Class 10 Maths Formulas (PDF) — All Important Formulas Free
Free class 10 maths formulas — all 44 key formulas on one page, downloadable as PDF for fast revision before CBSE board exams. No signup.
TL;DR: Free class 10 maths formulas — all 44 key formulas on one page, downloadable as PDF for fast revision before CBSE board exams. No signup.
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Essential formulas for CBSE Class 10 Maths covering Real Numbers, Polynomials, Linear/Quadratic Equations, AP, Triangles, Coordinate Geometry, Trigonometry, Circles, and Statistics. Master these formulas for board exams and competitive entrance tests.
Real Numbers & Polynomials
| Formula | Expression |
|---|---|
| Euclid's Division Algorithm | a = bq + r, where 0 ≤ r < b — For finding HCF of two numbers |
| Fundamental Theorem of Arithmetic | Every positive integer > 1 is a product of primes, unique up to order — Used for HCF and LCM |
| Quadratic Formula | x = (-b ± √(b² - 4ac)) / 2a — For ax² + bx + c = 0, discriminant D = b² - 4ac |
| Sum of roots | α + β = -b/a — For quadratic ax² + bx + c = 0 |
| Product of roots | αβ = c/a — For quadratic ax² + bx + c = 0 |
| Polynomial division identity | p(x) = g(x) × q(x) + r(x) — where deg(r) < deg(g) |
Linear Equations
| Formula | Expression |
|---|---|
| Linear equation in two variables | ax + by + c = 0 — General form; a ≠ 0 or b ≠ 0 |
| Slope-intercept form | y = mx + c — m = slope, c = y-intercept |
| Consistency condition | For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ (unique solution) — If ratios equal, lines are parallel or identical |
| Determinant method (Cramer's rule) | x = D₁/D, y = D₂/D where D = a₁b₂ - a₂b₁ — For system of two linear equations |
Arithmetic Progression
| Formula | Expression |
|---|---|
| nth term | aₙ = a + (n - 1)d — a = first term, d = common difference |
| Sum of n terms | Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l) — l = last term = a + (n - 1)d |
| Common difference | d = aₙ₊₁ - aₙ — Difference between consecutive terms |
| Arithmetic mean | A = (a + b)/2 — Mean of two numbers a and b |
Trigonometry
| Formula | Expression |
|---|---|
| Basic trigonometric ratios | sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent — cot θ = 1/tan θ, sec θ = 1/cos θ, cosec θ = 1/sin θ |
| Pythagorean identity | sin² θ + cos² θ = 1 — Fundamental trigonometric identity |
| Reciprocal identity | tan θ = sin θ/cos θ, cot θ = cos θ/sin θ — Relations between trigonometric ratios |
| Pythagorean variants | 1 + tan² θ = sec² θ, 1 + cot² θ = cosec² θ — Derived from sin² θ + cos² θ = 1 |
| Trigonometric values of standard angles | sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1 — Also sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3 |
| Complementary angle relations | sin(90° - θ) = cos θ, cos(90° - θ) = sin θ, tan(90° - θ) = cot θ — For complementary angles |
| Height and distance formula | tan θ = height/distance — Used in angle of elevation and depression problems |
Coordinate Geometry
| Formula | Expression |
|---|---|
| Distance formula | d = √[(x₂ - x₁)² + (y₂ - y₁)²] — Distance between points (x₁, y₁) and (x₂, y₂) |
| Section formula | P = ((m × x₂ + n × x₁)/(m + n), (m × y₂ + n × y₁)/(m + n)) — Point P divides line segment in ratio m:n internally |
| Midpoint formula | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) — Midpoint of line segment joining (x₁, y₁) and (x₂, y₂) |
| Area of triangle | Area = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| — Coordinates of vertices: (x₁, y₁), (x₂, y₂), (x₃, y₃) |
| Slope of line | m = (y₂ - y₁)/(x₂ - x₁) — Slope between points (x₁, y₁) and (x₂, y₂) |
Circles & Similarity
| Formula | Expression |
|---|---|
| Circumference of circle | C = 2πr — r = radius |
| Area of circle | A = πr² — r = radius |
| Arc length | l = (θ/360°) × 2πr = (θ/180°) × πr (in radians: l = rθ) — θ in degrees; r = radius |
| Sector area | A = (θ/360°) × πr² — θ in degrees |
| Segment area | A = r²/2 (θ - sin θ) where θ is in radians — Area between chord and arc |
| Similar triangles ratio | If △ABC ~ △DEF, then a/d = b/e = c/f = k (linear ratio), and Area(△ABC)/Area(△DEF) = k² — k = scale factor; areas in square of linear ratio |
Surface Areas & Volumes
| Formula | Expression |
|---|---|
| Cube | Surface Area = 6a², Volume = a³ — a = side length |
| Cuboid | Surface Area = 2(lb + bh + hl), Volume = l × b × h — l = length, b = breadth, h = height |
| Cylinder | Curved Surface Area = 2πrh, Total Surface Area = 2πr(r + h), Volume = πr²h — r = radius, h = height |
| Cone | Curved Surface Area = πrl, Total Surface Area = πr(r + l), Volume = 1/3 πr²h — r = radius, h = height, l = slant height = √(r² + h²) |
| Sphere | Surface Area = 4πr², Volume = 4/3 πr³ — r = radius |
| Hemisphere | Curved Surface Area = 2πr², Total Surface Area = 3πr², Volume = 2/3 πr³ — r = radius |
Statistics & Probability
| Formula | Expression |
|---|---|
| Mean (Arithmetic average) | Mean = (Σx)/n or Mean = (ΣfᵢXᵢ)/(Σfᵢ) — Σx = sum of all observations, n = number of observations, fᵢ = frequency |
| Mode | Mode = most frequently occurring value — For grouped data: Mode = l + (f₁ - f₀)/((2f₁ - f₀ - f₂)) × h |
| Median | For ungrouped: arrange in order, median is middle value. For grouped: Median = l + ((n/2 - cf)/f) × h — l = lower boundary of median class, cf = cumulative frequency, f = frequency of median class, h = class width |
| Variance | σ² = (Σ(xᵢ - mean)²)/n — Measure of dispersion |
| Standard deviation | σ = √[(Σ(xᵢ - mean)²)/n] — Square root of variance |
| Probability | P(E) = (Number of favourable outcomes)/(Total number of outcomes) — 0 ≤ P(E) ≤ 1; P(E) + P(not E) = 1 |
FAQs
What is the discriminant and why is it important?
The discriminant D = b² - 4ac determines the nature of roots in a quadratic equation ax² + bx + c = 0. If D > 0: two distinct real roots; D = 0: one repeated real root; D < 0: no real roots (two complex conjugate roots).
How do I solve a pair of linear equations?
Methods: (1) Substitution—express one variable in terms of other and substitute; (2) Elimination—multiply equations to eliminate one variable; (3) Graphical—plot lines and find intersection; (4) Determinant (Cramer's rule)—use determinants. Choose based on the form of equations.
What's the difference between sin, cos, and tan?
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. In a right-angled triangle, these ratios relate sides to the angle θ. Remember: SOH-CAH-TOA (Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent).
How do I find the area of a triangle given three vertices?
Use the coordinate geometry formula: Area = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|, where vertices are (x₁, y₁), (x₂, y₂), (x₃, y₃). The absolute value ensures the area is positive.
What's the formula for the sum of an arithmetic progression?
Sₙ = n/2 [2a + (n - 1)d] where n = number of terms, a = first term, d = common difference. Alternatively, Sₙ = n/2 (a + l) where l is the last term.