Maths Formulas for Class 9 – All Chapters

Maths formulas for Class 9 help students solve problems faster and improve accuracy. This page provides all important Class 9 Maths formulas in a clear, chapter-wise format. Students can use these formulas for homework, daily practice, and exam revision.

The list covers all key topics from the latest syllabus. Each formula is easy to understand and simple to apply in questions. Students can also download the Class 9 Maths formulas PDF for quick revision anytime.

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Must Buy: Oswal Publishers’ CBSE Class 9 Books

Class 9 Maths Formulas Chapter-wise

Here are a list of chapter-wise all formulas of class 9th:

Chapter 1: Number Systems

• Rational Numbers: A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0
• Irrational Numbers: Numbers that cannot be expressed as p/q (where p and q are integers and q ≠ 0)
• Real Numbers: The collection of all rational and irrational numbers
• Laws of Exponents:
  • a^m × a^n = a^(m+n)
  • a^m ÷ a^n = a^(m-n)
  • (a^m)^n = a^(m×n)
  • a^0 = 1 (where a ≠ 0)
  • a^(-n) = 1/a^n

TIP: When rationalising denominators with surds, multiply both numerator and denominator by the conjugate. For example, 1/(√2-1) = (√2+1)/(√2-1)(√2+1) = (√2+1)/(2-1) = √2+1

Chapter 2: Polynomials

• Polynomial: An algebraic expression of the form a₀xⁿ + a₁xⁿ⁻¹ + … + aₙ₋₁x + aₙ
• Degree of Polynomial: Highest power of the variable in the polynomial
• Remainder Theorem: If p(x) is divided by (x-a), then remainder = 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)
  • (a+b)³ = a³ + 3a²b + 3ab² + b³
  • (a-b)³ = a³ – 3a²b + 3ab² – b³
  • a³ + b³ = (a+b)(a² – ab + b²)
  • a³ – b³ = (a-b)(a² + ab + b²)

Chapter 3: Coordinate Geometry

• Cartesian Coordinate System: A system where any point is represented as an ordered pair (x, y)
• Distance Formula: Distance between points (x₁, y₁) and (x₂, y₂) = √[(x₂ – x₁)² + (y₂ – y₁)²]
• Section Formula: The coordinates of the point P that divides the line segment joining points A(x₁, y₁) and B(x₂, y₂) in the ratio m:n are:
  • x = (mx₂ + nx₁)/(m+n)
  • y = (my₂ + ny₁)/(m+n)
• Midpoint Formula: Midpoint of line segment joining (x₁, y₁) and (x₂, y₂) = ((x₁+x₂)/2, (y₁+y₂)/2)
• Area of Triangle: Area of triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

TIP: When finding the midpoint, simply take the average of x-coordinates and the average of y-coordinates.

Chapter 4: Linear Equations in Two Variables

• Standard Form: ax + by + c = 0, where a, b are not both zero
• General Form: y = mx + c, where m is the slope and c is the y-intercept
• Slope Formula: m = (y₂-y₁)/(x₂-x₁) for two points (x₁, y₁) and (x₂, y₂)
• Relation between slopes of parallel lines: m₁ = m₂
• Relation between slopes of perpendicular lines: m₁ × m₂ = -1
• Point-Slope Form: y – y₁ = m(x – x₁), where m is the slope and (x₁, y₁) is a point on the line
• Condition for consistency of two linear equations: Two equations are consistent if they have at least one common solution

Chapter 5: Introduction to Euclid’s Geometry

• Euclid’s Five Postulates:
  1. A straight line can be drawn between any two points
  2. A line segment can be extended indefinitely
  3. A circle can be drawn with any centre and any radius
  4. All right angles are equal
  5. Through a point not on a given line, exactly one line can be drawn parallel to the given line

NOTE: This chapter is primarily conceptual and doesn’t contain maths formulas for class 9 to memorise, but understanding these postulates is fundamental to geometric reasoning.

