CBSE Class 9 Maths Notes
Class 9th is an important phase of students’ life when they’re at the turning point of their life. The reason being class 9 is the foundation level to succeed in class 10. As you know, students must complete Class 9 in order to sit for Class 10 board examinations. Also, it lays the groundwork for the following classes. A kid who is well-versed in class 9 topics would find it simpler to perform well in competitive exams. Math and science are two subjects that demand a lot of practice to score in class 9. Hence we brought out the best from our resource treasury – CBSE Class 9 Maths Notes. GeeksforGeeks specially curated NCERT Notes for Class 9 Maths, compiled by experts.
These class 9th maths notes are prepared in an order to provide good knowledge of all the chapters based on the updated CBSE Class 9 Syllabus. GeeksforGeeks provided the following chapter-wise notes containing numbers of theorems with their proper explanations, formulae with their applications, and graphs that need to accelerate students’ knack for practicing problems and boost their interest in pursuing maths in higher classes.
CBSE Class 9 Maths Notes cover all the important chapters given in the revised NCERT textbooks that include some Important topics of Class 9 Maths like Number System, Polynomials in One Variable, Linear Equations, etc.
Class 9 Maths Notes cover some more important topics like Experimental Probability, Volumes of Cubes and Cuboids, Mean, Median, Mode, and Range, etc. Our experts have also covered Class 9 Maths Solutions like NCERT Solutions for Class 9 Maths, and RD Sharma Class 9 Solutions.
Chapter 1: Number Systems
The numeral or number system is the combination of natural, integers, rational, irrational, and real numbers. This lesson covers the entire concepts of the numeral system and its types, representation on the number line, laws of rational exponents, and integral powers. To simplify the concept of number systems, the technique of portraying numbers on a number line using certain rules and symbols is known as anumber system. A number line is a straight-line representation of integers with a set spacing between them. The Number System is used to do mathematical computations ranging from intricate scientific calculations to calculate how many chocolates are left in the box.
The major topics covered in Number systems chapter in Class 9 are Representation of natural numbers, integers, rational numbers on the number line, Rational numbers as recurring/ terminating decimals and Operations on real numbers. Some topics which have great importance in further chapters of Class 9 are Rationalization of real numbers and Laws of exponents for real numbers
Important rules that are used in CBSE Class 9 Maths Chapter 1- Number Systems are,
- √ab = √a × √b
- √(a/b)= √a/√b
- (√a + √b) × (√a – √b) = a−b
- (a + √b) × (a − √b) = a2 −b
- (√a+√b)2 =a2 + 2√ab +b
- ap × bq = (ab)p+q
- (ap)q = apq
- ap / aq = (a)p-q
- ap / bp = (ab)p
Chapter 1 of CBSE Class 9 Maths Notes covers the following topics:
- Number System
- Real Numbers
- Decimal Expansion of Real Numbers
- Operations on Real numbers
- Laws of exponents for real numbers
- Irrational numbers
- Rationalization of Denominators
- nth roots of a number
More resources for CBSE Class 9 Maths Chapter 1
Chapter 2: Polynomials
A polynomial expression is made up of variables, which are also known as indeterminates and coefficients in mathematics. The coefficients involve operations such as subtraction, addition, non-negative integer variable exponents, and multiplication. Both algebraic expression and polynomials in mathematics are made up of variables and constants, as well as arithmetic operations. The sole difference is that algebraic expressions include irrational numbers in their powers.
Topics covered in Class 9 Polynomial Chapters are the basics of polynomial in one variable (including the Coefficients of a polynomial, terms of a polynomial and zero polynomial), Degree of a polynomial and Types of Polynomials-Monomials, binomials, trinomials. Some important topics covered in this chapter are Factors and multiples, Zeros of a polynomial, and Factorization using the Factor Theorem.
Here is the list of the important theorems learned in CBSE Class 9 Maths Chapter 2- Polynomials,
- Remainder Theorem: If p(x) has the degree greater than or equal to 1 and p(x) when divided by the linear polynomial x – a will give the remainder as p(a).
- Factor Theorem: x – a will be the factor of the polynomial p(x), whenever p(a) = 0. The vice-versa also holds true every time.
Chapter 2 of CBSE Class 9 Maths Notes covers the following topics:
- Polynomials in One Variable
- Zeroes of a polynomial
- Remainder Theorem
- Multiplying Polynomials
- Algebraic Identities of Polynomials
- Standard Algebraic Identities
More resources for CBSE Class 9 Maths Chapter 2
Chapter 3: Coordinate Geometry
Coordinate geometry is important because it connects geometry with algebra using line graphs and curves. Because it allows us to find points on any plane, coordinate geometry is helpful in mathematics. It is also used in trigonometry, calculus, and other fields. Learn about the cartesian system, coordinate points, how to plot points on coordinate axes, quadrants with signs, and other concepts in coordinate geometry.
