📘 CBSE Class 9 Maths Complete Formula Cheat Sheet (2026)
Premium Revision Notes & Formula Master Guide covering all 15 Chapters! Perfect for Quick Revision, Exams, Unit Tests, and Last-Minute Preparation.
1. Number Systems
📐 Laws of Exponents
• am × an = am+n
• (am)n = amn
• am / an = am-n (where m > n)
• am × bm = (ab)m
• a0 = 1
• (am)n = amn
• am / an = am-n (where m > n)
• am × bm = (ab)m
• a0 = 1
🧠 Key Concepts
- Rational Numbers: Can be written as p/q (q ≠ 0). Decimals are terminating or repeating.
- Irrational Numbers: Cannot be written as p/q. Decimals are non-terminating and non-repeating (e.g., √2, π).
⚠️ Common Mistake
Assuming π is exactly 22/7. It is an approximation! π is irrational, but 22/7 is rational.
💡 Exam Tip
Rationalizing the denominator is a guaranteed question. Multiply numerator and denominator by the conjugate.
2. Polynomials
📐 High-Weightage Algebraic Identities
• (x + y)² = x² + 2xy + y²
• (x - y)² = x² - 2xy + y²
• x² - y² = (x + y)(x - y)
• (x + a)(x + b) = x² + (a+b)x + ab
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy(x + y)
• (x - y)³ = x³ - y³ - 3xy(x - y)
• x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
• (x - y)² = x² - 2xy + y²
• x² - y² = (x + y)(x - y)
• (x + a)(x + b) = x² + (a+b)x + ab
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy(x + y)
• (x - y)³ = x³ - y³ - 3xy(x - y)
• x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
⚡ Shortcut Trick
If x + y + z = 0, then immediately use x³ + y³ + z³ = 3xyz. This saves 5 minutes of calculation in exams!
3. Coordinate Geometry
| Concept | Property |
|---|---|
| Origin | (0, 0) |
| x-coordinate | Abscissa (Distance from y-axis) |
| y-coordinate | Ordinate (Distance from x-axis) |
| Equation of x-axis | y = 0 |
| Equation of y-axis | x = 0 |
🧭 Quadrants
I: (+, +) | II: (-, +) | III: (-, -) | IV: (+, -)
4. Linear Eq. in 2 Variables
🧠 Core Concept
- Standard Form: ax + by + c = 0
- A linear equation in two variables has infinitely many solutions.
- Graph of a linear equation is always a straight line.
⚡ Graph Shortcut
To find points quickly for a graph: First put x = 0 to find y. Then put y = 0 to find x.
5. Euclid's Geometry
🏛️ Important Axioms & Postulates
- Axiom 1: Things which are equal to the same thing are equal to one another.
- Axiom 2: If equals are added to equals, the wholes are equal.
- Postulate 1: A straight line may be drawn from any one point to any other point.
- Postulate 5 (VVIP): If a straight line falling on two straight lines makes interior angles on the same side less than 180°, the lines will meet on that side.
6. Lines and Angles
| Complementary | Sum is 90° |
| Supplementary | Sum is 180° |
| Linear Pair | Adjacent angles forming a line sum to 180° |
| Vertically Opposite | Always equal |
📏 Parallel Lines Transversal Rules
- Alternate Interior Angles are equal. (Z-Rule)
- Corresponding Angles are equal. (F-Rule)
- Co-interior Angles (same side) are supplementary (Sum = 180°).
⚠️ Common Mistake
Assuming any 'Z' shape creates equal angles. The lines MUST be explicitly given as parallel for alternate angles to be equal.
7. Triangles
📐 Congruence Criteria (The Big 5)
- SAS (Side-Angle-Side): Two sides and the included angle.
- ASA (Angle-Side-Angle): Two angles and the included side.
- AAS (Angle-Angle-Side): Two angles and non-included side.
- SSS (Side-Side-Side): All three sides.
- RHS (Right angle-Hypotenuse-Side): For right-angled triangles only.
💡 Exam Tip
Always write the congruence statement in the correct matching order. If ΔABC ≅ ΔPQR, then AB = PQ, not PR.
8. Quadrilaterals
⏹️ Properties of Parallelogram
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Diagonals bisect each other.
