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Class 10 Maths formula sheet 2026-27: every chapter, with a worked example each
This is every formula, identity and rule a Class 10 student uses in the 2026-27 NCERT Maths book, chapter by chapter in the order of the book. Each line says when to use it, each chapter has one short worked example of our own, and each chapter links to our 8 practice questions on it. Where the CBSE curriculum for 2026-27 limits what the board paper can ask, the chapter says so, with the page.
To print it, use your browser's Print option; the page prints without colours.
The layout, wording and examples on this page are our own, drafted with AI help. Every example was re-computed by a script and every formula was checked against the NCERT chapter, but no practising teacher has reviewed the page yet. If you find a mistake, please tell us.
Page numbers are the printed page numbers of NCERT Mathematics, Textbook for Class X (Reprint 2026-27). "CBSE limit" lines quote the Mathematics curriculum 2026-27 (PDF pages 4-7).
Chapter 1: Real Numbers
| Formula or rule | When to use it |
|---|---|
| Every composite number is a product of primes in exactly one way, if the order of the factors is ignored (Fundamental Theorem of Arithmetic, page 3) | Writing a number as powers of primes, the first step for HCF and LCM |
| HCF = product of the shared primes, each with its smaller power; LCM = product of every prime that appears, each with its larger power (page 4) | Finding HCF and LCM from prime factorisations |
| HCF(a, b) × LCM(a, b) = a × b, for two positive integers (page 4) | Finding the LCM when the HCF is known, or checking your answer. It does not work for three numbers |
| If a prime p divides a², then p divides a (Theorem 1.2, page 6) | The key step in proving √2, √3 or √5 irrational |
Example (our own). 12 = 2² × 3 and 18 = 2 × 3². HCF = 2 × 3 = 6 and LCM = 2² × 3² = 36. Check: 6 × 36 = 216 = 12 × 18.
CBSE limit: none on the formulas. The irrationality proofs to know are for numbers like √2, √3, √5 and 3 + 2√5 (curriculum, page 3).
Practise: Class 10 Real Numbers questions with answers.
Chapter 2: Polynomials
| Formula or rule | When to use it |
|---|---|
| The zeroes of p(x) are the x-coordinates where the graph of y = p(x) meets the x-axis (pages 12-13) | Reading the number of zeroes from a graph |
| A quadratic has at most 2 zeroes; a cubic has at most 3 (summary, page 23) | Checking how many zeroes to look for |
| For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a (page 20) | Checking zeroes, or building a quadratic from a given sum and product: x² − (sum)x + (product) |
| For ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a (page 21) | Cubics only |
Example (our own). For x² − 5x + 6 the zeroes are 2 and 3. Sum 2 + 3 = 5 = −(−5)/1, and product 2 × 3 = 6 = 6/1.
CBSE limit: the curriculum lists the zeroes-coefficients relation for quadratic polynomials only (curriculum, page 3); the cubic relation is in the book but not in that list.
Practise: Class 10 Polynomials questions with answers.
Chapter 3: Pair of Linear Equations in Two Variables
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 (Table 3.1, page 26; summary, page 37):
| Compare the ratios | Lines on a graph | Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | cross at one point | exactly one (consistent) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | parallel | none (inconsistent) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | the same line | infinitely many (dependent, consistent) |
Solve by substitution (make one variable the subject and put it in the other equation) or elimination (make one variable's coefficients equal, then add or subtract) (section 3.3, pages 30-37).
Example (our own). 2x + 3y − 7 = 0 and 4x + 6y − 14 = 0: a₁/a₂ = 2/4, b₁/b₂ = 3/6, c₁/c₂ = −7/−14, all equal to 1/2. Same line, so infinitely many solutions.
CBSE limit: graphical method, the conditions above, substitution and elimination, and simple situational problems (curriculum, page 4).
Practise: Class 10 Pair of Linear Equations questions with answers.
Chapter 4: Quadratic Equations
| Formula or rule | When to use it |
|---|---|
| Standard form: ax² + bx + c = 0, with a ≠ 0 (page 39) | Rearranging a word problem before solving |
| Factorisation: split the middle term, then set each factor to zero (section 4.3, pages 42-43) | When the roots are easy whole numbers or fractions |
| Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a, when b² − 4ac ≥ 0 (page 44) | Any quadratic with real roots |
| Discriminant D = b² − 4ac: D > 0 two different real roots; D = 0 two equal real roots, each −b/2a; D < 0 no real roots (page 45) | "Find the nature of the roots" and "find k so the roots are equal" |
Example (our own). x² − 2x − 8 = 0: D = (−2)² − 4 × 1 × (−8) = 36, so two different real roots, x = (2 ± 6) ÷ 2, that is 4 and −2.
