# NCERT Solutions Class 9 Maths

NCERT Solutions for Class 9 Maths includes solutions to all the questions given in the NCERT Textbook for Class 9. The students can download PDF of chapter-wise solutions to these problems from the links provided in this page. These solutions of Class 9 Maths NCERT Solutions cover all the topics included in the textbook like – Number System, Coordinate Geometry, Polynomials, Euclid’s Geometry, Quadrilaterals, Triangles, Circles, Constructions, Surface Areas and Volumes, Statistics, Probability, etc.

With the help of these solutions available for NCERT Books for Class 9 Maths, students can practice all types of questions of the chapters. These CBSE Board Class 9 solutions have been designed by our experts in a well-structured format to provide several possible methods of answering the problems and ensure a proper understanding of concepts. The students are suggested to practice all these solutions thoroughly for their exams. Also, it will help them in building a foundation for higher-level classes.

NCERT Solutions for Class 9 Maths are an essential asset for students of Class 9. These solutions of Maths include answers to all questions as per latest CBSE Board Syllabus. Mathematics is one of the necessary subjects which not only decides the careers of many young students but also enhances their ability of analytical and rational thinking. EDUGROSS provide chapter-wise NCERT Solutions to help students clear their doubts by giving in-depth knowledge of the concepts of the chapters.

These NCERT Solutions for Class 9 Maths have been structured by highly skilled teachers. Students who are aiming to score higher in Board Exams and also in other competitive exams must practice from NCERT Solutions MathEDUGROSS provides a wide range of illustrative problems and solutions. By practicing these problems, students can increase their efficiency in problem-solving in Maths. Students can easily download Maths NCERT Solutions PDF for free.

A list of chapters provided in Class 9 Maths NCERT Solutions free pdf.

##### 1) NCERT Solutions for Class 9 Maths Chapter 1 Number System

Maths NCERT Solutions of Class 9 Chapter 1 Number System is provided here. The number system or the numeral system is the system of naming or representing numbers. There are various types of number systems in Maths such as binary, decimal, etc. This lesson covers the entire concepts of the numeral system with their types, conversions and questions. In this chapter, students will learn about the following topics which are given below:-

• Introduction
• Irrational Numbers
• Real Numbers and their Decimal Expansions
• Representing Real Numbers on the Number Line
• Operations on Real Numbers
• Laws of Exponents for Real Numbers

Some important formulae from this chapter are given below-

• Let a > 0 be a real number and p and q be rational numbers. Then

(i) ap . aq = ap+q                                    (ii) (ap )q = apq

(iii) ap /aq  = ap-q                                  (iv) ap .bp = (ab)p

• A number r is called a rational number, if it can be written in the form p/q , where p and q are integers and q ≠ 0.
• A number s is called an irrational number, if it cannot be written in the form p/q , where p and q are integers and q ≠ 0.

Chapter 1 Number System

##### 2) NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

Class 9 NCERT Solutions Chapter 2 Polynomials are available here. Polynomials are the algebraic expressions which consist of variables and coefficients. Variables are also sometimes called indeterminate. We can perform the arithmetic operations such as addition, subtraction, multiplication and also positive integer exponents for polynomial expressions but not division by variable. A list of the topics from this chapter is given below:

• Introduction
• Polynomials in One Variable
• Zeroes of a Polynomial
• Remainder Theorem
• Factorisation of Polynomials
• Algebraic Identities

Some important formulae from this chapter are given below-

• (x + y)2 = x2 + 2xy + y2
• (x – y)2 = x2 – 2xy + y2
• x2 – y2 = (x + y) (x – y)

Chapter 2 Polynomials

##### 3) NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry

Class 9 NCERT Solutions Chapter 3 Coordinate Geometry are available here. Coordinate geometry (or analytic geometry) is defined as the study of geometry using the coordinate points. In this chapter students will learn about the following topics which are given below: –

• Introduction
• Cartesian System
• Plotting a Point in the Plane if its Coordinates are Given

