Class 10 Maths Formula Sheet (CBSE 2026-27)
How to memorise a formula sheet: do not read it top to bottom. Pick one chapter, cover the right-hand column with a sheet of paper, and write each formula from the name alone. Mark the ones you miss and revise only those the next day. Three short rounds beat one long reread.
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\n Real Numbers
\n Number Systems · 6 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Euclid’s division lemma | a = bq + r, 0 ≤ r < b | Used for HCF by repeated division. |
| HCF–LCM relation | HCF(a, b) × LCM(a, b) = a × b | Only for two positive integers, not three. |
| Terminating decimal test | q = 2m5n | After reducing p/q to lowest terms. |
| Prime factorisation | HCF = product of smallest powers; LCM = product of highest powers | Write both numbers as products of primes first. |
Watch out: Using HCF × LCM = product of numbers for three numbers. It is a two-number identity.
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\n Polynomials
\n Algebra · 20 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Quadratic zeroes | α + β = −b/a, αβ = c/a | Signs matter — the sum is negative b over a. |
| Build from zeroes | x² − (α + β)x + αβ | Multiply by any constant k for a family of answers. |
| Cubic zeroes | α + β + γ = −b/a; αβ + βγ + γα = c/a; αβγ = −d/a | Signs alternate: −, +, −. |
| Division algorithm | p(x) = g(x)q(x) + r(x) | deg r < deg g. |
Watch out: Writing α + β = b/a and losing the minus sign.
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\n Pair of Linear Equations in Two Variables
\n Algebra · 20 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Unique solution | a₁/a₂ ≠ b₁/b₂ | Intersecting lines, consistent. |
| Infinitely many | a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident lines, dependent. |
| No solution | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel lines, inconsistent. |
| Cross-multiplication | x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁) | Check the determinant is not zero first. |
Watch out: Comparing only the coefficients of x and y and forgetting the constant term when testing for infinite solutions.
\nRead the full Pair of Linear Equations in Two Variables notes →
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\n Quadratic Equations
\n Algebra · 20 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a | Works for every quadratic, including irrational roots. |
| Discriminant | D = b² − 4ac | D > 0 distinct real; D = 0 equal real; D < 0 no real roots. |
| Zeroes | α + β = −b/a, αβ = c/a | Useful for checking your answers. |
| Completing the square | (x + b/2a)² = (b² − 4ac)/4a² | The derivation of the formula. |
Watch out: Dropping the ± and reporting only one root.
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\n Arithmetic Progressions
\n Algebra · 20 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| nth term | aₙ = a + (n − 1)d | Check whether n is the term number or the term value. |
| Sum of n terms | Sₙ = n/2 [2a + (n − 1)d] | Use when d is known. |
| Sum with last term | Sₙ = n/2 (a + l) | Faster when the last term is given. |
| Number of terms | n = (l − a)/d + 1 | Handy for ‘how many multiples…’ questions. |
| Term from sums | aₙ = Sₙ − Sₙ₋₁ | Valid for n ≥ 2. |
Watch out: Using n instead of (n − 1) in the nth-term formula — the classic one-mark loss.
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\n Triangles
\n Geometry · 15 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Basic Proportionality Theorem | DE ∥ BC ⇒ AD/DB = AE/EC | Also true with AD/AB = AE/AC. |
| Area ratio | area(△ABC)/area(△PQR) = (AB/PQ)² | Square the side ratio. |
| Pythagoras | AC² = AB² + BC² | AC is the hypotenuse, opposite the right angle. |
| Converse of Pythagoras | If AC² = AB² + BC², then ∠B = 90° | Used to prove a triangle is right-angled. |
Watch out: Treating SSA as a similarity criterion.
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\n Coordinate Geometry
\n Coordinate Geometry · 6 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Distance | d = √((x₂ − x₁)² + (y₂ − y₁)²) | Squared distances avoid the square root. |
| Section (internal) | ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) | Ratio m₁ : m₂ measured from the first point. |
| Midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) | Special case of 1 : 1. |
| Area of triangle | ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| | Zero area ⇒ collinear points. |
Watch out: Dropping a negative sign when subtracting coordinates.
