NCERT New Book Class 9 Maths (2026-27)

NCERT has released the draft of the class 9 new maths book for the session 2026-27. Here is the complete analysis of the NCERT new book class 9 Maths. The new book of class 9 Maths has 15 chapters.

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Class 9 Maths New Book Chapters

As discussed, there are 15 chapters in the new book for class 9th Maths. Here are these chapters:

Chapter Name	Topics Included
Coordinate Geometry	– Brief history of coordinate geometry – The 2-D Cartesian coordinate system – Distance between two points in the 2-D plane – Midpoint of the distance between two points in the 2-D plane
Introduction to Polynomials	– Algebraic expressions – Definition of a polynomial. Degree of a polynomial – Introduction to linear polynomials and applications – Exploring linear patterns – Modelling linear growth and linear decay – Linear relationships – Visualising linear relationships – Slope and y-intercept of a line $\mathrm{y}=\mathrm{ax}+\mathrm{b}$
Number Systems	– Introduction to rational numbers – Representation of rational numbers on the number line – Density of rational numbers and its proof – Finding rational numbers between any two rational numbers – Decimal representation of rational numbers – Introduction to irrational numbers – Proof of irrationality of $\sqrt{2}$ and $\sqrt{3}$ – The square root spiral
Introduction to Euclid’s Geometry: Axioms and Postulates	– History of geometry – Constructing a square with a given side as described in the Baudhayana’s Sulbasutras – Discovering Euclid’s definitions – Axioms: Axioms of measurement and rules for geometric objects
Lines and Angles	– Rays and angles – Measures of angles – Intersecting lines and angles – Pairs of angles – Theorems and examples on intersecting lines – Theorems and examples on parallel lines
Sequences and Progressions	– Introduction to sequences – Explicit or general rule of a sequence – Recursive rule of a sequence – Arithmetic Progressions (AP): nth term, visualising an AP, and practical contexts leading to – APs – Sum of the first n natural numbers – Geometric Progressions (GP): nth term, visualising a GP, and practical contexts leading to GPs – Applications of GP in fractals – Tower of Hanoi puzzle
Triangles: Congruence Theorems	– Practical applications and uses of triangles – Conditions of congruence of triangles and their proofs – Theorems on triangles – Propositions and the converse of a proposition – Problems based on applications of theorems on triangles
Mensuration: Area and Perimeter	– Perimeter of shapes – Perimeter of a circle: Introduction to Pi and its irrationality – Length of an arc – Area of shapes: rectangles, parallelograms, and triangles – Heron’s formula – Squaring a rectangle: Proof from Baudhayana’s Sulbasutras – Area of a circle: derivation – Area of the sector of a circle – Brahmagupta’s formula for the area of a cyclic 4-gon – Heron’s formula as a special case of Brahmagupta’s formula
Exploring Algebraic Identities	– Revisiting algebraic identities – Visualising identities using geometrical models – Factorisation of algebraic expressions using identities – More identities and their applications – Visualising factorisation of quadratic expressions through algebra tiles – Factorisation without using algebra tiles – Finding new identities – Simplifying rational expressions
4-gons (Quadrilaterals)	– Properties of parallelograms – Important theorems related to parallelograms and their proof – The midpoint theorem and its applications – Understanding the notion of central symmetry in the context of parallelograms
Circles	– Practical applications and uses of circles – Definitions related to a circle-centre, diameter, and radius – Chords and the angles they subtend – Midpoints and perpendicular bisectors of chords – Distance of chords from the centre – Subtended angles by an arc – Cyclicity of points
Linear Equations in Two Variables	– Introduction to linear equations in two variables through practical examples – Solution of linear equation in two variables: graphical representation – Slope-intercept form of linear equation in two variables – Drawing graphs of linear equations when x and y assume only certain values – Pair of linear equations in two variables – Graphical method for solving a pair of linear equations in two variables – Nature of solutions: consistency and inconsistency – Algebraic methods of solving a pair of linear equations: method of substitution and method of elimination
Mensuration: Surface Area and Volume	– Surface areas and volumes of spheres (including hemispheres) and right circular cones
Statistics	– Graphical representation of data – Measures of central tendency
Introduction to Probability	– Concept of probability and randomness – The probability scale – Empirical probability: analysing statistical data and performing experiments – Theoretical probability: sample space and events – Representing probability through tree diagrams and tables

Learning Outcomes

Here are the learning outcomes for each chapter.