Important conclusions from CBSE Class 9 Maths Chapter 3- Coordinate Geometry are,
- The horizontal line is known as the x-axis and the vertical line is called the y-axis.
- The coordinates of a point are in the form of (+, +) in the first quadrant, (–, +) in the second quadrant, (–, –) in the third quadrant, and (+, –) in the fourth quadrant; where + and – denotes the positive and the negative real number respectively.
- The coordinates of the origin are (0, 0) and thereby it gets up to move in the positive and negative numbers.
Chapter 3 of CBSE Class 9 Maths Notes covers the following topics:
More resources for CBSE Class 9 Maths Chapter 3
Chapter 4: Linear Equations in Two Variables
Any equation which can be defined in the form ax + by + c = 0, where a, b and c arereal numbers, and a and b are not both zero, is called a linear equation in two variables. This chapter on Linear Equations in Two Variables is an essential subject in Mathematics since it allows us to define physical relationships between two variables, compute rates, perform conversions, and make predictions, among other things.
Students should pay special attention while solving and practicing the questions in this chapter because the majority of the questions in their examinations will require some experience in this area.
Some important formulas and identities from CBSE Class 9 Maths Chapter 4- Linear Equations in Two Variables are listed below,
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- (a + b) (a – b) = a2 -b2
- (x + a) (x + b) = x2 + (a + b) x + ab
- (x + a) (x – b) = x2 + (a – b) x – ab
- (x – a) (x + b) = x2 + (b – a) x – ab
- (x – a) (x – b) = x2 – (a + b) x + ab
- (a + b)3 = a3 + b3 + 3ab (a + b)
- (a – b)3 = a3 – b3 – 3ab (a – b)
- (x + y + z)2 = x2 + y2 + z2 + 2xy +2yz + 2xz
- (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
- (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
- (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
- x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz -xz)
- x2 + y2 = 1212 [(x + y)2 + (x – y)2]
- (x + a) (x + b) (x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
- x3 + y3 = (x + y) (x2 – xy + y2)
- x3 – y3 = (x – y) (x2 + xy + y2)
- x2 + y2 + z2 – xy – yz – zx = 1212 [(x – y)2 + (y – z)2 + (z – x)2]
Chapter 4 of CBSE Class 9 Maths Notes covers the following topics:
- Linear Equations and their solutions
- Graph of a linear equation in two variables
- Equations of Lines Parallel to the x-axis and y-axis
More resources for CBSE Class 9 Maths Chapter 4
Chapter 5: Introduction to Euclid’s Geometry
Euclidean geometry is the branch of geometry that deals with the study of geometrical shapes and figures based on different axioms and theorems. This study provides a brief explanation for flat surfaces. This chapter is the introduction to Euclid for Class 9 students.
This chapter is all about Euclid’s method of formalizing observed phenomena into rigorous Mathematics with definitions, axioms, postulates and theorems. Also includes the five postulates of Euclid, Equivalent versions of the fifth postulate, and a Representation of the relationship between axiom and theorem.
Here are the important rules from CBSE Class 9 Maths Chapter 5- Introduction to Euclid’s Geometry,
- Axioms: The basic facts which are taken for granted without proof are called axioms. Some of Euclid’s axioms are:
- Things which are equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- The whole is greater than the part.
- Postulates: Axioms are the general statements, postulates are the axioms relating to a particular field. Euclid’s five postulates are.
- A straight line may be drawn from anyone point to any other point.
- A terminated line can be produced indefinitely.
- A circle can be drawn with any center and any radius.
- All right angles are equal to one another.
- If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which the angles are less than two right angles.
Chapter 5 of CBSE Class 9 Maths Notes covers the following topics:
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Chapter 6: Lines and Angles
In geometry,Lines and Angles are defined as figures that are made up of infinite points extending indefinitely in both directions. Lines are straight and have length and breadth, while an angle is a figure from which two rays emerge from a common point.
To define it in simpler words, a line is defined as a row of closely spaced dots that spans in two directions indefinitely. It just has one dimension, which is its length. A line can be represented by a horizontal mark written on a sheet of paper. An angle is a figure formed by two rays that intersect at a shared terminus. A protractor is used to measure them in degrees. Lines and angles are present in all geometry forms.
