CBSE Class 10 Maths Notes

Last Updated : 10 Oct, 2026

Maths is an important subject in the CBSE Class 10 Board Exam, and steady preparation and revision can help you score well.

These chapter-wise notes cover all 14 chapters of the 2026-27 syllabus, with key concepts linked to detailed articles and the formulas worth memorising.

Chapter 1: Real Numbers

Real numbers combine rational and irrational numbers. This chapter covers the Fundamental Theorem of Arithmetic, HCF and LCM, and proofs that numbers such as √2 are irrational.

Must-know formulas and results

  • Fundamental theorem: every composite number is a product of primes, and this product is unique apart from the order of the factors.
  • HCF = product of the smallest power of each common prime factor. LCM = product of the greatest power of each prime factor that appears.
  • For two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. This does not hold for three numbers.
  • If a prime p divides a², then p divides a. This is the key step in every irrationality proof.
  • The sum or difference of a rational and an irrational number is irrational, and so is the product of a non-zero rational and an irrational number. So 3 + 2√5 is irrational.

Chapter 2: Polynomials

A zero of a polynomial is a value of x for which p(x) = 0, and on a graph it is where the curve meets the x-axis. This chapter links the zeros of a quadratic polynomial to its coefficients.

Chapter 3: Pair of Linear Equations in Two Variables

A pair of linear equations in two variables represents two lines that can intersect, be parallel or coincide. This chapter covers the graphical, substitution and elimination methods and the conditions for the number of solutions.

Must-know formulas

  • Standard pair: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
  • If a₁/a₂ ≠ b₁/b₂: one unique solution (the lines intersect, and the pair is consistent).
  • If a₁/a₂ = b₁/b₂ = c₁/c₂: infinitely many solutions (the lines coincide, and the pair is consistent).
  • If a₁/a₂ = b₁/b₂ ≠ c₁/c₂: no solution (the lines are parallel, and the pair is inconsistent).
  • Substitution: write one variable in terms of the other from one equation, then substitute into the second. Elimination: multiply to match the coefficients of one variable, then add or subtract.

Chapter 4: Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0 and is solved by factorisation or the quadratic formula. The discriminant tells you whether the roots are real and distinct, real and equal, or not real.

Must-know formulas

  • Standard form: ax² + bx + c = 0, where a ≠ 0
  • Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
  • Discriminant: D = b² − 4ac
  • If D > 0, there are two distinct real roots. If D = 0, there are two equal real roots, x = −b/2a. If D < 0, there are no real roots.
  • Sum of roots = −b/a and product of roots = c/a

Chapter 5: Arithmetic Progressions

An arithmetic progression is a list of numbers with a constant common difference. This chapter gives the formulas for the nth term and the sum of the first n terms.

Must-know formulas

  • AP: a, a + d, a + 2d, … where a is the first term and d = a₂ − a₁ is the common difference
  • nth term: aₙ = a + (n − 1)d
  • Sum of the first n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term
  • aₙ = Sₙ − Sₙ₋₁
  • Sum of the first n natural numbers = n(n + 1)/2
  • Three terms in AP are conveniently taken as a − d, a, a + d.

Chapter 6: Triangles

Similar triangles have equal corresponding angles and proportional corresponding sides. This chapter covers the Basic Proportionality Theorem and the criteria for similarity.

Must-know results

  • BPT (prove): if a line parallel to BC meets AB at D and AC at E in triangle ABC, then AD/DB = AE/EC.
  • Converse of BPT (state): if AD/DB = AE/EC, then DE is parallel to BC.
  • AAA: if the corresponding angles of two triangles are equal, the triangles are similar.
  • SSS: if the corresponding sides are proportional, the triangles are similar.
  • SAS: if one angle is equal and the sides including it are proportional, the triangles are similar.
  • For similar triangles ABC and DEF: AB/DE = BC/EF = CA/FD.

Chapter 7: Coordinate Geometry

Coordinate geometry uses (x, y) points to answer geometry questions with algebra. This chapter covers the distance formula and the section formula.

Must-know formulas

  • Distance between A(x₁, y₁) and B(x₂, y₂): AB = √[(x₂ − x₁)² + (y₂ − y₁)²]. The distance of (x, y) from the origin is √(x² + y²).
  • Section formula (internal division in the ratio m : n): P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
  • Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
  • Three points A, B and C are collinear if the largest of AB, BC and CA equals the sum of the other two.
  • Shape tests: a parallelogram has equal opposite sides (or diagonals with the same midpoint). A rhombus has four equal sides. A rectangle has equal opposite sides and equal diagonals. A square has four equal sides and equal diagonals.