CBSE limit: only real roots (curriculum, page 4).
Practise: Class 10 Quadratic Equations questions with answers. The whole chapter: NCERT Class 10 Maths Chapter 4, explained.
Chapter 5: Arithmetic Progressions
| Formula or rule | When to use it |
|---|---|
| A list is an AP if a(k+1) − a(k), the gap between neighbours, is the same all through; that gap is d (page 53) | Testing whether a list is an AP |
| nth term: aₙ = a + (n − 1)d (page 57) | Any one term, or "which term is ...?" |
| Sum of the first n terms: Sₙ = (n/2)[2a + (n − 1)d] (page 64) | When you know a, d and n |
| Sₙ = (n/2)(a + l), where l is the last term (page 64) | When you know the first and last terms |
Example (our own). For 3, 7, 11, ...: a = 3 and d = 4. a₁₀ = 3 + 9 × 4 = 39, and S₁₀ = (10/2)(3 + 39) = 210.
CBSE limit: the nth term and the sum of the first n terms, with daily-life problems (curriculum, page 4).
Practise: Class 10 Arithmetic Progressions questions with answers.
Chapter 6: Triangles
| Theorem or criterion | When to use it |
|---|---|
| Basic Proportionality Theorem: a line parallel to one side cuts the other two sides in the same ratio (Theorem 6.1, page 80) | Finding a missing length when a line inside a triangle is parallel to a side |
| Its converse: a line cutting two sides in the same ratio is parallel to the third side (Theorem 6.2, page 82) | Proving two lines parallel |
| AAA and AA: equal corresponding angles mean similar triangles (Theorem 6.3, page 87) | Two angle pairs are enough |
| SSS: all three side ratios equal means similar (Theorem 6.4, page 88) | Only side lengths are given |
| SAS: one equal angle, with the sides around it in the same ratio, means similar (Theorem 6.5, page 90) | One angle and the two sides that form it |
Example (our own). In triangle PQR, DE is parallel to QR, with D on PQ and E on PR. If PD = 4 cm, DQ = 6 cm and PE = 6 cm, then 4/6 = 6/ER, so ER = 9 cm.
CBSE limit: prove the Basic Proportionality Theorem; state without proof its converse and the similarity criteria (curriculum, page 5).
Practise: Class 10 Triangles questions with answers.
Chapter 7: Coordinate Geometry
| Formula | When to use it |
|---|---|
| Distance between (x₁, y₁) and (x₂, y₂) = √((x₂ − x₁)² + (y₂ − y₁)²) (page 102) | Lengths, and checking whether points form a given shape |
| Distance of (x, y) from the origin = √(x² + y²) (page 112) | One point and the origin |
| Point dividing the join of (x₁, y₁) and (x₂, y₂) internally in the ratio m₁ : m₂ = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) (pages 107 and 112) | Section formula: the point a given fraction of the way along |
| Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2) (page 107) | The ratio 1 : 1 |
Example (our own). The distance from (1, 2) to (4, 6) is √(3² + 4²) = 5. The point dividing the join of (2, 1) and (8, 7) in the ratio 1 : 2 is ((1 × 8 + 2 × 2)/3, (1 × 7 + 2 × 1)/3) = (4, 3), and their mid-point is (5, 4).
CBSE limit: section formula for internal division only (curriculum, page 5).
Practise: Class 10 Coordinate Geometry questions with answers.
Chapter 8: Introduction to Trigonometry
In a right triangle with the right angle at B, for the acute angle A (page 115):
| Ratio | Definition | Also equal to |
|---|---|---|
| sin A | opposite ÷ hypotenuse | 1 ÷ cosec A |
| cos A | adjacent ÷ hypotenuse | 1 ÷ sec A |
| tan A | opposite ÷ adjacent | sin A ÷ cos A |
| cosec A | hypotenuse ÷ opposite | 1 ÷ sin A |
| sec A | hypotenuse ÷ adjacent | 1 ÷ cos A |
| cot A | adjacent ÷ opposite | cos A ÷ sin A |
Values to know (Table 8.1, page 125):
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | not defined |
| cosec | not defined | 2 | √2 | 2/√3 | 1 |
| sec | 1 | 2/√3 | √2 | 2 | not defined |
| cot | not defined | √3 | 1 | 1/√3 | 0 |
Identities (pages 128-129): sin²A + cos²A = 1; 1 + tan²A = sec²A (for 0° ≤ A < 90°); cot²A + 1 = cosec²A (for 0° < A ≤ 90°). sin A and cos A are never more than 1; sec A and cosec A are never less than 1 (page 132).