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\n Introduction to Trigonometry
\n Trigonometry · 12 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Basic ratios | sin θ = P/H, cos θ = B/H, tan θ = P/B | ‘Pandit Badri Prasad Har Har’ is the usual mnemonic. |
| Reciprocals | cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ | sin pairs with cosec, not sec. |
| Quotient | tan θ = sin θ/cos θ, cot θ = cos θ/sin θ | Useful for converting everything to sin and cos. |
| Pythagorean identities | sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ | Divide the first by cos²θ or sin²θ to get the others. |
| Standard values | sin: 0, ½, 1/√2, √3/2, 1 for 0°, 30°, 45°, 60°, 90° | cos runs the same list backwards. |
Watch out: Reading sin²θ as sin(θ²).
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\n Some Applications of Trigonometry
\n Trigonometry · 12 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Basic relation | tan θ = height / horizontal distance | The workhorse of the chapter. |
| Useful values | tan 30° = 1/√3, tan 45° = 1, tan 60° = √3 | Standard exam angles. |
| Two triangles | h = x tan α = (x + d) tan β | Eliminate x to solve for h. |
Watch out: Putting the angle of elevation inside the triangle at the wrong vertex.
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\n Circles
\n Geometry · 15 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Tangent–radius | OP ⊥ tangent at P | Join the centre to the point of contact first. |
| Equal tangents | PA = PB from external point P | Creates two congruent right triangles. |
| Circumscribed quadrilateral | AB + CD = BC + AD | Useful in one-mark and case-study questions. |
| Pythagoras in tangent problems | OP² = OA² + AP² | O centre, A external point, P point of contact. |
Watch out: Assuming the tangent passes through the centre.
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\n Surface Areas and Volumes
\n Mensuration · 10 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Cylinder | CSA = 2πrh; TSA = 2πr(r + h); V = πr²h | r is the radius, h the height. |
| Cone | l = √(r² + h²); CSA = πrl; TSA = πr(l + r); V = ⅓πr²h | l is the slant height, not the vertical height. |
| Sphere | SA = 4πr²; V = 4/3 πr³ | Solid sphere. |
| Hemisphere | CSA = 2πr²; TSA = 3πr²; V = 2/3 πr³ | TSA adds the flat circular face. |
| Frustum | CSA = πl(r₁ + r₂); TSA = πr₁² + πr₂² + πl(r₁ + r₂); V = ⅓πh(r₁² + r₂² + r₁r₂) | l = √(h² + (r₁ − r₂)²). |
| Cube / cuboid | Cube: 6a², a³; cuboid: 2(lb + bh + hl), lbh | Check whether the top is open. |
Watch out: Using the diameter as r in every formula.
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\n Statistics
\n Statistics and Probability · 11 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Mean (direct) | x̄ = Σfᵢxᵢ / Σfᵢ | xᵢ is the class mark. |
| Mean (step deviation) | x̄ = a + h × (Σfᵢuᵢ / Σfᵢ), uᵢ = (xᵢ − a)/h | a is the assumed mean, h the class size. |
| Median | l + ((n/2 − cf)/f) × h | l lower limit, cf cumulative frequency before the class, f frequency of the class. |
| Mode | l + ((f₁ − f₀)/(2f₁ − f₀ − f₂)) × h | f₁ is the modal frequency. |
| Empirical relation | 3 Median = Mode + 2 Mean | Only for moderately skewed data. |
Watch out: Using class limits instead of class marks in the mean.
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\n Probability
\n Statistics and Probability · 11 marks\n\n \n
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\n | Formula / rule | Statement | Notes |
|---|---|---|
| Theoretical probability | P(E) = n(E)/n(S) | S is the sample space. |
| Range | 0 ≤ P(E) ≤ 1 | A probability outside this range is wrong. |
| Complement | P(E) + P(not E) = 1 | Use for ‘at least one’ problems. |
| Two dice | 36 equally likely outcomes | Sums from 2 to 12. |
| Deck of cards | 52 cards = 4 suits × 13; 12 face cards | Ace is not a face card. |
Watch out: Counting the same outcome twice when two dice or two coins are used.
\n \nFormula-sheet habits that save marks
- Write the formula before substituting numbers. Examiners give method marks for the correct formula even when the arithmetic slips.
- Keep units attached to every substituted value, and convert to a single unit before calculating.
- Check whether the question expects π = 22/7 or 3.14; the answer changes and so does the marking.
- For trigonometry, write the angle in every line. Dropping “θ” is the most common silent error.
- At the end, ask whether the answer is physically reasonable — a negative length or a probability above 1 is always wrong.
Last updated: 22 October 2026 · Author: Bazid Ahmed Laskar · Independent study material.