers
- Multiplication of Decimal Numbers
- Division of Decimal Numbers
- Problems Based on Decimal Numbers
Data Handling
Simple Equations
Lines and Angles
The Triangle and Its Properties
- Basic Concepts of Triangles
- Classification of Triangles based on Sides
- Classification of Triangles based on Angles
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Basic Properties of a Triangle
- Right-angled Triangles and Pythagoras Property
Comparing Quantities
- Ratio
- Concept of Equivalent Ratios
- Proportion
- Unitary Method
- Basic Concept of Percentage
- Estimation in Percentages
- Interpreting Percentages
- Conversion between Percentage and Fraction or Decimal
- Ratios to Percents
- Increase Or Decrease as Percent
- Basic Concepts of Profit and Loss
- Profit or Loss as a Percentage
- Calculation of Interest
Congruence of Triangles
- Similarity and Congruency of Figures
- Congruence Among Line Segments
- Congruence of Angles
- Congruence of Triangles
- Criteria for Congruence of Triangles
- Criteria for Similarity of Triangles
- SAS Congruence Criterion
- ASA Congruence Criterion
- RHS Congruence Criterion
- Exceptional Criteria for Congruence of Triangles
Rational Numbers
- Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
Perimeter and Area
- Basic Concepts in Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangle
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Parallelogram
- Area of a Triangle
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter
- Problems based on Area
Practical Geometry
- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)
Algebraic Expressions
Exponents and Powers
- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Crores
Symmetry
Visualizing Solid Shapes
- Introduction
- Formula: Area of a Rectangle
- Example 1
- Example 2
- Key Points Summary
Introduction
A rectangle is a shape with four straight sides and four right angles.
- Opposite sides are equal
-
The length (l) is usually the longer side.
-
The breadth (b) or Width is the shorter side.
- The area of a rectangle is always measured in square units (cm², m², km²), because length and breadth are both measured in linear units and multiplying them gives a squared unit.
Formula: Area of a Rectangle
Area of a rectangle = length × breadth
Written as: A = l × b
l = `A/b` i.e., length = `"Area"/"Breadth"`
and, b = `A/l` i.e., breadth = `"Area"/"Length"`
Example 1
Solution:
standard formula:
Area = Length × Breadth
= 5 × 3 sq cm
= 15 sq cm
The area of the given rectangle is 15 square centimetres (sq cm).
Example 2
Question: A rectangular garden has an area of 30 m² and a length of 6 m. What is its width/breadth?
Given Information:
-
Area (A) = 30 m²
-
Length (l) = 6 m
-
Width (b) = ?
Solution:
We know: A = l × b
Rearranging: b = A ÷ l
-
b = 30 ÷ 6
-
b = 5 m
Check: 6 m × 5 m = 30 m²
Key Points Summary
1.Formula: Area of Rectangle = Length × Breadth (A = l × b).
2. Always Use Square Units: Write cm², m², km², etc., never just cm or m.
3. Order Doesn't Matter: 5 × 3 = 3 × 5 = 15 cm² (both give the same area).
4. Reverse Problems: If you know area and one side, divide to find the other: b = A ÷ l.
Example Question 1
Find the area of a rectangle whose length and breadth are 12 cm and 4 cm respectively.
Length of the rectangle = 12 cm
Breadth of the rectangle = 4 cm
Area of the rectangle = length × breadth
= 12 cm × 4 cm
= 48 sq cm.
Example Question 2
The area of a rectangular piece of cardboard is 36 sq cm and its length is 9 cm. What is the width of the cardboard?
Area of the rectangle = 36 sq cm
Length = 9 cm
Width = ?
Area of a rectangle = length × width
So, width = `"Area"/"Length" = 36/9` = 4 cm
Thus, the width of the rectangular cardboard is 4 cm.
Example Question 3
Find the area in square metre of a piece of cloth 1m 25 cm wide and 2 m long.
Length of the cloth = 2 m
Breadth of the cloth = 1 m 25 cm = 1 m + 0. 25 m = 1.25 m
(since 25 cm = 0.25m)
Area of the cloth = length of the cloth × breadth of the cloth
= 2 m × 1.25 m = 2.50 sq m.