Important definitions from CBSE Class 9 Maths Chapter 6- Lines and Angles are,
- Acute angle: An acute angle measures between 0° and 90°.
- Right angle: A right angle is exactly equal to 90°.
- Obtuse angle: An angle greater than 90° but less than 180°.
- Straight angle: A straight angle is equal to 180°.
- Reflex angle: An angle that is greater than 180° but less than 360° is called a reflex angle.
- Complementary angles: Two angles whose sum is 90° are called complementary angles. Let one angle be x, then its complementary angle is (90°−x).
- Supplementary angles: Two angles whose sum is 180° are called supplementary angles. Let one angle be x, then its supplementary angle is (180°−x).
- Adjacent angles: Two angles are Adjacent when they have a common side and a common vertex (corner point) and don’t overlap.
- Linear pair: A linear pair of angles is formed when two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of a straight angle is 180°, so a linear pair of angles must add up to 180°.
- Vertically opposite angles: Vertically opposite angles are formed when two lines intersect each other at a point. Vertically opposite angles are always equal.
Chapter 6 of CBSE Class 9 Maths Notes covers the following topics:
- Basic terms and definitions
- Pairs of Angles
- Parallel Lines and Transversals
- Lines parallel to the same line and Angle Sum Property
More resources for CBSE Class 9 Maths Chapter 6
Chapter 7: Triangles
Geometrically, a triangle is defined as a three-sided polygon consisting of three edges and three vertices. The most important and applied property of a triangle is its Angle sum property which means the sum of the internal angles of a triangle is equal to 180 degrees only.
Important rules covered in CBSE Class 9 Maths Chapter 7- Triangles are,
Congruence Rules: Here is the list of some important congruence rules of triangles,
- Side angle side (SAS) Congruence
- Angle Side Angle (ASA) Congruence
- Angle angle side (AAS) Congruence
- Side side side (SSS) Congruence
- Right-angle Hypotenuse Side (RHS) Congruence
Chapter 7 of CBSE Class 9 Maths Notes covers the following topics:
- Congruence of Triangles
- Properties of triangles
- Angle Sum Property of a Triangle
- Inequalities in a triangle
- Theorem – Angle opposite to equal sides of an isosceles triangle are equal
More resources for CBSE Class 9 Maths Chapter 7
Chapter 8: Quadrilateral
A quadrilateral is a plane geometrical figure which has four sides and four corners or vertices. Typically, quadrilaterals are rectangle, square, trapezoid, and kite or irregular and uncharacterized figures with four sides.
The topics covered in this chapter will help students to learn all the concepts of Quadrilateral thoroughly, They are the Angle sum property of a Quadrilateral, types of quadrilaterals, properties of a parallelogram, and the mid-point theorem.
Important properties that are covered in CBSE Class 9 Maths Chapter 8- Quadrilateral are,
- The Sum of all angles of a quadrilateral is 360°.
- A diagonal of a parallelogram divides it into two congruent triangles.
- In a parallelogram,
- diagonals bisect each other.
- opposite angles are equal.
- opposite sides are equal
- Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
- A line through the mid-point of a side of a triangle parallel to another side bisects the third side. (Midpoint theorem)
- The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the third side.
- In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.
- If a diagonal of a parallelogram bisect one of the angles of a parallelogram it also bisects the second angle.
- The angle bisectors of a parallelogram form a rectangle.
- Each of the four angles of a rectangle is the right angle.
- The diagonals of a rhombus are perpendicular to each other.
Chapter 8 of CBSE Class 9 Maths Notes covers the following topics:
- Types of quadrilateral
- Properties of Parallelograms
- Angle Sum Property of a Quadrilateral
- MidPoint Theorem
- Kite – Quadrilaterals
More resources for CBSE Class 9 Maths Chapter 8
Chapter 9: Areas of Parallelograms and Triangles
The area of a plane figure is described as the amount of the planar surface covered by a closed geometric figure like a rectangle, square, etc. In this chapter, we’ll try to strengthen our understanding of the equations for calculating the areas of various figures by looking at relationships between the areas of geometric shapes that have the same base and parallels. This study will also help in the understanding of several findings about ‘triangle similarity.’
The important topics covered in this chapter are the area of two or more triangles and parallelograms with the same base between the same parallels and finding the area of triangles that are split by a median as well as the area of congruent figures.
