NCERT Solutions for Class 9 Maths 2026-27 | Ganita Manjari Part 1 & Old Textbook – Free PDF

For other State Boards

• Chapter 13. Surface Areas and Volumes
• Chapter 14. Statistics
• Chapter 15. Probability

Ganita Manjari Part 1 is the new Class 9 Mathematics textbook introduced by NCERT for session 2026-27 under NEP 2020. It has 8 chapters that take a more conceptual and activity-based approach compared to the previous book. Solutions for all chapters and exercises are available below, free, in English and Hindi medium.

• Chapter 1: Orienting Yourself — The Use of Coordinates
• Chapter 2: Introduction to Linear Polynomials
• Chapter 3: The World of Numbers
• Chapter 4: Exploring Algebraic Identities
• Chapter 5: I’m Up and Down, and Round and Round
• Chapter 6: Measuring Space: Perimeter and Area
• Chapter 7: The Mathematics of Maybe — Introduction to Probability
• Chapter 8: Predicting What Comes Next? — Exploring Sequences and Progressions

This chapter introduces the Cartesian coordinate system and teaches students how to locate points on a 2D plane, calculate distances and find midpoints. The exercises focus on spatial reasoning and real-life problems rather than routine calculation, making it one of the most foundational topics in the new syllabus.
What you will learn in this chapter:

• The 2D Cartesian coordinate system and its history
• How to locate and plot points on a coordinate plane
• Calculating the distance between two points
• Finding the midpoint of a line segment
• Real-life applications of coordinates.

Students begin by revisiting algebraic expressions and then move into polynomials — their definition, degree and structure. The chapter focuses on linear polynomials and how they model real-world situations like growth and decay. A key highlight is learning to interpret the slope and y-intercept of y = ax + b graphically, which builds strong intuition for coordinate geometry.
What you will learn in this chapter:

• Definition of a polynomial and its degree
• Introduction to linear polynomials
• Modelling linear growth and linear decay
• Visualising linear relationships through graphs
• Understanding slope and y-intercept of y = ax + b.

This chapter takes a careful, proof-based look at the number system. Students explore why √2 and √3 cannot be expressed as fractions, how rational numbers are densely packed on the number line and how to represent irrational numbers geometrically using the square root spiral. The approach is more rigorous than the previous textbook, encouraging students to think like mathematicians.
What you will learn in this chapter:

• Rational numbers and their representation on the number line
• Density of rational numbers and its proof
• Decimal representation of rational numbers
• What makes a number irrational
• Proof of irrationality of √2 and √3
• Constructing the square root spiral.

Rather than memorising identities, students here understand where they come from — using geometric models and algebra tiles to visualise why identities work. The chapter then moves into factorisation and simplifying rational expressions, with an emphasis on multiple approaches to the same problem.
What you will learn in this chapter:

• Revisiting and extending algebraic identities
• Visualising identities using geometric models
• Factorisation using algebra tiles
• Factorisation of quadratic expressions without algebra tiles
• Discovering new identities
• Simplifying rational expressions

This chapter connects mathematics to oscillatory and circular patterns found in nature and everyday life. Students explore how these patterns can be described and predicted mathematically, building an intuitive feel for periodic behaviour before formally studying trigonometry in higher classes.
What you will learn in this chapter:

• Recognising oscillatory and circular patterns in real life
• Describing periodic behaviour mathematically
• Representing up-down and circular motion through graphs
• Connecting patterns to mathematical sequences.

This chapter goes deeper into mensuration — introducing π properly and explaining its irrationality, deriving the area of a circle, working with arc length and sectors and applying Heron’s formula. It also includes Brahmagupta’s formula for cyclic quadrilaterals, giving students a glimpse into classical Indian mathematics. Solutions walk through each derivation so students understand the reasoning, not just the result.
What you will learn in this chapter:

• Perimeter of standard shapes
• Introduction to π and why it is irrational
• Length of an arc and area of a sector
• Area of rectangles, parallelograms and triangles
• Heron’s formula for area of a triangle
• Derivation of the area of a circle
• Brahmagupta’s formula for cyclic quadrilaterals.

Probability is introduced through experiments and observation rather than abstract definitions. Students first develop an intuitive sense of likelihood, then learn to compute empirical probability from data and theoretical probability using sample spaces, tree diagrams and tables. The chapter makes students comfortable with uncertainty as a mathematical idea.
What you will learn in this chapter:

• The concept of probability and randomness
• The probability scale from 0 to 1
• Empirical probability through data and experiments
• Theoretical probability using sample space and events
• Representing probability with tree diagrams and tables.

This chapter covers arithmetic and geometric progressions — how to find the nth term, visualise the pattern and apply formulas in practical contexts. The inclusion of the Tower of Hanoi puzzle and fractals as applications of GP shows students that sequences are tools for describing real complexity, not just abstract exercises.
What you will learn in this chapter:

• Introduction to sequences and their rules
• Arithmetic Progressions (AP): nth term and sum of n terms
• Visualising and applying AP in real-life situations
• Geometric Progressions (GP): nth term and applications
• Sum of the first n natural numbers
• GP in fractals and the Tower of Hanoi puzzle.

