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 : Perimeter of a Rectangle
- Example 1
- Example 2
- Example 3
- Key Points Summary
Introduction
Perimeter is the distance you travel when you walk around the outside of a shape. Think of it as the length of a fence that surrounds your garden or the border around a picture frame.
For a rectangle, the perimeter is the total distance around all four sides.
Formula : Perimeter of a Rectangle
Perimeter of a rectangle = 2 × length + 2 × breadth
P = 2(1 + b) ⇒ (i) l = `P/2` − b, i.e., length = `"Perimeter"/2` − breadth
(ii) l = `P/2` − l, i.e., breadth = `"Perimeter"/2` − length
Example 1
The length of the rectangle below is 7 cm, and its breadth is 3 cm. Let us find its perimeter.
Solution:
Perimeter of rectangle PQRS = 2 × length + 2 × breadth
= 2 × 7 + 2 × 3
= 14 + 6
= 20
Therefore, the perimeter of the rectangle is 20 cm.
Example 2
Find Perimeter:
Solution:
Opposite sides of a rectangle are always equal. So, AB = CD and AD = BC
= AB + BC + AB + BC
= 2 × AB + 2 × BC
= 2 × (AB + BC)
= 2 × (12 cm + 8 cm)
= 2 × (20 cm)
= 40 cm.
Example 3
Find Perimeter
Solution:
The perimeter of the rectangle = Sum of the lengths of its four sides.
= AB + BC + CD + DA
= AB + BC + AB + BC...... (Opposite sides of a rectangle are equal)
= 2 × AB + 2 × BC
= 2 × (AB + BC)
= 2 × (15 cm + 9 cm)
= 2 × (24 cm)
= 48 cm
Key Points Summary
-
Perimeter = distance around a shape
-
A rectangle has 4 sides (opposite sides are equal)
-
Formula: P = 2 × (l + b)
-
Always use the same units for length and width
-
Answer must include units (cm, m, etc.)
Example Question 1
An athlete takes 10 rounds of a rectangular park, 50 m long and 25 m wide. Find the total distance covered by him.
Length of the rectangular park = 50 m
Breadth of the rectangular park = 25 m
Total distance covered by the athlete in one round will be the perimeter of the park.
Now, perimeter of the rectangular park = 2 × (length + breadth)
= 2 × (50 m + 25 m)
= 2 × 75 m
= 150 m
So, the distance covered by the athlete in one round is 150 m.
Therefore, distance covered in 10 rounds = 10 × 150 m = 1500m
The total distance covered by the athlete is 1500 m.
Example Question 2
A farmer has a rectangular field of length and breadth 240 m and 180 m respectively. He wants to fence it with 3 rounds of rope as shown in the figure. What is the total length of rope he must use?
The farmer has to cover three times the perimeter of that field.
Therefore, the total length of rope required is thrice its perimeter.
Perimeter of the field = 2 × (length + breadth)
= 2 × ( 240 m + 180 m)
= 2 × 420 m
= 840 m
Total length of rope required = 3 × 840 m = 2520 m.
Example Question 3
Find the cost of fencing a rectangular park of length 250 m and breadth 175 m at the rate of Rs. 12 per metre.
Length of the rectangular park = 250 m
Breadth of the rectangular park = 175 m
Perimeter of the rectangle = 2 × (length + breadth)
= 2 × (250 m + 175 m)
= 2 × (425 m)
= 850 m
Cost of fencing 1m of park = Rs. 12
Therefore, the total cost of fencing the park = Rs. 12 × 850 = Rs. 10200