Coordinate Geometry

Specify locations and the position of one point relative to another point using coordinates.
Represent a floor plan on a grid using coordinates.
Compute the distance between two points using coordinates.
Determine whether three points lie in a straight line using coordinates.
Compute the position of the midpoint of a line segment using coordinates.
Check whether a triangle is right-angled using coordinates.
Apply computational thinking to model situations on the coordinate plane and verify geometric properties through systematic reasoning.
Relevant CGs: CG-4, C-4.5, CG-9

Introduction to Polynomials

Understand the meaning of an algebraic expression.
Define a polynomial.
Identify the degree, terms and coefficients of terms in a polynomial.
Model linear growth and decay using linear polynomials.
Explain and identify patterns in linear relationships.
Identify the slope and y-intercept of a linear equation in two variables.
Graph a linear equation in two variables.
Use computational thinking to identify patterns, construct linear expressions, and systematically represent and analyse linear relationships using equations and graphs.
Relevant CGs: CG-3, C-3.2 , CG-9

Number Systems

Understand the concept of a rational number.
Represent rational numbers on the number line.
Understand the properties of rational numbers.
Explain the concept of density of rational numbers.
Compute decimal representation of rational numbers.
Understand the concept of irrational numbers.
Prove the irrationality.
Construct the square root spiral.
Apply computational thinking to represent rational and irrational numbers through algorithms and visual models, generate decimal expansions systematically, and reason about numbers using step-by-step logical procedures.
Relevant CGs: CG-1, C-1.1, CG-9

Introduction to Euclid’s Geometry: Axioms and Postulates

Describe how geometry grew from the practical needs ancient civilisations.
Describe contributions of India, Egypt and Greece to the development of geometric ideas.
Understand the role of definitions, axioms, and postulates.
Explain that there are elements of plane geometry (point, line, surface) for which we have an intuitive sense.
State the 5 postulates of Euclidean geometry.
Define parallelism of straight lines.
Explain the construction of a square as given in the Sulbasutras.
Justify simple constructions using the axioms.
Relevant CGs: CG-7, C-7.1, C-7.3

Lines and Angles

Explain the notion of an angle.
Explain the notion of a ray.
Explain that angles are formed between two rays with a common starting point.
State that a straight angle equals two right angles and measures $180^{\circ}$ while a right angle measures $90^{\circ}$.
Classify angles as acute, right, obtuse, or reflex.
Define parallelism.
State and apply the linear pair theorem and its converse.
Follow proof by contradiction in geometry.
Prove that vertically opposite angles are equal.
Identify corresponding, alternate, and interior angles.
Explain transitivity of parallelism.
Explain why a triangle must have at least two acute angles; why it cannot have two obtuse angles, or all three angles less than $60^{\circ}$
Apply computational thinking to analyse geometric ideas by breaking constructions into ordered steps, using axioms and postulates as rules, and justifying geometric results through logical step-by-step reasoning.
Relevant CGs: CG-7, C-7.1, C-7.3, CG-9

Sequences and Progressions

Understand the concept of a sequence of numbers.
Identify the pattern in a sequence and predict the next few terms.
Determine the recursive and explicit rules for different sequences.
Obtain the terms of a sequence given its recursive and explicit rule.
Identify Arithmetic Progressions (AP).
Determine the nth term of an AP.
Visualise an AP graphically.
Identify Geometric Progressions (GP).
Determine the nth term of a GP.
Visualise a GP graphically.
Analyse attributes of fractals using GP.
Solve the Tower of Hanoi puzzle.
Use computational thinking to identify patterns, write step-by-step rules, and model patterns in sequences and progressions.
Relevant CGs: CG-11, C-8.1, CG-9

Triangles: Congruence Theorems

Explain that a triangle is rigid, unlike a quadrilateral.
Identify uses of triangle rigidity.
Explain why triangles give strength and stability to structures.
Describe what it means for two triangles to be congruent.
Identify correspondence between the vertices, sides, and angles of two congruent triangles.
Use the SAS congruence axiom.
Use the SSS congruence condition.
Use the ASA congruence condition.
Use the RHS congruence condition.
Use the AAS congruence condition.
Prove the basic properties of isosceles triangles.
Explain the notion of a proposition.
Explain the notion of converse of a proposition.
Identify the converse of a given proposition.
Explain that not all converses are true; use counter examples to show that some converses are false.
Explain why SSA is not, in general, a valid congruence condition.
Identify the situations where SSA is a valid congruence condition.
Justify the role of diagram accuracy.
Relevant CGs: CG-4, C 4.1, C-7.3

Mensuration: Area and Perimeter

Define perimeter as the length around the boundary of any shape.
Explain that the circumference-to-diameter ratio is constant for all circles.
List historical approximations to $\pi$ (from Archimedes, Aryabhata, and Zu Chongzhi).
Compute the circumference of a circle and the length of an arc.
Apply ideas of circle perimeter and arc-length to real-world contexts.
Explain why a median of a triangle divides it into two triangles of equal area.
Use Heron’s formula to compute the area of a triangle from its sides.
Explain the classical problem of ‘squaring’ a given shape.
Explain how ancient civilisations approximated the area of a circle.
Compute the area of a circle using the formula.
Explain and use the formula for area of a sector of a circle.
Solve problems on areas of sectors and segments of circles.
State Brahmagupta’s formula for the area of a cyclic quadrilateral in terms of its sides.
Explain why Heron’s formula is a ‘special case’ of Brahmagupta’s formula.
Explain the notion of ‘special case’ and ‘generalisation’ in mathematics.
Use computational thinking to break down shapes, apply step-by-step methods to calculate perimeter and area, recognise patterns across formulae, and understand generalisation and special cases in geometry.
Relevant CGs: CG-5, C-5.1, CG-9