Students preparing for Class 9 Maths exams need quick access to all important formulas. The list below covers the most essential formulas from each chapter of the NCERT Class 9 Maths syllabus 2026–27, organized for fast revision.

• √(a/b) = √a / √b
• √(ab) = √a × √b
• (√a + √b)(√a − √b) = a − b
• (√a + √b)² = a + 2√(ab) + b
• aᵐ × aⁿ = aᵐ⁺ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• aᵐ / aⁿ = aᵐ⁻ⁿ (m > n)

• Remainder Theorem: p(a) is the remainder when p(x) is divided by (x − a)
• Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0
• (x + y)² = x² + 2xy + y²
• (x − y)² = x² − 2xy + y²
• (x + y)(x − y) = x² − y²
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy(x + y)
• (x − y)³ = x³ − y³ − 3xy(x − y)
• x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx)

• Semi-perimeter: s = (a + b + c) / 2
• Area = √[s(s−a)(s−b)(s−c)]

• Mean = Sum of all observations ÷ Total number of observations
• Mode = Value that appears most frequently in the data set
• Median = Middle value of data arranged in ascending order
• For grouped data: Mean = Σ(fᵢxᵢ) / Σfᵢ

Download the complete Class 9 Maths Formula Sheet from our app for offline revision without internet access.

Practicing the right questions is as important as studying solutions. The following are the most frequently examined topics and question types from each key chapter of Class 9 Maths, based on CBSE school exam patterns under the NCF 2023 syllabus.

• Represent √3, √5, and √10 on the number line using geometric construction.
• Rationalize the denominator of 1/(√7 − √6) and simplify.
• Prove that √2 is irrational.
• Express 0.3̄ and 1.2̄7̄ in the form p/q, where p and q are integers and q ≠ 0.

• Find the remainder when x³ + 3x² + 3x + 1 is divided by (x + 1) using the Remainder Theorem.
• Factorize: 8x³ + y³ + 27z³ − 18xyz.
• Using the Factor Theorem, determine whether g(x) = x − 2 is a factor of p(x) = x³ − 4x² + x + 6.

• Prove: If a transversal intersects two parallel lines, alternate interior angles are equal.
• In a figure where two lines intersect, prove that vertically opposite angles are equal.

• Prove that angles opposite to equal sides of an isosceles triangle are equal.
• In ΔABC, AB = AC and D is any point on BC. Prove that AD bisects angle A.
• Prove: The sum of any two sides of a triangle is greater than the third side.

• Prove that equal chords of a circle subtend equal angles at the center.
• Prove that the angle subtended by an arc at the center is double the angle subtended at any point on the remaining arc.
• Prove that angles in the same segment of a circle are equal.

• A cone of height 24 cm and base radius 6 cm is melted and recast into a sphere. Find the radius of the sphere.
• A hemispherical bowl of internal diameter 36 cm contains a liquid. How many cylindrical bottles of radius 3 cm and height 6 cm can be filled from it?

• The mean of 10 observations is 15. If one observation 20 is replaced by 30, find the new mean.
• Draw a frequency polygon for a given frequency distribution without drawing a histogram.

For complete step-by-step solutions to all important questions, visit each chapter page from the table above. You can also test yourself using the MCQ Online Tests linked per chapter.

Class 9 Mathematics for the academic session 2026–27 is designed according to the latest NCF 2023 guidelines, focusing on conceptual understanding rather than rote learning. The subject now emphasizes reasoning, logical thinking and real-life application of mathematical concepts.

Students are encouraged to explore multiple methods of solving problems, develop mathematical proofs and connect Maths with everyday situations. This approach helps build a strong foundation for higher classes and competitive exams.

The following table provides a detailed overview of the Class 9 Maths Ganita Manjari Part 1 syllabus for 2026–27, including all 8 chapters (From 1. Coordinate Geometry to 8) and their key concepts as per the latest NCF 2023 guidelines. This structured syllabus helps students understand what topics will be covered and how each chapter contributes to building strong mathematical concepts.

The updated syllabus includes the following key areas:

• Number Systems: Understanding real numbers, rational and irrational numbers, and their properties.
• Algebra: Algebraic expressions, identities and factorisation with a focus on patterns and relationships.
• Geometry: Study of quadrilaterals, circles, symmetry and basic theorems with logical reasoning and proofs.
• Data Handling: Collection, representation, and interpretation of data using graphs like bar graphs and histograms.
• Probability: Introduction to probability through real-life examples and experiments.

This syllabus is aligned with the latest educational reforms and focuses on practical understanding.

