Memorising formulas is only half the battle — you need to know when and how to use them. This cheatsheet covers every formula from the CBSE Class 10 Maths syllabus that has appeared in board exams over the last 10 years.
Don't just read this list. Write each formula 3 times, then solve at least 2 NCERT questions that use it. Understanding > memorisation.
Chapter 1 — Real Numbers
| Formula | Use |
|---|---|
| HCF × LCM = Product of two numbers | Finding HCF or LCM when one is given |
| Euclid's Division: a = bq + r (0 ≤ r < b) | Finding HCF step-by-step |
Key theorem: If p is prime and p divides a², then p divides a.
Types of decimals:
- Terminating: denominator has only 2 or 5 as prime factors
- Non-terminating recurring: denominator has any other prime factor
Chapter 2 — Polynomials
For a quadratic p(x) = ax² + bx + c with roots α and β:
| Relation | Formula |
|---|---|
| Sum of roots | α + β = −b/a |
| Product of roots | αβ = c/a |
For a cubic p(x) = ax³ + bx² + cx + d with roots α, β, γ:
| Relation | Formula |
|---|---|
| α + β + γ | = −b/a |
| αβ + βγ + γα | = c/a |
| αβγ | = −d/a |
Chapter 3 — Pair of Linear Equations
Condition for pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
| Condition | Result |
|---|---|
| a₁/a₂ ≠ b₁/b₂ | Unique solution (intersecting lines) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinite solutions (coincident lines) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | No solution (parallel lines) |
Chapter 4 — Quadratic Equations
Quadratic Formula:
x = [−b ± √(b² − 4ac)] / 2a
Discriminant (D = b² − 4ac):
| D value | Nature of roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots |
| D < 0 | No real roots |
Always calculate D first before deciding your method (factorisation vs formula). If D is a perfect square, factorisation is faster.
Chapter 5 — Arithmetic Progressions
For an AP with first term a, common difference d, and n terms:
| Formula | What it gives |
|---|---|
| aₙ = a + (n−1)d | nth term |
| Sₙ = n/2 [2a + (n−1)d] | Sum of n terms |
| Sₙ = n/2 [a + l] | Sum when last term l is known |
Middle term of AP with odd n: a_
Chapter 6 — Triangles (Similar Triangles)
Basic Proportionality Theorem (Thales): If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Areas of similar triangles: (Area₁/Area₂) = (Side₁/Side₂)²
Pythagoras theorem: AC² = AB² + BC² Converse: If AC² = AB² + BC², angle B = 90°
Chapter 7 — Coordinate Geometry
| Formula | Use |
|---|---|
| Distance = √[(x₂−x₁)² + (y₂−y₁)²] | Distance between 2 points |
| Section formula: x = (m₁x₂ + m₂x₁)/(m₁+m₂) | Point dividing AB in ratio m₁:m₂ |
| Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2) | Midpoint of a line segment |
| Area of triangle = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| | Area using coordinates |
If area of triangle = 0, the three points are collinear (on the same line).
Chapter 8 — Introduction to Trigonometry
Basic ratios (for angle θ in a right triangle):
| Ratio | Formula |
|---|---|
| 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 |
Standard values table:
| Angle | 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 | — |
Identities:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Chapter 9 — Applications of Trigonometry
Height & Distance formulas:
| Scenario | Formula |
|---|---|
| Height of tower (angle of elevation α, distance d) | h = d × tan α |
| Angle of elevation to top of height h from distance d | tan α = h/d |
Always draw a diagram before solving. Label the right angle, angle of elevation/depression, and known sides.
Chapter 10 — Circles
Key theorems:
- Tangent to a circle is perpendicular to the radius at point of contact
- From an external point, two tangents are equal in length
- Tangent-chord angle = Angle in alternate segment
Chapter 11 — Areas Related to Circles
| Shape | Formula |
|---|---|
| Area of circle | πr² |
| Circumference | 2πr |
| Area of sector (angle θ) | θ/360 × πr² |
| Length of arc | θ/360 × 2πr |
| Area of segment | Area of sector − Area of triangle |
Use π = 22/7 unless question says otherwise.
Chapter 12 — Surface Areas and Volumes
| Solid | Surface Area | Volume |
|---|---|---|
| Cube (side a) | 6a² | a³ |
| Cuboid (l,b,h) | 2(lb+bh+hl) | lbh |
| Cylinder (r,h) | 2πr(r+h) | πr²h |
| Cone (r,l,h) | πr(r+l) where l=√(r²+h²) | ⅓πr²h |
| Sphere (r) | 4πr² | 4/3 πr³ |
| Hemisphere (r) | 3πr² | 2/3 πr³ |
When one solid is melted to form another: Volume stays the same. Set V₁ = V₂ and solve.
Chapter 13 — Statistics
| Measure | Formula |
|---|---|
| Mean (direct method) | Σfᵢxᵢ / Σfᵢ |
| Mean (assumed mean) | a + (Σfᵢdᵢ / Σfᵢ) where dᵢ = xᵢ − a |
| Mean (step deviation) | a + h(Σfᵢuᵢ / Σfᵢ) where uᵢ = (xᵢ−a)/h |
| Mode | l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h |
| Median | l + [(n/2 − cf)/f] × h |
Empirical relation: Mode = 3 Median − 2 Mean
Chapter 14 — Probability
| Formula | Use |
|---|---|
| P(E) = n(E)/n(S) | Probability of event E |
| P(E) + P(Ē) = 1 | Complementary probability |
| P(impossible event) = 0 | |
| P(certain event) = 1 |
A standard deck has 52 cards: 4 suits × 13 cards each. A die has 6 faces. A coin has 2 faces. These sample spaces appear in 80% of probability questions.
Quick Revision — Formulas Exam Papers Test Most
Based on 10 years of CBSE Class 10 board papers, these formulas appear every year:
- Quadratic formula + discriminant
- nth term and sum of AP
- Distance formula + section formula
- All 3 trigonometric identities
- Area of sector and segment
- Volume of cone, cylinder, sphere
Master these 6 areas and you've covered 40–50% of the marks in Paper 2 (Standard Maths).
Practice these formulas with real CBSE questions at JoyOfExams.in — our AI tracks which formulas you struggle with and gives targeted practice.