Chapter 3 Coordinate Geometry

##### 4) NCERT Solutions for Class 9 Maths Chapter 4 Linear Equations in Two Variables

NCERT Solutions of Class 9 Maths Chapter 4 Linear Equations in Two Variables is provided here. If a, b and r are real numbers (and if a and b are not both equal to 0) then ax + by = r is called a linear equation in two variables. Students will learn the following topics in this chapter: –

• Introduction
• Linear Equations
• Solution of a Linear Equation
• Graph of a Linear Equation in Two Variables
• Equations of Lines Parallel to the x-axis and y-axis

Some important formulae from this chapter are given below-

• An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.
• x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
• The graph of x = a is a straight line parallel to the y-axis.
• The graph of y = a is a straight line parallel to the x-axis.
• An equation of the type y = mx represents a line passing through the origin.

NCERT Solutions Maths Chapter 4 Linear Equations in Two Variables download free pdf.

Chapter 4 Linear Equations in Two Variables

##### 5) NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclids Geometry

Class 9 NCERT Solutions Chapter 5 Introduction to Euclids Geometry has been provided here in an easy and simple manner. A list of the topics from this chapter is given below: –

• Introduction
• Euclid’s Definitions, Axioms and Postulates
• Equivalent Versions of Euclid’s Fifth Postulate

NCERT Solutions Maths Chapter 5 Introduction to Euclids Geometry download free pdf.

Chapter 5 Introduction to Euclids Geometry

##### 6) NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

NCERT Solutions of Class 9 Maths Chapter 6 Lines and Angles is provided here. An angle is formed when two lines intersect each other. We represent an angle by the symbol . An angle involves two legs and one common vertex at which two lines meet. For example: ∠AOD is formed when line AB and CD intersect with each other. In this chapter students will learn about the following topics which are given below: –

• Introduction
• Basic Terms and Definitions
• Intersecting Lines and Non-Intersecting Lines
• Pairs of Angles
• Parallel Lines and a Transversal
• Lines Parallel to the Same Line
• Angle Sum Property of a Triangle

Some important Properties from this chapter are given below-

• If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice versa. This property is called as the Linear pair axiom.
• If two lines intersect each other, then the vertically opposite angles are equal.
• Lines which are parallel to a given line are parallel to each other.
• The sum of the three angles of a triangle is 180°.
• If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

Chapter 6 Lines and Angles

##### 7) NCERT Solutions for Class 9 Maths Chapter 7 Triangles

Chapter 7 Triangles for Class 9 NCERT Solutions are provided here in a less complicated manner. A triangle has three sides, three angles and three vertices. For example, in triangle ABC (denoted as ∆ ABC) – AB, BC, CA are the three sides, ∠ A, ∠ B, ∠ C are the three angles and A, B, C are three vertices. In this chapter students will learn about the following topics which are given below: –

• Introduction
• Congruence of Triangles
• Criteria for Congruence of Triangles
• Some Properties of a Triangle
• Some More Criteria for Congruence of Triangles
• Inequalities in a Triangle

Some important Properties from this chapter are given below-

• Angles opposite to equal sides of a triangle are equal.
• Sides opposite to equal angles of a triangle are equal.
• Each angle of an equilateral triangle is of 60°.
• If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent (SSS Congruence Rule).
• If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS Congruence Rule).
• In a triangle, angle opposite to the longer side is larger (greater).
• In a triangle, side opposite to the larger (greater) angle is longer.
• Sum of any two sides of a triangle is greater than the third side.

Chapter 7 Triangles

##### 8) NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Here, students will learn about quadrilateral. NCERT Solutions of Class 9 Maths Chapter 8 Quadrileteral is provided here, at EDUGROSS with wide range of problems and their solutions. A list of the topics from this chapter is given below: –

• Introduction
• Angle Sum Property of a Quadrilateral
• Properties of a Parallelogram
• Another Condition for a Quadrilateral to be a Parallelogram
• The Mid-point Theorem

Some important Properties from this chapter are given below-

• Sum of the angles of a quadrilateral is 360°.
• A diagonal of a parallelogram divides it into two congruent triangles.
• Diagonals of a rectangle bisect each other and are equal and vice-versa.
• Diagonals of a rhombus bisect each other at right angles and vice-versa.
• Diagonals of a square bisect each other at right angles and are equal, and vice-versa.

