In short: This is the complete chapter-wise list of every formula, identity, and theorem you need for Class 9 Maths 2027. Based on the new NCERT "Ganita Manjari Part 1" textbook. Bookmark this page, print the formula tables, and revise daily. Students who revise formulas every day score 15-20 marks more than those who try to remember formulas during the exam. Works for CBSE, UP Board, Bihar Board, MP Board, Rajasthan Board, and all NCERT-following boards.
How to Use This Formula Sheet
- Print or screenshot the tables you need
- Revise daily — 5 minutes before sleep is enough
- Practice with problems — a formula without practice is useless
- Understand the formula — know what each variable represents
- Write from memory — test yourself by writing formulas without looking
Number Systems
Real Numbers
| Concept | Formula / Rule |
|---|---|
| Rational numbers | Numbers of the form p/q where q ≠ 0 and p, q are integers |
| Irrational numbers | Numbers that cannot be written as p/q (e.g., √2, √3, π) |
| Real numbers | All rational + irrational numbers together |
Laws of Exponents
| Law | Formula |
|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ |
| Power of a product | (ab)ᵐ = aᵐ × bᵐ |
| Power of a quotient | (a/b)ᵐ = aᵐ / bᵐ |
| Zero exponent | a⁰ = 1 (where a ≠ 0) |
| Negative exponent | a⁻ⁿ = 1/aⁿ |
| Fractional exponent | a^(1/n) = ⁿ√a |
| General fractional | a^(m/n) = ⁿ√(aᵐ) |
Rationalisation
| Expression | Rationalised Form |
|---|---|
| 1/√a | √a / a |
| 1/(√a + √b) | (√a - √b) / (a - b) |
| 1/(√a - √b) | (√a + √b) / (a - b) |
Algebraic Identities — The Most Important Section
Standard Identities
| # | Identity |
|---|---|
| 1 | (a + b)² = a² + 2ab + b² |
| 2 | (a - b)² = a² - 2ab + b² |
| 3 | a² - b² = (a + b)(a - b) |
| 4 | (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca |
| 5 | (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) |
| 6 | (a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ - b³ - 3ab(a - b) |
| 7 | a³ + b³ = (a + b)(a² - ab + b²) |
| 8 | a³ - b³ = (a - b)(a² + ab + b²) |
| 9 | a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) |
Special Case
If a + b + c = 0, then a³ + b³ + c³ = 3abc
This identity is frequently tested. Remember the condition: sum = 0 → cubes sum = 3abc.
Polynomial Theorems
| Theorem | Statement |
|---|---|
| Remainder Theorem | If polynomial p(x) is divided by (x - a), the remainder is p(a) |
| Factor Theorem | (x - a) is a factor of p(x) if and only if p(a) = 0 |
Linear Equations
| Concept | Formula |
|---|---|
| Standard form | ax + by + c = 0 |
| Slope of line y = mx + c | m (coefficient of x) |
| y-intercept of y = mx + c | c (constant term) |
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Coordinate Geometry
| Formula | Expression |
|---|---|
| Distance between (x₁,y₁) and (x₂,y₂) | d = √[(x₂ - x₁)² + (y₂ - y₁)²] |
| Midpoint of (x₁,y₁) and (x₂,y₂) | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) |
| Origin | (0, 0) |
| Collinearity test | Three points are collinear if area of triangle = 0 |
Quadrant Rules
| Quadrant | x-sign | y-sign | Example |
|---|---|---|---|
| I | + | + | (3, 4) |
| II | - | + | (-3, 4) |
| III | - | - | (-3, -4) |
| IV | + | - | (3, -4) |
Geometry — Theorems and Properties
Lines and Angles
| Theorem / Property | Statement |
|---|---|
| Vertically opposite angles | Are always equal |
| Linear pair | Sum = 180° |
| Angles on a straight line | Sum = 180° |
| Angles around a point | Sum = 360° |
| Corresponding angles (parallel lines) | Are equal |
| Alternate interior angles (parallel lines) | Are equal |
| Co-interior angles (parallel lines) | Sum = 180° (supplementary) |
| Angle sum of triangle | Sum of all angles = 180° |
| Exterior angle theorem | Exterior angle = sum of two opposite interior angles |
Triangle Properties
| Property | Statement |
|---|---|
| Angle sum | ∠A + ∠B + ∠C = 180° |
| Exterior angle | = Sum of remote interior angles |
| Isosceles triangle | Angles opposite equal sides are equal |
| Equilateral triangle | All angles = 60°, all sides equal |
| Triangle inequality | Sum of any two sides > third side |
| Pythagoras theorem | In right triangle: a² + b² = c² (hypotenuse) |
Congruence Rules
| Rule | What to Check |
|---|---|
| SSS | Three sides of one = Three sides of other |
| SAS | Two sides and included angle |
| ASA | Two angles and included side |
| AAS | Two angles and one non-included side |
| RHS | Right angle + Hypotenuse + One side |
Quadrilateral Properties
| Shape | Properties |
|---|---|
| Parallelogram | Opposite sides equal and parallel, opposite angles equal, diagonals bisect each other |
| Rectangle | All angles 90°, diagonals equal, opposite sides equal |
| Rhombus | All sides equal, diagonals bisect at 90°, opposite angles equal |
| Square | All sides equal, all angles 90°, diagonals equal and bisect at 90° |
| Trapezium | One pair of parallel sides |
| Kite | Two pairs of adjacent sides equal |
Midpoint Theorem
The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
If D and E are midpoints of AB and AC: DE ∥ BC and DE = ½ BC
Circle Theorems
| Theorem | Statement |
|---|---|
| Equal chords | Equal chords are equidistant from the centre |
| Perpendicular from centre | Perpendicular from centre to chord bisects the chord |
| Angle at centre | = 2 × angle at circumference (same arc) |
| Angles in same segment | Are equal |
| Angle in semicircle | = 90° |
| Cyclic quadrilateral | Opposite angles are supplementary (sum = 180°) |
Mensuration — Area, Surface Area, Volume
2D Shapes — Area and Perimeter
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Triangle | ½ × b × h | a + b + c |
| Equilateral Triangle | (√3/4) × a² | 3a |
| Circle | πr² | 2πr (circumference) |
| Semicircle | ½πr² | πr + 2r |
| Parallelogram | b × h | 2(a + b) |
| Trapezium | ½ × (a + b) × h | Sum of all sides |
| Rhombus | ½ × d₁ × d₂ | 4a |
Heron's Formula
For a triangle with sides a, b, c:
Semi-perimeter: s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]
This formula works for any triangle when you know all three sides.