Important formulas used in CBSE Class 9 Maths Chapter 9- Areas of Parallelograms and Triangles are,
Chapter 9 of CBSE Class 9 Maths Notes covers the following topics:
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Chapter 10: Circles
Be it a bottle cap or the merry-go-round – the circle is a part of our day-to-day life and is included in everything we saw. But how exactly circle came to be? To explain it in mathematical words, a circle is a geometrical shape that is defined as the locus of points that moves in a plane so that its distance from a fixed point is always constant. This fixed point is the Centre of the circle while the fixed distance from it is called the radius of the circle.
Some important properties covered in CBSE Class 9 Maths Chapter 10- Circles are,
Chord: The chord of the circle is a line segment that connects any two locations on a circle. Some important properties of Chords of a circle are:
- The diameter of a circle is defined as a chord that passes across its center.
- A circle’s diameter divides it into two equal sections, which are called arcs. A semi-circle is made up of these two arcs.
- If two arcs of a circle have the same degree of measure, they are said to be congruent.
- When two arcs have the same length, their associated chords are likewise the same length.
- The chord is bisected by a perpendicular drawn from the center to the chord of the circle, and vice versa.
- Three non-collinear points are intersected by one and only one circle.
- Equal circle chords are equidistant from the center.
- The line across the centers of two circles intersecting in two points is perpendicular to the common chord.
- An arc’s angle at the center of the circle is double the angle it has throughout the rest of the circumference.
- Any two angles in the same circle segment are equal.
- A circle’s equal chords form an equal angle at the center.
- The greater chord of a circle is closer to the center than the smaller chord.
- The semicircle has a right angle. At the circle’s center, equal chords subtend an equal angle.
Cyclic Quadrilateral: A quadrilateral is said to be cyclic if all of its vertices are on the perimeter of a circle.
- The sum of opposing angles in a cyclic quadrilateral is 180°, and vice versa.
- A cyclic quadrilateral’s exterior angle is equal to its inner opposite angle.
Chapter 10 of CBSE Class 9 Maths Notes covers the following topics:
- Circles and its Related Terms
- Circle Theorems
- Theorem – There is one and only one circle passing through three given non-collinear points
- Theorem – The sum of opposite angles of a cyclic quadrilateral is 180°
- Inscribed Shapes in a Circle – Problem Solving
More resources for CBSE Class 9 Maths Chapter 10
Chapter 11: Constructions
Construction helps to understand the approach to construct different types of triangles for different given conditions using a ruler and compass of required measurements. Constructions are based on Geometry which is the foundation for comprehending fundamental arithmetic principles used in many professions.
Geometry form construction is a necessary ability that necessitates a thorough understanding of their qualities. As a result, students must thoroughly research this subject. The NCERT Solutions Class 9 Maths Chapter 11 Constructions is an excellent resource for learning about this geometry topic. These solutions serve as study aids for students.
Important constructional rules discussed in CBSE Class 9 Maths Chapter 11- Constructions,
- Construction of bisector of a line segment
- Construction of bisector of a given angle
- Construction of Equilateral triangle
- Construction of a triangle when its base, sum of the other two sides and one base angle are given
- Construction of a triangle when its base, difference of the other two sides and one base angle are given
- Construction of a triangle of given perimeter and two base angles
Chapter 11 of CBSE Class 9 Maths Notes covers the following topics:
More resources for CBSE Class 9 Maths Chapter
Chapter 12: Heron’s Formula
In this chapter, a formula called Heron’s formula is introduced which helps to determine the area of the triangle when three sides of its being given. The application of this formula also helps to find the area of other different polygons also. Heron’s formula is a useful technique to calculate the area of a triangle when the length of all three sides is given. These Class 9 maths NCERT Notes Chapter 12 Heron’s Formula will help students to understand this concept in detail.
Important formulas covered in CBSE Class 9 Maths Chapter 12- Heron’s Formula are
Chapter 12 of CBSE Class 9 Maths Notes covers the following topics:
More resources for CBSE Class 9 Maths Chapter 12
Chapter 13: Surface Areas and Volumes
Surface area and volume are the measures calculated for a three-dimensional geometrical shape like a cube, cuboid, sphere, etc. The surface area of any given object is the area occupied by the surface of the object while volume is the amount of space available in an object.
