Class 9 Mathematics formula sheet and revision guide
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06 Aug, 2026
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Class 9 Mathematics: Quick Revision & Formula Sheet
Chapter 1: Number Systems
- Real Numbers: Every rational or irrational number corresponds to a unique point on the number line.
- Laws of Exponents (for a, b > 0, m and n are rational numbers):
- am × an = am + n
- am ÷ an = am - n
- (am)n = am × n
- am × bm = (a × b)m
- a-m = 1 / am
- a0 = 1
- Square Root Identity: (√a + √b)(√a - √b) = a - b
Chapter 2: Polynomials
- Remainder Theorem: If a polynomial p(x) is divided by (x - a), the remainder is p(a).
- Factor Theorem: (x - a) is a factor of polynomial p(x) if and only if p(a) = 0.
- Algebraic Identities:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- a2 - b2 = (a + b)(a - b)
- (x + a)(x + b) = x2 + (a + b)x + ab
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a + b)3 = a3 + b3 + 3ab(a + b)
- (a - b)3 = a3 - b3 - 3ab(a - b)
- a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
- Special Condition: If a + b + c = 0, then a3 + b3 + c3 = 3abc
Chapter 3 & 4: Coordinate Geometry & Linear Equations
- Cartesian Plane: Point coordinates (x, y) represent (abscissa, ordinate).
- Quadrant I: (+, +)
- Quadrant II: (-, +)
- Quadrant III: (-, -)
- Quadrant IV: (+, -)
- Linear Equation in Two Variables: Standard form is ax + by + c = 0 (where a, b ≠ 0). It has infinitely many solutions, and its graphical representation is a straight line.
Chapter 6 & 7: Lines, Angles & Triangles
- Linear Pair Axiom: The sum of adjacent angles on a straight line is equal to 180°.
- Parallel Lines & Transversal: Corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side of transversal sum to 180°.
- Angle Sum Property: Sum of interior angles in a triangle is 180°.
- Triangle Congruence Rules: SAS, ASA, AAS, SSS, and RHS.
- Isosceles Triangle Property: Angles opposite to equal sides are equal, and vice versa.
Chapter 8 & 11: Quadrilaterals & Circles
- Mid-point Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length (DE ∥ BC and DE = ½ BC).
- Circle Theorems:
- Equal chords subtend equal angles at the center.
- The perpendicular from the center of a circle to a chord bisects the chord.
- Angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle (∠AOB = 2 ∠ACB).
- Angles in the same segment of a circle are equal.
- Opposite angles of a cyclic quadrilateral sum to 180°.
Chapter 12: Heron's Formula
- Semi-perimeter (s): s = (a + b + c) / 2
- Area of Triangle: Area = √[s(s - a)(s - b)(s - c)]
- Area of Equilateral Triangle: Area = (√3 / 4) a2
Chapter 10: Surface Areas and Volumes
| Shape | Curved / Lateral Area (LSA/CSA) | Total Surface Area (TSA) | Volume |
|---|---|---|---|
| Cube | 4a2 | 6a2 | a3 |
| Cuboid | 2h(l + b) | 2(lb + bh + lh) | l × b × h |
| Right Cylinder | 2πrh | 2πr(r + h) | πr2h |
| Right Cone | πrl (where l = √(r2 + h2)) | πr(r + l) | ⅓πr2h |
| Sphere | 4πr2 | 4πr2 | &frac43;πr3 |
| Hemisphere | 2πr2 | 3πr2 | ⅔πr3 |
Chapter 9 & 14: Statistics & Probability
- Mean (¯x): ¯x = (∑ xi) / n or (∑ fixi) / (∑ fi)
- Median:
- If n is odd: Median = [½(n + 1)]th observation
- If n is even: Median = Average of (n/2)th and [(n/2) + 1]th observations
- Probability P(E):
P(E) = (Number of trials in which the event happened) / (Total number of trials)
Range: 0 ≤ P(E) ≤ 1