The new Class 9 Maths curriculum introduces several important changes:

• Emphasis on concept-based learning
• Real-life application of mathematical ideas
• Development of logical and analytical thinking
• Inclusion of visual tools like graphs and diagrams
• Encouragement of multiple problem-solving strategies
• Integration of computational thinking

This makes learning Maths more engaging and meaningful for students.

At Tiwari Academy, we provide detailed and step-by-step NCERT Solutions for Class 9 Maths based on the latest syllabus. Each chapter is explained in a simple and easy-to-understand manner to help students grasp concepts quickly.

These solutions are prepared by subject experts and cover all exercises from the NCERT textbook. Students can use them for homework, revision and exam preparation.

All answers follow a logical approach and include proper steps, diagrams, and explanations to ensure clarity.

The learning approach in the new syllabus focuses on understanding rather than memorization. Students are encouraged to:

• Explore mathematical concepts through activities
• Use visual aids like graphs and models
• Work collaboratively in groups
• Apply Maths in real-life situations
• Develop a problem-solving mindset

This method helps students build confidence and improves their overall mathematical skills.

To score well in Class 9 Maths, students should follow a structured approach:

• Start with understanding the concept before solving questions
• Practice all NCERT examples and exercises
• Revise formulas and theorems regularly
• Use diagrams wherever required
• Solve additional questions for better clarity

Consistency and regular practice are key to mastering Mathematics.

The latest syllabus introduces several new elements:

1. Focus on competency-based learning
2. Reduced emphasis on rote memorization
3. Greater importance of reasoning and proofs
4. Inclusion of real-world problem-solving
5. Activity-based and experiential learning

These changes aim to make Maths more practical and student-friendly.

After completing Class 9 Maths, students will be able to:

1. Understand and apply real number concepts
2. Solve algebraic expressions and equations
3. Prove basic geometrical theorems
4. Interpret data using graphs and charts
5. Apply probability in everyday situations

These outcomes ensure a strong conceptual base for future studies.

• 100% free NCERT solutions
• Updated as per latest syllabus
• Easy language explanations
• Step-by-step answers
• Suitable for all students

Solving Class 9 Maths chapter/syllabus-wise has several strong benefits — both for exam performance and long-term learning:

1. Strong Conceptual Foundation
   When you study chapter-by-chapter in NCERT order, each topic builds on the previous one.
   Example: Algebraic identities in Polynomials help in solving factorisation problems later.
2. Systematic Coverage Without Gaps
   Syllabus-wise practice ensures no topic is skipped — which is important because Class 9 syllabus is the base for Class 10 Board Exam & competitive exams like NTSE, Olympiads, JEE Foundation.
3. Better Exam Preparation
   Questions in school exams often come chapter-wise or mixed but balanced.
   If you solve syllabus-wise, you’ll be ready for both section-wise tests and the final paper.
4. Time Management
   Breaking the mathematics syllabus into chapters helps set daily/weekly goals and track progress.
   You avoid last-minute cramming and can revise multiple times before exams.
5. Linking Theory with Application
   Class 9 Maths has both concept-based (like Geometry, Algebra) and application-based (like Mensuration, Statistics) chapters.
   Syllabus-wise solving lets you understand where to apply which formula in real-world contexts.
6. Confidence Boost
   Completing chapters one by one gives a sense of accomplishment.
   This keeps motivation high and reduces exam stress.
7. Better for Revision
   When you solve chapter-wise, you can later revise topic-wise formula sheets, tricks and important questions quickly before exams.

Class 9 Maths Solutions simply means the step-by-step answers to all the problems and exercises given in the NCERT Class 9 Mathematics textbook (or any other prescribed book). It should includes:

1. Detailed Step-by-Step Solutions
   ► Each question is solved in a logical order, showing how to reach the final answer.
   ► Explains formulas, methods, and reasoning.
2. Chapter-wise Arrangement
   ► All 12 chapters (like Number Systems, Polynomials, Geometry, Statistics, etc.) are solved separately for easy reference.
3. NCERT Exercise Questions
   ► Covers every In-text Question, Exercise Question, and sometimes Additional Questions.
4. Multiple Methods (where possible)
   ► Sometimes a question is solved using more than one method to improve understanding.
5. Diagrams and Graphs
   ► For Geometry and Coordinate Geometry, proper figures are drawn for clarity.

In Class 9 Maths NCERT Chapter 3 – Coordinate Geometry, students explore the Cartesian plane, plotting points, and the basics that lead to the distance formula. These NCERT Class 9 Maths solutions provide clear, step-by-step explanations to strengthen mathematical reasoning and analytical skills. With detailed derivations of formulas and systematic problem-solving methods, even complex equations and theorems become easy to understand. This chapter is essential for building a strong foundation for higher-level geometry and algebra in later classes. The availability of Class 9 NCERT Mathematics textbook solutions in PDF format ensures students can conveniently access these resources for a way to effective study.