🔥 Mid-Point Theorem (VVIP)
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it.
9. Areas, 10. Circles & 12. Heron's Formula
📐 Areas (Parallelograms & Triangles)
- Theorem: Parallelograms on the same base and between same parallels have equal area.
- Triangle Area: 1/2 × base × height.
- If a triangle and parallelogram are on the same base/parallels, Area of Triangle = 1/2 (Area of Parallelogram).
⭕ Circles Key Theorems
- Equal chords subtend equal angles at the center.
- Perpendicular from center to a chord bisects the chord.
- Angle subtended by an arc at the center is double the angle subtended at the remaining part.
- Cyclic Quadrilateral: Sum of opposite angles is 180°.
🔺 Heron's Formula
Semi-perimeter: s = (a + b + c) / 2
Area = √s(s-a)(s-b)(s-c)
Area = √s(s-a)(s-b)(s-c)
⚠️ Error Alert: Students often forget the square root sign during intermediate steps. Write step-by-step!
13. Surface Areas and Volumes
| Shape | Curved/Lateral Surface Area (CSA/LSA) | Total Surface Area (TSA) | Volume (V) |
|---|---|---|---|
| Cube (side a) | 4a² | 6a² | a³ |
| Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cylinder (r, h) | 2πrh | 2πr(r + h) | πr²h |
| Cone (r, h, l) | πrl (Slant l = √(r² + h²)) |
πr(r + l) | (1/3)πr²h |
| Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |
⚡ Memory Trick
Volume Trick: Cylinder is πr²h. A Cone is exactly 1/3 of that!
CSA vs TSA: CSA = Curved only. TSA = Total outside area including top/bottom caps.
14. Statistics
📊 Key Formulas
- Mean: Sum of observations / Total number of observations.
- Median: Middle value of sorted data. If n is even, average of the two middle terms.
- Mode: Observation with maximum frequency.
15. Probability
🎲 Formulas & Rules
P(E) = (No. of favorable outcomes) / (Total outcomes)
- Probability always lies between 0 and 1: 0 ≤ P(E) ≤ 1
- P(Sure Event) = 1
- P(Impossible Event) = 0
🔥 BONUS: MASTER REVISION SECTION
⭐ Top 10 Frequently Asked Concepts
- Rationalizing denominators involving roots.
- Factorizing polynomials using the Factor Theorem.
- Proving triangles congruent using SAS/ASA.
- Applying the Mid-Point Theorem in quadrilaterals.
- Finding angles using cyclic quadrilateral properties.
- Calculating curved/total surface area of cones and hemispheres.
- Drawing histograms and calculating mean/median.
- Evaluating algebraic identities (especially cubic ones).
- Plotting coordinates and identifying quadrants.
- Calculating probability of dice/coin/card events.
📝 During The Exam Tips
- Draw Diagrams: Always draw neat figures for Geometry and Mensuration questions. It fetches partial marks!
- Units Matter: Deducting 0.5 marks for missing cm² or cm³ is very common. Double-check units.
- Step Marking: Write the formula explicitly before putting in the values.
⚠️ Common Exam Mistakes
- Sign errors: Recheck steps.
- Wrong units: Always write units at the end.
- Formula confusion: Learn with shapes and diagrams.
- Calculation errors: Solve slowly and double-check addition/multiplication.
📅 7-Day Maths Revision Strategy
| Day | Focus Area | Action Item |
|---|---|---|
| Day 1 | Number Systems & Polynomials | Practice identities and rationalization. |
| Day 2 | Geometry (Lines, Triangles, Euclid) | Revise theorems and congruence criteria. |
| Day 3 | Quadrilaterals & Circles | Solve 10 proofs involving cyclic quads & mid-point thm. |
| Day 4 | Mensuration (Surface Area & Vol) | Memorize formulas; solve 15 complex calculation questions. |
| Day 5 | Algebra & Coordinate Geo | Graph plotting and linear equation solutions. |
| Day 6 | Stats & Probability | Draw histograms; practice mean/median formulas. |
| Day 7 | Full Mock Test | Attempt a 3-hour paper. Review mistakes using this Cheat Sheet. |