##### 9) NCERT Solutions for Class 9 Maths Chapter 9 Area of Parallelograms and Triangles

Class 9 NCERT Solutions Chapter 9 Areas of Parallelograms and Triangles has been provided here in an understandable pattern. Students will learn the following topics in this chapter: –

• Introduction
• Figures on the Same Base and Between the Same Parallels
• Parallelograms on the same Base and Between the same Parallels
• Triangles on the same Base and between the same Parallels

Some important formulae from this chapter are given below-

• Parallelograms on the same base (or equal bases) and between the same parallels are equal in area.
• Area of a parallelogram is the product of its base and the corresponding altitude.
• Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels.
• If a parallelogram and a triangle are on the same base and between the same parallels, then area of the triangle is half the area of the parallelogram.
• Triangles on the same base (or equal bases) and between the same parallels are equal in area.
• Area of a triangle is half the product of its base and the corresponding altitude.
• Triangles on the same base (or equal bases) and having equal areas lie between the same parallels.
• A median of a triangle divides it into two triangles of equal areas.

NCERT Solutions Maths Chapter 9 Area of Parallelograms and Triangles download free pdf.

Chapter 9 Area of Parallelograms and Triangles

##### 10) NCERT Solutions for Class 9 Maths Chapter 10 Circles

Chapter 10 Circles for Class 9 Maths NCERT Solutions are provided here in a less complicated manner. A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The following topics from this chapters are given below:-

• Introduction
• Circles and Its Related Terms: A Review
• Angle Subtended by a Chord at a Point
• Perpendicular from the Centre to a Chord
• Circle through Three Points
• Equal Chords and Their Distances from the Centre
• Angle Subtended by an Arc of a Circle

Some important formulae and properties from this chapter are given below-

• The perpendicular from the centre of a circle to a chord bisects the chord.
• The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
• There is one and only one circle passing through three non-collinear points.
• Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).
• Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
• Angles in the same segment of a circle are equal.
• Angle in a semicircle is a right angle.
• If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.
• The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
• If sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

Chapter 10 Circles

##### 11) NCERT Solutions for Class 9 Maths Chapter 11 Constructions

Class 9 Maths NCERT Solutions Class 9 Chapter 11 Constructions are provided here in an effortless pattern so that students can easily understand the method of solving problems. In this chapter students will learn about the following topics which are given below:-

• Introduction
• Basic Constructions
• Some Constructions of Triangles

Chapter 11 Constructions

##### 12) NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula

Maths NCERT Solutions Chapter 12 Heron’s Formula is provided here. In Geometry, Heron’s formula (sometimes called Hero’s formula), named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first. Students will learn the following topics in this chapter: –

• Introduction
• Area of a Triangle — by Heron’s Formula
• Application of Heron’s Formula in Finding Areas of Quadrilaterals

Some important formulae and properties from this chapter are given below-

• Area of a triangle with its sides as a, b and c is calculated by using Heron’s formula, stated as        Area of triangle = [s (s – a) (s – b) (s – c)]1/2              ;    where   S = (a + b + c )/2
• Area of a quadrilateral whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and using the Heron’s formula.