Example (our own). If sin A = 3/5, then cos A = √(1 − 9/25) = 4/5 and tan A = (3/5) ÷ (4/5) = 3/4. Check: 9/25 + 16/25 = 1.
CBSE limit: proof and use of sin²A + cos²A = 1, and only simple identities (curriculum, page 6).
Practise: Class 10 Trigonometry questions with answers. The whole chapter: NCERT Class 10 Maths Chapter 8, explained.
Chapter 9: Some Applications of Trigonometry
| Term or rule | When to use it |
|---|---|
| Angle of elevation: between the horizontal and your line of sight when you look up (page 133) | The object is above eye level |
| Angle of depression: between the horizontal and your line of sight when you look down (page 134) | You look down from a height |
| In the right triangle formed, tan θ = height ÷ horizontal distance (from Chapter 8) | Most heights-and-distances questions; use sin or cos when the slant length is given or wanted |
Example (our own). From a balcony 12 m above the ground, a cat on the ground is seen at an angle of depression of 30°. The cat's horizontal distance from the building is 12 ÷ tan 30° = 12√3 ≈ 20.78 m.
CBSE limit: no more than two right triangles, and angles of elevation or depression of 30°, 45° and 60° only (curriculum, page 6).
Practise: Class 10 Applications of Trigonometry questions with answers.
Chapter 10: Circles
| Theorem | When to use it |
|---|---|
| The radius drawn to the point where a tangent touches the circle is at right angles to that tangent (Theorem 10.1, page 146) | Any tangent question: it gives you a right angle, so use Pythagoras |
| From one point outside a circle, both tangents to the circle have the same length (Theorem 10.2, page 149) | Two tangents from one point, and polygons drawn around a circle |
Example (our own). A circle has radius 8 cm, and a point P is 17 cm from its centre. Each tangent from P is √(17² − 8²) = √225 = 15 cm long, and both tangents are 15 cm by Theorem 10.2.
CBSE limit: prove both theorems (curriculum, page 6).
Practise: Class 10 Circles questions with answers.
Chapter 11: Areas Related to Circles
For a circle of radius r and a sector with angle θ in degrees (pages 155-156; summary, page 160):
| Formula | When to use it |
|---|---|
| Area of the sector = (θ/360) × πr² | Pizza-slice shapes, a clock hand sweeping |
| Length of the arc = (θ/360) × 2πr | The curved edge only |
| Area of the segment = area of the sector − area of the triangle formed by the two radii | The region between a chord and its arc |
Example (our own). For r = 6 cm and θ = 60°, with π = 3.14: sector = (60/360) × 3.14 × 36 = 18.84 cm²; arc = (60/360) × 2 × 3.14 × 6 = 6.28 cm. The triangle is equilateral with side 6 cm, area (√3/4) × 36 ≈ 15.59 cm², so the segment ≈ 18.84 − 15.59 = 3.25 cm².
CBSE limit: segment questions use central angles of 60°, 90° and 120° only (curriculum, page 7).
Practise: Class 10 Areas Related to Circles questions with answers.
Chapter 12: Surface Areas and Volumes
The chapter joins solids whose surface areas and volumes you learnt in Class IX (page 161), and its worked examples (pages 162-169) build on these formulas:
| Solid | Curved or lateral surface | Total surface | Volume |
|---|---|---|---|
| Cube, edge a | 4a² | 6a² | a³ |
| Cuboid, l × b × h | 2h(l + b) | 2(lb + bh + hl) | lbh |
| Cylinder, radius r, height h | 2πrh | 2πr(r + h) | πr²h |
| Cone, radius r, height h, slant l = √(r² + h²) | πrl | πr(l + r) | (1/3)πr²h |
| Sphere, radius r | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere, radius r | 2πr² | 3πr² | (2/3)πr³ |
The rule for a joined solid: add the volumes, but for the surface add only the parts you can still see; the faces that touch are hidden (pages 162 and 167).
Example (our own). A cylinder of radius 3 cm and height 10 cm has a hemisphere of radius 3 cm on top. Volume = π × 9 × 10 + (2/3) × π × 27 = 108π ≈ 339.12 cm³ (π = 3.14). Outside surface, with the base closed = 2π × 3 × 10 + 2π × 9 + π × 9 = 87π ≈ 273.18 cm².