• Helps students understand concepts clearly.
• Acts as a self-check tool after solving questions.
• Useful for exam preparation, homework help and doubt clearing.
• Builds a strong base for Class 10 and higher classes.

To enhance understanding further, students can utilize interactive resources like NCERT Class 9 Maths Video Lectures and MCQ Online Tests. These are complemented by Mock Tests, MCQs and Doubt Clearing Sessions, ensuring that all learning gaps are addressed. For exam preparation, Class 9 NCERT Math Exam Pattern and Syllabus guides offer clarity on what to expect. Whether through Online Coaching or Interactive Learning Platforms, these solutions ensure students excel academically while enjoying the process of learning mathematics.

From a Class 9 Maths Solutions perspective, mobile apps can sometimes feel more comfortable and convenient than websites. Apps are often designed for quick access, offline use and smooth navigation, which makes it easier for students to find and practice solutions without distractions. They also allow features like bookmarking important questions, receiving instant updates and accessing solutions anytime, even without internet connectivity. This makes apps particularly useful for students who prefer studying on mobile devices over browsing through a website.
The benefits of using an app over a website especially relevant for something like Class 9 Maths Solutions:

1. Offline Access: Tiwari Academy Class 9 Maths apps let you download content so you can study without internet, unlike most websites.
2. Faster Performance: Apps usually load quicker and run smoother because they’re optimized for mobile devices.
3. Personalized Experience: Apps can remember your progress, bookmarks and preferences, offering a tailored learning journey.
4. Push Notifications: Get instant alerts for updates, new solutions or reminders without having to check manually.
5. Better User Interface on Mobile: Apps are designed specifically for small screens, making navigation and reading more comfortable.
6. Extra Features: Some apps offer in-built calculators, practice tests, progress trackers or gamified learning features.
7. Continuous Access: You can open an app instantly from your home screen without typing URLs or searching in a browser.

NCERT Solutions for Class 9 Maths are an essential study resource for students aiming to build strong mathematical fundamentals. These solutions follow the latest CBSE Class 9 Maths syllabus and provide chapter-wise, step-by-step explanations for clear understanding. Students can boost their exam preparation by practising Sample Papers, Important Questions and Previous Year Papers. With added support from Maths Formulas, Revision Notes and NCERT Class 9 Maths Worksheets, learners can revise key concepts quickly and strengthen weak areas through regular practice. This complete set of Class 9 Maths study materials ensures effective learning and improved performance in exams.

The Need of a Perfect Study Plan for Class 9 Maths

Following a study timetable in NCERT Class 9 Maths ensures consistent learning, better time management, and reduced stress. It helps prioritize subjects, focus on weak areas and balance schoolwork with leisure. A structured plan builds discipline, enhances productivity and prepares students effectively for mathematics exams, setting a strong foundation for future academic success in math.

Success is the sum of small efforts, repeated day in and day out. – Robert Collier

To support diverse learning needs, students can explore NCERT Class 9 Math Video Lectures and participate in Online Tests to build their confidence. For personalized guidance, Doubt Clearing Sessions and Online Coaching help clarify challenging topics. Tools like NCERT Maths MCQs and Interactive Learning Platforms make preparation engaging and interactive. With access to Study Material, Assignments and insights into the Exam Pattern, students gain a well-rounded understanding of the subject. Whether it’s solving complex problems or learning through Tips and Tricks, these resources ensure that Class 9 Mathematics students excel in math and approach the subject with confidence.
NCERT solutions for Class 9 Mathematics are incredibly useful for students because they are created by subject experts based on the CBSE curriculum. If you’re searching for the revised Class 9 Maths UP Board NCERT Solutions for the 2026-27 exams, Tiwari Academy has you covered. They offer updated solutions for all chapters, making it easier for students to understand key concepts and score higher in exams. These solutions not only strengthen mathematical understanding but also help students excel in their studies.

NCERT Solution Apps for 9th Class
You can get to know about various concepts while solving Class 9 Maths NCERT Solutions, which will be useful in your higher classes. Get Class 9th Maths question answers for the new Academic year based on latest NCERT book syllabus.

Our Educational App offers notifications and reminders for important dates, exams and study schedules, helping students stay organised and focused. The app also allows students to track their scores, monitor exercise completion and identify areas for improvement, making it a valuable tool for staying on track with their studies.

For a Class 9 student aiming to excel in Maths, a structured approach to learning and practice is the key. Start by thoroughly understanding the current syllabus and the fundamental concepts it covers.

Physical exercise, hobbies or short walks can refresh your mind and improve focus while studying. Staying positive and believing in yourself helps you grow. Regular quizzes or timed math practice show weak areas and build confidence.

These Chapter Wise Class 9 Mathematics Solutions App is well designed so that class 9 students can easily understand each sum’s solutions. Class 9th NCERT Math textbook has a total of 12 chapters which are divided into six units.
Solving Class 9th Maths solution in each chapter will ensure positive results. Chapters covered in the NCERT 9th Math textbook are Number System, Polynomials, Introduction to Euclids Geometry, Quadrilaterals, Surface Areas, Heron’s Formula and Volumes, Statistics etc.