Some important formulas in CBSE Class 9 Maths Chapter 13- Surface Areas and Volumes are
- TSA of a Cuboid = 2(l x b) +2(b x h) +2(h x l)
- TSA of a Cube = 6a2
- TSA of a Right circular Cylinder = 2πr(h+r)
- TSA of a Right circular Cone = πr(l+r)
- TSA of a Sphere = 4πr2
- CSA of a Cuboid = 2h(l+b)
- CSA of a Cube = 4a2
- CSA of a Right circular Cylinder = 2πrh
- CSA of a Right circular Cone = πrl
- Volume of a Cuboid = l x b x h
- Volume of a Cube = a3
- Volume of a Right circular Cylinder = πr2h
- Volume of a Right circular Cone = 1/3πr2h
- Volume of a Sphere = 4/3πr3
Here, l is the length, b is the breadth, h is the height, r is the radius and a is the side of the respective geometrical figure.
Chapter 13 of CBSE Class 9 Maths Notes covers the following topics:
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Chapter 14: Statistics
Statistics is the study of the representation, collection, interpretation, analysis, presentation, and organization of data. In other words, it is a mathematical way to collect and summarize data. The representation of data differently along with the frequency distribution.
Students will have a good understanding of the significance of well-organized data, as well as the three measures of central tendency for ungrouped data, namely, mean, median, and mode, from NCERT notes for class 9 mathematics chapter 14. After studying this topic, students will be able to apply these formulae to a wide range of problems.
Some important formulae and terms studied in CBSE Class 9 Maths Chapter 14- Statistics are
- Class mark = (Lower Limit + Upper Limit)/2
- The three central tendencies are measured as:
- Mean (x‾) = Sum of all observations (∑xn) / Total Number of observation (N)
- Median = The median for even number of observation is equal to the middlemost observation whole for the odd number of observation it is equal to value of ((n+1)/2)th observation.
- Mode = It is equal to observation which occurs the most or have the maximum frequency in the given data.
Chapter 14 of CBSE Class 9 Maths Notes covers the following topics:
- Presentation of data
- Graphical representation of Data
- Bar graphs and Histograms
- Central Tendency
- Mean, Median, Mode, and Range
More resources for CBSE Class 9 Maths Chapter 14
Chapter 15: Probability
Tossing coin yields either an up or a down result, which is easily predicted. But what if you toss two coins at once? The end product might be a head and tail combo. In the latter instance, the correct answer cannot be found, therefore only the probability of a result may be predicted. Probability is the name given to this prediction. Probability is frequently employed in all aspects of daily life, such as sports, weather forecasts, blood tests, statics, etc. In this chapter, we will study probability in-depth.
The Probability in this class includes basic probability theory, which is also used in the probability distribution, to learn the possibility of outcomes for a random experiment and to find the probability of a single event to occur, when the total number of possible outcomes.
Important terms used in CBSE Class 9 Maths Chapter 15- Probability are
- Probability P (E) = Number of favorable outcomes / Total Number of outcomes
- The probability of any event only lies between 1 and 0.
- Trial: It is defined as the set of observations of event in which one or more outcomes are observed.
- Event: It is defined as the collection of observation performed to observe an experiment.
Chapter 15 of CBSE Class 9 Maths Notes covers the following topics:
More resources for CBSE Class 9 Maths Chapter 15
Important Resources for CBSE Class 9th provided by GeeksforGeeks
- NCERT Solutions Maths Class 9
- RD Sharma Solutions Maths Class 9
- CBSE Class 9 Maths Formulas
- CBSE Class 9 Physics Notes
- CBSE Class 9 Chemistry Notes
Frequently Asked Questions (FAQs)
Question 1: Why is Class 9 Maths NCERT Chapter 1 Important?
Since the number systems chapter covers rational and irrational numbers, real numbers and their expansion, decimal form, and the rule of exponents, it is a good way to start. As a result, the NCERT solutions for class 9 maths are important for exams.
Question 2: How to score good marks in the Class 9 Maths exam?
- Finish the CBSE Class 9 syllabus as early as possible.
- Do not waste time as you study. Study seriously for the 5-6 hours daily right from the beginning.
- Solve questions every time you finish a chapter
- Do not blindly mug up the theories. Understand them first. Then you will be able to write the answers in your own words. It will also have better retention.
- Solve sample papers after the entire syllabus is finished.
- Stay calm and confident. Allocate time for your entertainment as well. This will help you stay focused.
Question 3: How many hours should I study in Class 9?
On average, to prepare well for the exam, students should study for 6 hours per day on a regular basis. Six hours of preparation a day is more than enough for Class 9 CBSE preparation.
Question 4: Where and how can I download the complete NCERT Class 9 Syllabus, solutions, and notes?
The students can find the complete NCERT Syllabus of CBSE Class 9 easily on the GeeksforGeeks platform. Students can also find the books, notes, and complete solutions on the GeeksforGeeks platform at no cost, to help the students prepare well.