Exploring Algebraic Identities

Visualise algebraic identities using geometric models.
Determine the factors of algebraic expressions using identities.
Interpret factors of quadratic expressions through geometric models.
Find simplified versions of rational expressions.
Use computational thinking strategies, such as decomposition and step-by-step procedures to visualise algebraic identities, factor expressions, and simplify rational expressions.
Relevant CGs: CG-7, C-7.2, CG-9

4-gons (Quadrilaterals)

Frame a precise definition of a 4-gon.
Prove various characterisations of a parallelogram.
Prove the midpoint theorem.
Prove a converse of the midpoint theorem.
Prove that the medians of a triangle are concurrent and each median is divided in the ratio 2:1 at the point of concurrence.
Prove that the 4-gon formed by joining the midpoints of a given 4-gon is a parallelogram.
Find the coordinates of the midpoint of a line segment given its end points and find the coordinates of the fourth vertex of a parallelogram given the other three.
Understand reflection and rotation symmetries of 4-gons.
Understand how any 4-gon can tile a plane.
Practice forming logical converses of statements and asking questions guided by converses of theorems.
Engage in drawing, measurement and paper manipulation activities to discover geometric patterns involving triangles and 4-gons.
Relevant CGs: CG-4, C-4.2, C-7.3

Circles

State the definition of a circle.
Explain the meanings of the terms ‘chord’, ‘diameter’, ‘radius’, ‘arc’, ‘segment’, and ‘sector’.
Explain why there exists a unique circle through three non-collinear points.
Construct the circumcircle and circumcentre of a triangle.
Describe the location of the circumcentre for acute, obtuse, and right-angled triangles.
Explain what ‘angle subtended by an arc at the centre’ means.
Explain why ‘equal chords subtend equal angles at the centre’.
Explain why ‘chords that subtend equal angles at the centre are equal’.
Explain why ‘the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord’.
Explain why ‘a perpendicular from the centre to a chord bisects the chord’.
State the relationship between length of a chord and its distance from the centre of the circle.
Explain why ‘equal chords are equidistant from the centre (and conversely)’.
Explain why ‘among unequal chords, the longer chord is closer to the centre’.
Explain why ‘the diameter is the longest chord’.
Explain why ‘the angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the remaining part of the circle’.
Explain why ‘angles in the same segment of a circle are equal’.
Explain why ‘the angle in a semicircle is a right angle’.
Determine when four given points are concyclic.
Explain why ‘a quadrilateral with supplementary opposite angles is cyclic, and conversely’.
Explain how circular wheels have influenced transport, farming, building, and technology.
Identify cultural motifs involving circles, for example, the Dharmachakra, Ashoka Chakra, Sudarshan Chakra.
Use computational thinking to break down circle-related problems, apply geometric rules step-by-step, and verify properties of figures, such as chords, angles, and cyclic quadrilaterals through systematic reasoning.
Relevant CGs: CG-4, C-7.3, CG-9

Linear Equations in Two Variables

Understand the concept of a linear equation in two variables.
Graph a pair of linear equations.
Solve a pair of linear equations graphically.
Solve a pair of linear equations through the methods of substitution and elimination.
Determine the nature of solutions of a pair of linear equations.
Model and solve contextualised problems using a pair of linear equations and draw conclusions.
Model daily-life phenomena using representations, such as graphs, tables, and equations.
Use computational thinking to systematically represent, solve, and interpret pairs of linear equations through graphs, tables, and step-by-step procedures.
Relevant CGs: CG-3, C-3.2, C-8.1, CG-9

Mensuration: Surface Area and Volume

Recognise cuboids and cubes in real-life situations.
Compute the surface area and volume of a cuboid.
Explain how a cube is a ‘special case’ of a cuboid.
Describe a right circular cylinder using its radius and height.
Compute the surface area and volume of a cylinder.
Recognise cones in daily life, and describe them using radius and height.
Compute the surface area and volume of a cone.
Recognise a pyramid, and identify its base and apex.
Compute the surface area and volume of a pyramid.
Recognise spheres in real-life situations.
Compute the surface area and volume of a sphere.
Use computational thinking to systematically calculate, and compare surface areas and volumes of 3-D shapes by varying dimensions and analysing patterns.
Relevant CGs: CG-5, C-5.1, CG-9

Statistics

Collect, organise, visualise and interpret data to answer a statistical investigative question.
Compute and apply weighted average in different settings.
Read and interpret stacked bar graphs and 100\% stacked bar graphs.
Apply computational thinking strategies to analyse real-life data, create appropriate graphical representations, and interpret mean, median and mode for decision-making.
Relevant CGs: CG-6, C-6.1, CG – 9

Introduction to Probability

Understand the concept of randomness.
Describe the likelihood of an event using the probability scale.
Estimate the empirical probability of the occurrence of an event by analysing statistical data.
Define theoretical probability of an event.
Apply the definition of theoretical probability to compute the probability of an event.
Compute probability of events with the help of tree diagrams and tables.
Use computational thinking strategies, such as pattern recognition and simulation, to model random experiments and estimate probabilities.
Relevant CGs: CG-6, C-6.2, CG – 9