Students who are facing any trouble in solving tough Class 9th Maths problems can get best help from these CBSE Maths of Class 9 Solutions of NCERT (https://ncert.nic.in/) textbooks. The syllabus contains 12 chapters with important topics. Here all solutions build your fundamental very clear and efficiency by solving theses exercise-wise solutions.

NCERT Solutions for Class 9 Maths Chapter 1 introduces key topics like rational numbers and irrational numbers. You will learn how to represent integers, rational numbers and irrational numbers on a number line. This chapter provides a comprehensive understanding of the number line, helping students grasp these fundamental concepts effectively.

Class 9 Maths Chapter 1 for CBSE Board
Class 9 Maths Exercise 1.1 in English
Class 9 Maths Exercise 1.2 in English
Class 9 Maths Exercise 1.3 in English
Class 9 Maths Exercise 1.4 in English
Class 9 Maths Exercise 1.5 in English

Class 9 Maths Chapter 1 Solution for State Boards
Class 9 Maths Chapter 1 Exercise 1.1
Class 9 Maths Chapter 1 Exercise 1.2
Class 9 Maths Chapter 1 Exercise 1.3
Class 9 Maths Chapter 1 Exercise 1.4
Class 9 Maths Chapter 1 Exercise 1.5
Class 9 Maths Chapter 1 Exercise 1.6

Class 9 Maths Chapter 1 in Hindi Medium
Class 9 Maths Exercise 1.1 in Hindi
Class 9 Maths Exercise 1.2 in Hindi
Class 9 Maths Exercise 1.3 in Hindi
Class 9 Maths Exercise 1.4 in Hindi
Class 9 Maths Exercise 1.5 in Hindi

There are five exercises in this NCERT Solutions for Class 9 Maths Chapter 1 Number System, where you can also find all word problems based on 9th Class Maths NCERT Book. You can learn from these 9th Class Maths Book Solution, the representation of terminating and non-terminating recurring decimals, presentation of square roots of 2, 3, other non-rational numbers on the number line.

Topics to Study in Class 9 Maths Chapter 1 Number Systems

Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.
2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number.
3. Definition of nth root of a real number.
4. Rationalization (with precise meaning) of real numbers of the type 1/(a + b√x) and 1/(√x + √y) (and their combinations) where x and y are natural number and a and b are integers.
5. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.)

Get NCERT Solutions for Class 9 Maths Chapter 2 here with free of cost, and all solutions are solved by the professional Maths teacher.

Class 9 Maths Chapter 2 for CBSE Board
Class 9 Maths Exercise 2.1 in English
Class 9 Maths Exercise 2.2 in English
Class 9 Maths Exercise 2.3 in English
Class 9 Maths Exercise 2.4 in English

Class 9 Maths Chapter 2 Solution for State Boards
Class 9 Maths Chapter 2 Exercise 2.1
Class 9 Maths Chapter 2 Exercise 2.2
Class 9 Maths Chapter 2 Exercise 2.3
Class 9 Maths Chapter 2 Exercise 2.4
Class 9 Maths Chapter 2 Exercise 2.5

Class 9 Maths Chapter 2 in Hindi Medium
Class 9 Maths Exercise 2.1 in Hindi
Class 9 Maths Exercise 2.2 in Hindi
Class 9 Maths Exercise 2.3 in Hindi
Class 9 Maths Exercise 2.4 in Hindi

This chapter Polynomial consists of variables and coefficients, its operations of addition, subtraction, multiplication and also non-negative integer exponents of variables.

Topics of Class 9 Maths Chapter 2 Polynomials

Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax² + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem.
Recall of algebraic expressions and identities. Verification of identities:
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
(x + y)³ = x³ + y³ + 3xy(x + y)
(x – y)³ = x³ – y³ – 3xy(x – y)
x³ + y³ = (x + y)(x² – xy + y²)
x³ – y³ = (x – y)(x² + xy + y²)
x³ + y³ + z³ – 3xyz = (x + y)(x² + y² + z² – xy – yz – 2zx)
and their use in factorization of polynomials.

Students can get NCERT exercises solutions for Class 9 Maths Chapter 3 with free of cost. Chapter 3 introduces the concepts of the Cartesian plane, including coordinates of a point, terms, coordinate plane notations, x-axis, y-axis, x-coordinate, y-coordinate, quadrants, origin and more. It provides a thorough understanding of these fundamental terms, helping students navigate and plot points on the coordinate plane effectively.