3D Shapes — Surface Area and Volume
| Shape | Lateral/Curved SA | Total SA | Volume |
|---|---|---|---|
| Cuboid | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cube | 4a² | 6a² | a³ |
| Cylinder | 2πrh | 2πr(r + h) | πr²h |
| Cone | πrl (l = slant height) | πr(r + l) | ⅓πr²h |
| Sphere | — | 4πr² | ⁴⁄₃πr³ |
| Hemisphere | 2πr² | 3πr² | ⅔πr³ |
Slant Height of Cone
l = √(r² + h²)
where r = radius, h = height, l = slant height
Diagonal of Cuboid
d = √(l² + b² + h²)
Statistics
| Measure | Formula |
|---|---|
| Mean (direct) | x̄ = Σxᵢ / n |
| Mean (frequency) | x̄ = Σ(fᵢxᵢ) / Σfᵢ |
| Median (ungrouped) | Middle value when arranged in order |
| Median (odd n) | ((n+1)/2)th value |
| Median (even n) | Average of (n/2)th and (n/2 + 1)th values |
| Mode | Most frequently occurring value |
| Range | Maximum value - Minimum value |
Relationship Between Mean, Median, Mode
Mode = 3 × Median - 2 × Mean (approximate, for moderately skewed data)
Probability
| Concept | Formula |
|---|---|
| Probability of event E | P(E) = Number of favourable outcomes / Total outcomes |
| Complementary event | P(not E) = 1 - P(E) |
| Certain event | P(E) = 1 |
| Impossible event | P(E) = 0 |
| Range | 0 ≤ P(E) ≤ 1 |
Common Probability Values
| Experiment | Total Outcomes | Useful Facts |
|---|---|---|
| Tossing 1 coin | 2 (H, T) | P(H) = P(T) = 1/2 |
| Tossing 2 coins | 4 (HH, HT, TH, TT) | P(both H) = 1/4 |
| Rolling 1 die | 6 (1,2,3,4,5,6) | P(any number) = 1/6 |
| Rolling 2 dice | 36 | P(sum 7) = 6/36 = 1/6 |
| Deck of cards | 52 | 4 suits, 13 each |
Quick Revision Checklist
Use this checklist before exams:
- All 9 algebraic identities written from memory
- Laws of exponents (8 laws)
- Distance and midpoint formulas
- All triangle congruence rules (5 rules)
- Quadrilateral properties (6 shapes)
- Circle theorems (6 theorems)
- Heron's formula
- All 3D shape formulas (6 shapes — SA + Volume)
- Mean, median, mode formulas
- Probability formula and complementary rule
Frequently Asked Questions
How many formulas do I need to memorise for Class 9 Maths?
Approximately 60-70 formulas, identities, and theorems. This sounds like a lot, but many are interconnected. If you understand the pattern (like all surface area formulas follow a similar logic), memorisation becomes much easier.
What is the best way to memorise Maths formulas?
Write them out by hand every day (not just read). Test yourself by writing formulas from memory. Use them in practice problems — formulas remembered through practice last longer than those memorised in isolation.
Which formulas are most important for the exam?
Algebraic identities (tested in every exam), Heron's formula, Surface Area and Volume formulas, and the three equations of motion from Physics overlap. Geometry theorems are tested as proofs worth 5 marks each.
Are these formulas the same for all boards?
Yes. The core NCERT formulas are the same across CBSE, UP Board, Bihar Board, MP Board, and Rajasthan Board. The exam pattern varies, but the mathematical content is identical.
Should I make a separate formula notebook?
Yes. A dedicated formula notebook that you revise daily is one of the most effective study tools. Write formulas on one page, practice problems on the opposite page. Carry it everywhere for quick revision.
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