CBSE limit: solids made by joining any two of cubes, cuboids, spheres, hemispheres, cylinders and cones (curriculum, page 7).
Practise: Class 10 Surface Areas and Volumes questions with answers.
Chapter 13: Statistics
For grouped data with class marks xᵢ (the mid-point of each class) and frequencies fᵢ (pages 173-179; summary, pages 200-201):
| Formula | When to use it |
|---|---|
| Direct method: mean = Σfᵢxᵢ ÷ Σfᵢ | Small, easy class marks |
| Assumed mean: mean = a + Σfᵢdᵢ ÷ Σfᵢ, where dᵢ = xᵢ − a | Large class marks; a is one of the class marks |
| Step deviation: mean = a + h × (Σfᵢuᵢ ÷ Σfᵢ), where uᵢ = (xᵢ − a)/h | Equal class widths h |
| Mode = l + h × (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) (page 184) | l = lower limit of the modal class (the class with the highest frequency), f₁ its frequency, f₀ and f₂ the classes just before and after |
| Median = l + h × (n/2 − cf) ÷ f (page 195) | l = lower limit of the median class (the first whose cumulative frequency reaches n/2), cf = cumulative frequency of the class before it, f its own frequency |
| 3 Median = Mode + 2 Mean, approximately (page 197) | A quick check, or finding one measure from the other two |
Example (our own). Classes 0-10, 10-20, 20-30, 30-40 with frequencies 2, 7, 5, 1 (n = 15). Mean = (2 × 5 + 7 × 15 + 5 × 25 + 1 × 35) ÷ 15 = 275 ÷ 15 ≈ 18.33. Mode: modal class 10-20, so 10 + 10 × (7 − 2) ÷ (14 − 2 − 5) ≈ 17.14. Median: n/2 = 7.5 falls in 10-20 (cumulative frequencies 2, 9, ...), so 10 + 10 × (7.5 − 2) ÷ 7 ≈ 17.86. Check: 3 × 17.86 ≈ 53.6 and 17.14 + 2 × 18.33 ≈ 53.8, close, as expected.
CBSE limit: mean, median and mode of grouped data, no bimodal data (curriculum, page 7).
Practise: Class 10 Statistics questions with answers.
Chapter 14: Probability
| Formula or rule | When to use it |
|---|---|
| P(E) = number of outcomes that favour E ÷ number of all possible outcomes, when every outcome is equally likely (page 203) | Dice, coins, cards, balls in a bag |
| 0 ≤ P(E) ≤ 1; a sure event has probability 1 and an impossible event 0 (pages 207 and 217) | Checking an answer: a probability above 1 is wrong |
| P(E) + P(not E) = 1 (page 206) | "Not", "at least one" or "does not" questions |
Example (our own). A bag has 3 red and 5 blue balls. P(red) = 3/8, so P(not red) = 1 − 3/8 = 5/8.
CBSE limit: the classical definition and simple problems (curriculum, page 7).
Practise: Class 10 Probability questions with answers.
Practise more
- Free practice, marked instantly: the practice page (pick Class 10) and printable Class 10 worksheets, in English, Hindi and Marathi.
- How the formulas are tested: our CBSE Class 10 Maths sample paper 2026-27 guide and the Class 10 Maths syllabus 2026-27 guide.
- A plan for the months before the exam: Class 10 boards, a 60-day revision plan.
Where CMDLearn fits
CMDLearn is building an AI tutor for Classes 1 to 10 that explains a step and checks the child's working, but leaves the writing to the child. It is still in development, and no child uses it yet. If you would like to hear when there is something to try, you can join the waitlist.
Sources
All opened and read on 3 November 2026. NCERT page numbers are printed page numbers; CBSE page numbers are PDF pages.
- NCERT, Mathematics, Textbook for Class X (Reprint 2026-27), Chapters 1-14, printed pages 1-217: ncert.nic.in/textbook/pdf/jemh101.pdf to ncert.nic.in/textbook/pdf/jemh114.pdf, and the contents: ncert.nic.in/textbook/pdf/jemh1ps.pdf
- CBSE, Mathematics curriculum, Class X, 2026-27 (course structure and the topic limits, pages 3-7; prescribed book, page 10): cbseacademic.nic.in/web_material/CurriculumMain27/SecPart1/Maths_SecP1X_2026-27.pdf
The formulas and theorems are the textbook's; the layout, the "when to use it" notes and the examples are our own, and every example was re-computed by script. This page is general information; follow your teacher, the textbook and CBSE's own notices.
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