Class 9 Maths Chapter 3 for CBSE Board
Class 9 Maths Exercise 3.1 in English
Class 9 Maths Exercise 3.2 in English

Class 9 Maths Chapter 3 for State Boards
Class 9 Maths Chapter 3 Exercise 3.1
Class 9 Maths Chapter 3 Exercise 3.2
Class 9 Maths Chapter 3 Exercise 3.3

Class 9 Maths Chapter 3 in Hindi Medium
Class 9 Maths Exercise 3.1 in Hindi
Class 9 Maths Exercise 3.2 in Hindi

In NCERT Solutions for Class 9 Maths Chapter 3, Coordinate Geometry, students learn about the coordinate plane and the mutually perpendicular lines, known as axes. They can download the NCERT Solutions for Class 9 Maths PDF to study offline anytime, making it convenient for learning on the go. Get Chapter Number System Class 9 Maths Solutions in Hindi medium and English medium.

Class 9 Maths Chapter 3 Coordinate Geometry Main Points

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations.

NCERT Solutions for Class 9 Maths Chapter 4 4 covers linear equations in one variable and introduces linear equations in two variables, like ax + by + c = 0. In this chapter, students will also learn how to plot the graph of a linear equation in two variables, gaining a solid understanding of these key concepts.

Class 9 Maths Chapter 4 for CBSE Board
Class 9 Maths Exercise 4.1 in English
Class 9 Maths Exercise 4.2 in English

Class 9 Maths Chapter 4 for State Boards
Class 9 Maths Chapter 4 Exercise 4.1
Class 9 Maths Chapter 4 Exercise 4.2
Class 9 Maths Chapter 4 Exercise 4.3
Class 9 Maths Chapter 4 Exercise 4.4

Class 9 Maths Chapter 4 in Hindi Medium
Class 9 Maths Exercise 4.1 in Hindi
Class 9 Maths Exercise 4.2 in Hindi

NCERT Solutions for Class 9 Maths Chapter 9, Linear Equations in Two Variables, includes just two exercises. Students can download the free NCERT Solutions for Class 9 Maths PDF to study anytime, anywhere. These solutions help students clear all concepts for each math problem. Download unlimited free solutions for Chapter 9 Linear Equations in Two Variables in both Hindi Medium and English Medium.

Topics to be covered in Class 9 Maths Chapter 4 Linear Equations in Two Variables

Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax + by + c = 0. Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line.

Get NCERT Solutions for Class 9 Maths Chapter 5 for all exercises. This chapter covers Euclid’s approach to geometry, which forms the foundation of present-day geometry. In the NCERT exercises for Class 9 Maths, Introduction to Euclid’s Geometry, students will learn about defining common geometrical shapes and terms, providing a solid understanding of the basics of geometry.

Class 9 Maths Chapter 5 for CBSE Board
Class 9 Maths Exercise 5.1 in English

Class 9 Maths Chapter 5 for State Boards
Class 9 Maths Chapter 5 Exercise 5.1
Class 9 Maths Chapter 5 Exercise 5.2

Class 9 Maths Chapter 5 in Hindi Medium
Class 9 Maths Exercise 5.1 in Hindi

This 9th Class Maths Book Solution includes only one exercise that provides a deeper understanding of the relationship between axioms, postulates, and theorems. Download Chapter Euclid’s Geometry Class 9 Maths Solutions in both Hindi Medium and English Medium for free to enhance your learning experience.

Syllabus of Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

History – Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Showing the relationship between axiom and theorem, for example:
(Axiom) 1. Given two distinct points, there exists one and only one line through them. (Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common.

NCERT Solutions for Class 9 Maths Chapter 6 revolves around the theorems present in the topics of Lines and Angles.

Class 9 Maths Chapter 6 Solutions for CBSE Board
Class 9 Maths Exercise 6.1 in English
Class 9 Maths Exercise 6.2 in English

Class 9 Maths Chapter 6 Solutions for State Boards
Class 9 Maths Chapter 6 Exercise 6.1
Class 9 Maths Chapter 6 Exercise 6.2
Class 9 Maths Chapter 6 Exercise 6.3

Class 9 Maths Chapter 6 in Hindi Medium
Class 9 Maths Exercise 6.1 in Hindi
Class 9 Maths Exercise 6.2 in Hindi

The 9th Maths Book Solutions provided here are carefully reviewed for accuracy. Get free NCERT Solutions for Class 9 Maths in PDF format for offline study. Practice these solutions thoroughly to score your best in the final exams. Download Chapter Lines and Angles Class 9 Maths Solutions in both Hindi and English medium.

Class 9 Maths Chapter 6 Lines and Angles Topics

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse.
2. (Prove) If two lines intersect, vertically opposite angles are equal.
3. (Motivate) Lines which are parallel to a given line are parallel.

Get NCERT Solutions for Class 9 Maths Chapter 7 and students will study detail knowledge about the triangle, congruence of triangles, rules of congruence and its properties of triangles and many more.