Chapter 12 Heron’s Formula

##### 13) NCERT Solutions for Class 9 Maths Chapter 13 Surface Areas and Volumes

Maths NCERT Solutions Chapter 13 Surface Areas and Volumes is provided here. The surface area is the area that describes the material that will be used to cover a geometric solid. Students will learn the following topics in this chapter:-

• Introduction
• Surface Area of a Cuboid and a Cube
• Surface Area of a Right Circular Cylinder
• Surface Area of a Right Circular Cone
• Surface Area of a Sphere
• Volume of a Cuboid
• Volume of a Cylinder
• Volume of a Right Circular Cone
• Volume of a Sphere

Some important formulae and properties from this chapter are given below-

• Surface Area of a Cuboid = 2(lb + bh + hl)
• Surface Area of a Cube = 6a2
• Curved Surface Area of a Cylinder = 2πrh
• Total Surface Area of a Cylinder = 2πr(r + h)
• Curved Surface Area of a Cone = 1 /2 × (l × 2πr) = πrl
• Total Surface Area of a Cone = πrl + πr2 = πr(l + r)
• Surface Area of a Sphere = 4 π r2
• Curved Surface Area of a Hemisphere = 2πr2
• Total Surface Area of a Hemisphere = 3πr2
• Volume of a Cuboid = base area × height = length × breadth × height
• Volume of a Cube = edge × edge × edge = a3
• Volume of a Cylinder = π r2 h
• Volume of a Cone = 1/3(π r2 h)
• Volume of a Sphere = 4/3 (π r3)
• Volume of a Hemisphere = 2/3 ( π r3 )

NCERT Solutions Maths Chapter 13 Surface Areas and Volumes download free pdf.

Chapter 13 Surface Areas and Volumes

##### 14) NCERT Solutions for Class 9 Maths Chapter 14Statistics

Numbers that have been collected in order to provide information about something is called statistics. Class 9 Maths NCERT Solutions for Chapter 14 Statistics has been provided here with complete explanation of problem solving. A list of the topics from this chapter is given below:-

• Introduction
• Collection of Data
• Presentation of Data
• Graphical Representation of Data
• Measures of Central Tendency
• Median : It is the value of the middle-most observation (s).
• If n is an odd number, the median = Value of the (n + 1/2) th observation.
• If n is an even number, the median = Mean of the values of the (n/2) th and (n/2 +1) th observations.
• Mode : The mode is the most frequently occurring observation.

Chapter 14 Statistics

##### 15) NCERT Solutions for Class 9 Maths Chapter 15 Probability

Chapter 15 Probability for Class 9 NCERT Solutions are provided here in a less complicated manner. Probability is the branch of Mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. Students will learn the following topics in this chapter:-

• Introduction
• Probability – an Experimental Approach
• The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.
• P(E) = Number of trials in which E has happened / Total number of trials.

Chapter 15 Probability

#### Advantages of Class 9 Maths NCERT Solutions

• All the questions are solved strictly based on the NCERT (CBSE) Syllabus and Books. Therefore, mastering these solutions will help students to score good marks in the examination.
• Solving these NCERT Solutions will help students clearing all their doubts with the help of materials. Students who are preparing for their upcoming exams are advised to practice these NCERT Solutions regularly to score better marks in their exam.
• While studying in CBSE Board Schools, students always get confused while deciding the right study material so you can take help from NCERT Solutions.
• For CBSE students, the best option is NCERT Solutions as it covers the whole CBSE Syllabus for Class 9 Maths.
• NCERT Solutions help in clearing the tough concepts by explaining it simply.
• Students should refer to these NCERT Solutions before their exams because these NCERT Solutions focus on basics to help students with concepts.
• NCERT Solutions give significant learning and also helps students to upgrade their skills.

#### Why prefer EDUGROSS for Class 9 Maths NCERT Solutions?

A lot of times students get stuck to a particular question. These solutions that we are providing here, at EDUGROSS, develop an interest in the students towards their studies. These solutions are designed by a group of experts so that every student can understand the concept in a simple way without further complications. Here, we provide you with the most reliable solutions.

Maths is one of those subjects which students find hard but here, at EDUGROSS, we provide you with the most accurate and easiest methods to solve various questions. Moreover, these solutions help students to develop their reasoning and logical skills. All study material here is completely based on the latest pattern and syllabus that is prescribed by the CBSE Board. All the basics of Class 9 Maths with exercises and solutions are thoroughly covered in the study material that is provided at EDUGROSS.

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