Class 10 Maths Formulas Handbook (Chapter-wise)

Every key formula and result from the Class 10 Maths NCERT syllabus, organised by chapter, in one place — covering all 12 chapters of the current 2026-27 CBSE syllabus, each cross-linked to our full Chapter-wise Solutions.

Class 10 Maths Formulas Handbook

Chapter 1: Real Numbers

• Euclid’s Division Lemma: a = bq + r, where 0 ≤ r < b
• HCF × LCM = Product of the two numbers (for exactly two numbers)
• Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation (order aside)
• Proof-of-irrationality method: assume rational (p/q, coprime, q≠0) → derive a common factor in p and q → contradiction
• See full Chapter 1 Solutions

Chapter 2: Polynomials

• Quadratic polynomial: p(x) = ax² + bx + c (a≠0); zeroes are the values of x where p(x) = 0
• Sum of zeroes α + β = −b/a (−(coefficient of x)/(coefficient of x²))
• Product of zeroes αβ = c/a (constant term/coefficient of x²)
• Forming a quadratic from sum/product: x² − (sum)x + (product) = 0
• See full Chapter 2 Solutions

Chapter 3: Pair of Linear Equations in Two Variables

• General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
• Intersecting (unique solution): a₁/a₂ ≠ b₁/b₂
• Parallel (no solution, inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Coincident (infinite solutions): a₁/a₂ = b₁/b₂ = c₁/c₂
• Methods: substitution, elimination; reciprocal substitution (u = 1/(x−y), v = 1/(x+y)) for speed/time word problems

Chapter 4: Quadratic Equations

• Standard form: ax²+bx+c = 0 (a≠0)
• Quadratic formula: x = [−b±√(b²−4ac)]/2a
• Discriminant D = b²−4ac: D>0 distinct real roots, D=0 equal real roots, D<0 no real roots
• Sum of roots = −b/a, Product of roots = c/a
• Factorisation: split the middle term bx into two terms whose product = a×c×x² and sum = bx

Chapter 5: Arithmetic Progressions

• nth term: aₙ = a + (n−1)d
• Sum (a,d known): Sₙ = (n/2)[2a+(n−1)d] | Sum (a,l known): Sₙ = (n/2)(a+l)
• See full Chapter 5 Solutions

Chapter 6: Triangles

• BPT: if DE∥BC in ΔABC (D on AB, E on AC), then AD/DB = AE/EC
• Similarity criteria: AA(AAA), SSS, SAS
• Ratio of areas of similar triangles = (ratio of corresponding sides)²
• Pythagoras theorem (via similarity): in right ΔABC right-angled at B, AB²+BC²=AC²
• See full Chapter 6 Solutions

Chapter 7: Coordinate Geometry

• Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²)
• Distance from origin: √(x²+y²)
• Section formula (ratio m:n): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
• Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2)
• Area of a triangle: ½|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)| — also the collinearity test (area=0)
• See full Chapter 7 Solutions

Chapter 8: Introduction to Trigonometry

• sinθ=opp/hyp, cosθ=adj/hyp, tanθ=sinθ/cosθ; cosec,sec,cot are reciprocals.
• sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.
• Complementary: sin(90°−θ)=cosθ, tan(90°−θ)=cotθ, sec(90°−θ)=cosecθ.
• Standard values at 0°,30°,45°,60°,90° — see full revision notes.

Chapter 9: Some Applications of Trigonometry

• height = distance × tan(theta); distance = height / tan(theta)
• line-of-sight length = height / sin(theta) = distance / cos(theta)
• Complementary angles from distances p, q: height = sqrt(pq)
• Moving-observer formula: height = a.tan(theta).tan(phi) / (tan(phi) – tan(theta))

Chapter 10: Circles

• Tangent is perpendicular to the radius at the point of contact
• Lengths of tangents from an external point to a circle are equal
• Tangent length = sqrt(d^2 – r^2) for a point at distance d from centre, circle radius r
• Parallelogram circumscribing a circle is a rhombus

Chapter 11: Areas Related to Circles

• Circumference = 2πr; Area of circle = πr²
• Arc length = (θ/360) × 2πr
• Area of sector = (θ/360) × πr² = (1/2) × l × r
• Area of segment = Area of sector − (1/2)r²sinθ
• Area of ring = π(R² − r²)
• Full Chapter 11 Solutions →

Chapter 12: Surface Areas and Volumes

• Cube: TSA = 6a², Volume = a³
• Cuboid: TSA = 2(lb+bh+hl), Volume = lbh
• Cylinder: CSA = 2πrh, TSA = 2πr(r+h), Volume = πr²h
• Cone: CSA = πrl, TSA = πr(l+r), Volume = (1/3)πr²h; slant height l = √(r²+h²)
• Sphere: SA = 4πr², Volume = (4/3)πr³
• Hemisphere: CSA = 2πr², TSA = 3πr², Volume = (2/3)πr³
• Combination solids: add curved/visible surfaces only for area; add (or subtract, for cavities) volumes directly.

Chapter 13: Statistics

• Mean (Direct): x̄ = Σfᵢxᵢ/Σfᵢ
• Mean (Assumed Mean): x̄ = a + Σfᵢdᵢ/Σfᵢ
• Mean (Step-Deviation): x̄ = a + (Σfᵢuᵢ/Σfᵢ)×h
• Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h
• Median = l + [(n/2-cf)/f] × h
• Empirical relationship: Mode = 3 Median − 2 Mean
• Full Chapter 13 Solutions →

Chapter 14: Probability

• P(E) = favourable outcomes / total outcomes
• 0 ≤ P(E) ≤ 1; P(E) + P(not E) = 1
• Two dice together: 36 outcomes; deck of cards: 52 (12 face cards)
• Full Chapter 14 Solutions →

How to Use This Handbook

Formulas alone won’t get you full marks — CBSE examiners award marks for showing correct working, not just stating a result. Use this page as a quick lookup while solving problems in our chapter-wise Solutions and Extra Questions, not as a replacement for practicing the full method.

Frequently Asked Questions

Is this handbook complete for the full Class 10 Maths syllabus?
Yes — this handbook now covers all 12 chapters of the current 2026-27 Class 10 Maths syllabus, from Real Numbers through Surface Areas and Volumes.

Can I download this as a PDF?
Yes — use the “Download PDF” button on this page to get a free, printable PDF of the full handbook. The page itself stays live and editable so formulas stay accurate as the syllabus changes.

Written by Satish

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