Class 9 Maths Chapter 7 Solutions for CBSE Board
Class 9 Maths Exercise 7.1 in English
Class 9 Maths Exercise 7.2 in English
Class 9 Maths Exercise 7.3 in English

Class 9 Maths Chapter 7 Solutions for State Boards
Class 9 Maths Chapter 7 Exercise 7.1
Class 9 Maths Chapter 7 Exercise 7.2
Class 9 Maths Chapter 7 Exercise 7.3
Class 9 Maths Chapter 7 Exercise 7.4
Class 9 Maths Chapter 7 Exercise 7.5

Class 9 Maths Chapter 7 in Hindi Medium
Class 9 Maths Exercise 7.1 in Hindi
Class 9 Maths Exercise 7.2 in Hindi
Class 9 Maths Exercise 7.3 in Hindi

In this 9th Maths Book Solution, students will prove the properties of triangles learned in previous classes. While solving problems, they will apply various congruence rules. This chapter includes eight theorems, and there are three exercises covered in the NCERT Solutions for Class 9 Maths Chapter 7, Triangles. Download Chapter Triangles Class 9 Maths Solutions in both Hindi Medium and English Medium for a comprehensive understanding.

Class 9 Maths Chapter 7 Triangles Main Points to Study

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).
2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).
3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).
4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence)
5. (Prove) The angles opposite to equal sides of a triangle are equal.
6. (Motivate) The sides opposite to equal angles of a triangle are equal.

Find Solutions for Class 9 Maths Chapter 8, which covers the topic Quadrilaterals. The joining four points in order is called a quadrilateral.

Class 9 Maths Chapter 8 for CBSE and State Boards
Class 9 Maths Exercise 8.1 in English
Class 9 Maths Exercise 8.2 in English

Class 9 Maths Chapter 8 in Hindi Medium
Class 9 Maths Exercise 8.1 in Hindi
Class 9 Maths Exercise 8.2 in Hindi

Class 9 Maths Chapter 8 includes two exercises and focuses on proving one theorem. These solutions cover important concepts such as the angle sum property of a quadrilateral, properties of a parallelogram, and different types of quadrilaterals. Download Chapter Quadrilaterals Class 9 Maths Solutions in both Hindi Medium and English Medium for free.

Class 9 Maths Chapter 8 Quadrilaterals Focus Points

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.
2. (Motivate) In a parallelogram opposite sides are equal, and conversely.
3. (Motivate) In a parallelogram opposite angles are equal, and conversely.
4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.
5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.
6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.

Get well-explained Solutions for Class 9 Maths Chapter Circles for free, with no need to log in or register. Access solutions for all exercises at no cost, making learning easy and convenient. In this NCERT Solutions for Class 9 Maths Chapter 9 Circle, learn that circle is a collection of all the points in a plane at a fixed distance.

Class 9 Maths Chapter 9 for CBSE Board
Class 9 Maths Exercise 9.1 in English
Class 9 Maths Exercise 9.2 in English
Class 9 Maths Exercise 9.3 in English

Class 9 Maths Chapter 9 Solutions for State Boards
Class 9 Maths Chapter 9 Exercise 9.1
Class 9 Maths Chapter 9 Exercise 9.2
Class 9 Maths Chapter 9 Exercise 9.3
Class 9 Maths Chapter 9 Exercise 9.4

Class 9 Maths Chapter 9 in Hindi Medium
Class 9 Maths Exercise 9.1 in Hindi
Class 9 Maths Exercise 9.2 in Hindi
Class 9 Maths Exercise 9.3 in Hindi

Learn about circles, including Chord at a Point and Equal Chords, with twelve theorems in total. Download NCERT Solutions for Class 9 Maths for free, with no registration needed. Get Chapter 9 Circles Solutions in both Hindi and English Medium at no cost.

Class 9 Maths Chapter 9 Circles Curriculum for 2026-27

1.(Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse.
2.(Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.
4. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
5. (Motivate) Angles in the same segment of a circle are equal.
6. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.
7. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

Students get Solutions for Class 9 Maths Chapter 10, which covers Heron’s Formula, which helps calculate the area of a triangle when the lengths of all three sides are given. With Heron’s Formula, you can find the area of a triangle without needing to calculate the heights or distances directly.

Class 9 Maths Chapter 10 for CBSE Board
Class 9 Maths Exercise 10.1 in English

Class 9 Maths Chapter 10 for State Boards
Class 9 Maths Chapter 10 Exercise 10.1
Class 9 Maths Chapter 10 Exercise 10.2
Class 9 Maths Chapter 10 Exercise 10.3
Class 9 Maths Chapter 10 Exercise 10.4
Class 9 Maths Chapter 10 Exercise 10.5
Class 9 Maths Chapter 10 Exercise 10.6

Class 9 Maths Chapter 10 in Hindi Medium
Class 9 Maths Exercise 10.1 in Hindi

Heron’s Formula is used to find the areas of quadrilaterals, triangles, and other polygons. There is only one exercise in NCERT Solutions for Class 9 Maths Chapter 10, Heron’s Formula. You can download the NCERT Solutions for Class 9 Maths Chapter 12 PDF for free and study anytime, offline, on your desktop or mobile.