Chapter 8: Introduction to Trigonometry

Trigonometry relates the angles of a right-angled triangle to the ratios of its sides. This chapter covers the six ratios, their values at standard angles, and the identity sin²θ + cos²θ = 1.

Must-know formulas

  • In a right triangle, for an acute angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
  • cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. Also tan θ = sin θ/cos θ and cot θ = cos θ/sin θ.
  • Identity: sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ. Dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Memory aid. For sin at 0°, 30°, 45°, 60°, 90°, write √0/2, √1/2, √2/2, √3/2, √4/2. The cos values are the same list in reverse.

Chapter 9: Some Applications of Trigonometry

This chapter applies the trigonometric ratios to find heights and distances using angles of elevation and depression, with angles of 30°, 45° and 60°.

Must-know ideas and formulas

  • Line of sight: the line from the observer’s eye to the object being viewed.
  • Angle of elevation: the angle between the horizontal and the line of sight when you look up at an object.
  • Angle of depression: the angle between the horizontal and the line of sight when you look down at an object.
  • The angle of elevation of an object from a point equals the angle of depression of that point from the object, because they are alternate angles.
  • Tools: tan θ = height/horizontal distance, sin θ = height/line of sight, cos θ = horizontal distance/line of sight.
  • Values to remember: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Chapter 10: Circles

A tangent touches a circle at exactly one point. This chapter covers the two tangent theorems and the number of tangents that can be drawn from a point.

Must-know results

  • Theorem 1 (prove): the tangent at any point of a circle is perpendicular to the radius through the point of contact.
  • Theorem 2 (prove): the lengths of the two tangents drawn from an external point to a circle are equal.
  • Number of tangents from a point: none from a point inside the circle, exactly one from a point on the circle, and exactly two from a point outside.
  • If P is at distance d from the centre of a circle of radius r, the length of the tangent from P is √(d² − r²), because the radius and tangent form a right angle at the point of contact.
  • A circle has at most two parallel tangents, one at each end of a diameter.

This chapter extends the area and circumference of a circle to sectors and segments, using angles in degrees.

Must-know formulas

  • Circumference = 2πr and area = πr²
  • For a sector of angle θ (in degrees): arc length l = (θ/360) × 2πr, and area of the sector = (θ/360) × πr², which also equals ½ × l × r.
  • Area of a minor segment = area of the sector − area of the triangle formed by the two radii and the chord.
  • Area of the major segment = πr² − area of the minor segment.
  • Triangle for a segment: for θ = 60° it is equilateral with area (√3/4)r². For θ = 90° the area is ½r². For θ = 120° the area is (√3/4)r².
  • A wheel covers a distance equal to its circumference in one revolution. Number of revolutions = total distance / circumference.

Chapter 12: Surface Areas and Volumes

This chapter covers the surface area and volume of cubes, cuboids, cylinders, cones, spheres and hemispheres, and of combinations of two solids.

Chapter 13: Statistics

Statistics summarises grouped data with the mean, median and mode. This chapter covers the direct, assumed mean and step deviation methods, and the formulas for the median and mode.

Must-know formulas

  • Class mark: xᵢ = (upper class limit + lower class limit)/2
  • Direct method: mean = Σfᵢxᵢ / Σfᵢ
  • Assumed mean method: mean = a + Σfᵢdᵢ / Σfᵢ, where dᵢ = xᵢ − a
  • Step deviation method: mean = a + h × (Σfᵢuᵢ / Σfᵢ), where uᵢ = (xᵢ − a)/h
  • Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
  • Median = l + [(n/2 − cf) / f] × h
  • Relation: Mode = 3 Median − 2 Mean
  • Symbols: l is the lower limit of the modal or median class, h is the class size, f₁ is the frequency of the modal class, f₀ and f₂ are the frequencies of the classes just before and after it, n = Σfᵢ, cf is the cumulative frequency of the class before the median class, and f is the frequency of the median class.

Chapter 14: Probability

Probability measures how likely an event is, on a scale from 0 to 1. This chapter uses the classical definition, in which all outcomes are equally likely.

Must-know formulas and facts

  • P(E) = number of outcomes favourable to E / total number of equally likely outcomes
  • 0 ≤ P(E) ≤ 1. A sure event has probability 1 and an impossible event has probability 0.
  • P(E) + P(not E) = 1
  • The probabilities of all elementary events of an experiment add up to 1.
  • Standard sample spaces: a coin has 2 outcomes, a die has 6, and two dice together have 36. A pack of 52 cards has 26 red and 26 black cards, 4 suits of 13, 12 face cards and 4 aces.
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