Class 9 Maths Chapter 10 Heron’s Formula Main Points To Study

Area of a triangle using Heron’s formula (without proof)

NCERT Solutions for Class 9 Maths Chapter 11, learn to find the surface areas and volumes of cuboids and cylinders in detail. In NCERT Solutions for Class 9 Maths Chapter 11, Surface Areas and Volumes, you will also explore the surface areas of other solids, such as cones and spheres.

Class 9 Maths Chapter 11 for CBSE Board
Class 9 Maths Exercise 11.1 in English
Class 9 Maths Exercise 11.2 in English
Class 9 Maths Exercise 11.3 in English
Class 9 Maths Exercise 11.4 in English

Class 9 Maths Chapter 11 Solutions for State Boards
Class 9 Maths Chapter 11 Exercise 11.1
Class 9 Maths Chapter 11 Exercise 11.2

Class 9 Maths Chapter 11 in Hindi Medium
Class 9 Maths Exercise 11.1 in Hindi
Class 9 Maths Exercise 11.2 in Hindi
Class 9 Maths Exercise 11.3 in Hindi
Class 9 Maths Exercise 11.4 in Hindi

In this chapter, students will deepen their understanding of surface areas and volumes, building on concepts from earlier classes. Chapter 11 includes four exercises covering the surface areas and volumes of various solids such as cubes, cuboids, hemispheres, spheres, cylinders and cones. Download the NCERT Solutions for Class 9 Maths PDF for free without any login or registration required. Get Chapter Surface Areas and Volumes Class 9 Maths Solutions in Hindi medium and English medium.

The Points to be covered in Class 9 Maths Chapter 11 Surface Areas and Volumes

Surface areas and volumes of spheres (including hemispheres) and right circular cones.

These best easy NCERT Solutions for Class 9 Maths Chapter 12, deals with Statistics which the extraction of meaningful information is studied is known as Statistics.

Class 9 Maths Chapter 12 for CBSE Board
Class 9 Maths Exercise 12.1 in English

Class 9 Maths Chapter 12 Solutions for State Boards
Class 9 Maths Chapter 12 Exercise 12.1
Class 9 Maths Chapter 12 Exercise 12.2

Class 9 Maths Chapter 12 in Hindi Medium
Class 9 Maths Exercise 12.1 in Hindi

Statistics plays an important role in daily life, influencing various aspects like people’s lives and state functions. NCERT Solutions for Class 9 Maths Chapter 12, Statistics, teaches how to present data in different ways, including frequency distribution. You’ll also learn how to organize data in tabular and row form, group it in regular intervals and calculate the mean, mode and median. Get Chapter Statistics Class 9 Maths Solutions in Hindi medium and English medium.

Class 9 Maths Chapter 12 Statistics Main Points

Bar graphs, histograms (with varying base lengths) and frequency polygons.

Tiwari Academy offers best NCERT solutions on Surface Areas and Volumes, based on the state board syllabus for the 2026-27 session, in both Hindi and English mediums. Master geometry with Tiwari Academy’s expert NCERT Solutions for Class 9 Maths Exercise 13.8, specifically designed for students of the current session, available in both languages.

Class 9 Maths Chapter 13 for State Boards
Class 9 Maths Chapter 13 Exercise 13.1
Class 9 Maths Chapter 13 Exercise 13.2
Class 9 Maths Chapter 13 Exercise 13.3
Class 9 Maths Chapter 13 Exercise 13.4
Class 9 Maths Chapter 13 Exercise 13.5
Class 9 Maths Chapter 13 Exercise 13.6
Class 9 Maths Chapter 13 Exercise 13.7
Class 9 Maths Chapter 13 Exercise 13.8
Class 9 Maths Chapter 13 Exercise 13.9

Access Class 9 Maths Chapter 14 Statistics solutions in both Hindi and English at Tiwari Academy. Perfect for CBSE students, our resources are comprehensive, user-friendly and designed to help you improve in your studies.

Class 9 Maths Chapter 1 Solution for State Boards
Class 9 Maths Chapter 14 Exercise 14.1
Class 9 Maths Chapter 14 Exercise 14.2
Class 9 Maths Chapter 14 Exercise 14.3
Class 9 Maths Chapter 14 Exercise 14.4

Master Class 9 Maths Chapter 15, Probability, with our solutions in English and Hindi. Clear and concise, these expert resources help you excel in CBSE exams and understand Probability easily.

Class 9 Maths Chapter 15 Solutions for State boards
Class 9 Maths Chapter 15 Exercise 15.1

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Following these discussions, the final draft of the solutions is prepared, adhering to the new syllabus. We offer well-structured materials with a classical approach to geometry and constructions, making key concepts easier to grasp. For UP Board High School students, complete solutions, sample papers and notes are available for the 2026-27 syllabus in Hindi and English. These resources support exam success and concept mastery.