Chapter 6: Lines and Angles

• Vertically Opposite Angles: Vertically opposite angles are equal
• Supplementary Angles: Two angles whose sum is 180° (a + b = 180°)
• Complementary Angles: Two angles whose sum is 90° (a + b = 90°)
• Angle Sum Property of a Triangle: Sum of all angles in a triangle = 180°
• Parallel Lines and Transversal Rules:
  • Corresponding angles are equal
  • Alternate interior angles are equal
  • Alternate exterior angles are equal
  • Interior angles on the same side of the transversal are supplementary (sum to 180°)

Chapter 7: Triangles

• Congruence Criteria for Triangles:
  • SSS (Side-Side-Side): All three sides are equal
  • SAS (Side-Angle-Side): Two sides and the included angle are equal
  • ASA (Angle-Side-Angle): Two angles and the included side are equal
  • RHS (Right angle-Hypotenuse-Side): For right-angled triangles, when the hypotenuse and one side are equal
• Angle Sum Property: Sum of angles in a triangle = 180°
• Exterior Angle Property: An exterior angle of a triangle equals the sum of the two interior opposite angles
• Pythagoras Theorem: In a right-angled triangle, (Hypotenuse)² = (Base)² + (Perpendicular)²
• Median: A line from a vertex to the midpoint of the opposite side
• Altitude: A perpendicular from a vertex to the opposite side (or its extension)
• Incentre: Point of intersection of angle bisectors
• Circumcentre: Point of intersection of perpendicular bisectors of sides
• Orthocentre: Point of intersection of altitudes
• Centroid: Point of intersection of medians (divides each median in the ratio 2:1 from the vertex)

Chapter 8: Quadrilaterals

• Angle Sum Property: Sum of all angles in a quadrilateral = 360°
• Parallelogram Properties:
  • Opposite sides are equal and parallel
  • Opposite angles are equal
  • Diagonals bisect each other
  • Each diagonal divides the parallelogram into two congruent triangles
• Rectangle Properties (in addition to parallelogram properties):
  • All angles are 90°
  • Diagonals are equal
• Rhombus Properties (in addition to parallelogram properties):
  • All sides are equal
  • Diagonals bisect each other at right angles
• Square Properties (combines rectangle and rhombus properties):
  • All sides are equal
  • All angles are 90°
  • Diagonals are equal and bisect each other at right angles
• Trapezium: A quadrilateral with exactly one pair of parallel sides
• Kite: A quadrilateral with two pairs of adjacent sides equal

Chapter 9: Areas of Parallelograms and Triangles

• Area of Parallelogram: base × height
• Area of Triangle: (1/2) × base × height
• Area of Triangle using Semiperimeter: √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the sides
• Area of Triangle using Coordinates: (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
• Parallelograms on the same base and between the same parallels have equal areas
• Triangles on the same base and between the same parallels have equal areas

EXAMPLE: If a triangle and parallelogram share the same base and height, the area of the triangle is exactly half the area of the parallelogram.

Chapter 10: Circles

• Circle: The set of all points in a plane that are equidistant from a fixed point (centre)
• Radius: The distance from the centre to any point on the circle
• Diameter: A line segment passing through the centre and connecting two points on the circle (= 2 × radius)
• Chord: A line segment connecting two points on the circle
• Arc: A portion of the circumference of a circle
• Central Angle: An angle formed at the centre by two radii
• Inscribed Angle: An angle formed by two chords meeting at a point on the circle
• Properties:
  • The perpendicular from the centre to a chord bisects the chord
  • Equal chords are equidistant from the centre
  • The angle in a semicircle is a right angle
  • Angles in the same segment are equal
  • The angle subtended by an arc at the centre is twice the angle subtended by it at any point on the alternate segment

Chapter 11: Constructions

This chapter focuses on geometric constructions rather than maths formulas for class 9. You’ll learn how to:

• Bisect angles and line segments
• Construct angles of specified measures (60°, 30°, 90°, 45°, etc.)
• Construct triangles given:
  • Three sides (SSS)
  • Two sides and the included angle (SAS)
  • Two angles and the included side (ASA)
• Construct parallel lines to a given line through a point not on the line

Chapter 12: Heron’s Formula

• Heron’s Formula: Area of triangle = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the sides
• Area of Quadrilateral (using diagonals d₁ and d₂): Area = (1/2) × d₁ × d₂ × sin(angle between diagonals)
• Semi-perimeter (s): s = (a+b+c)/2, where a, b, c are the sides of the triangle

EXAMPLE: For a triangle with sides 3, 4, and 5: s = (3+4+5)/2 = 6 Area = √[6(6-3)(6-4)(6-5)] = √[6×3×2×1] = √36 = 6 square units

Chapter 13: Surface Areas and Volumes

• Cube:
  • Surface Area = 6a², where a is the edge length
  • Volume = a³
• Cuboid:
  • Surface Area = 2(lb + bh + hl), where l, b, h are length, breadth, and height
  • Volume = l × b × h
• Right Circular Cylinder:
  • Curved Surface Area = 2πrh, where r is radius and h is height
  • Total Surface Area = 2πr(r+h)
  • Volume = πr²h
• Right Circular Cone:
  • Curved Surface Area = πrl, where r is radius and l is slant height
  • Total Surface Area = πr(l+r)
  • Volume = (1/3)πr²h, where h is height
• Sphere:
  • Surface Area = 4πr², where r is radius
  • Volume = (4/3)πr³
• Hemisphere:
  • Curved Surface Area = 2πr²
  • Total Surface Area = 3πr²
  • Volume = (2/3)πr³

TIP: Remember that π (pi) is approximately equal to 3.14 or 22/7 for your calculations.

Chapter 14: Statistics

• Mean (for ungrouped data): Mean = Sum of all observations / Number of observations
• Mean (for grouped data):
  • Direct Method: Mean = Σfx / Σf
  • Assumed Mean Method: Mean = a + Σfd / Σf
  • Step Deviation Method: Mean = a + h(Σfx / Σf)
• Median (for ungrouped data): Middle value when observations are arranged in ascending or descending order
• Median (for grouped data): Median = l + [(n/2 – cf) / f] × h
• Mode: Most frequently occurring value
• Mode (for grouped data): Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h
• Empirical Relationship: 3 Median = 2 Mode + Mean

EXAMPLE: For data 2, 3, 3, 4, 5, 5, 5, 6, 7: Mean = (2+3+3+4+5+5+5+6+7)/9 = 40/9 = 4.44 Median = 5 (middle value) Mode = 5 (occurs most frequently)

Chapter 15: Probability

• Probability: P(E) = Number of favourable outcomes / Total number of possible outcomes
• Properties of Probability:
  • 0 ≤ P(E) ≤ 1
  • P(Sure Event) = 1
  • P(Impossible Event) = 0
  • P(Not E) = 1 – P(E)
• Empirical Probability: Number of times event occurs / Total number of trials

EXAMPLE: When rolling a fair die, the probability of getting a 4 is 1/6, as there’s only one favourable outcome out of six possible outcomes.

How to Memorise Class 9 Maths Formulas Effectively

Mastering maths formulas for class 9 doesn’t need to be overwhelming! Try these proven techniques:

1. Create Flashcards: Write maths formulas for class 9 on cards with examples on the reverse. Review them regularly during short breaks throughout the day.
   
2. Use Visual Associations: Create mental images connecting maths formulas for class 9 to their applications. For example, imagine a triangle split into rectangles to remember the area formula.
   
3. Formulate Mnemonics: Create memorable phrases using the first letters of concepts. For Pythagoras’ theorem, “Harry Potter Has Read Books Perfectly” (Hypotenuse² = Base² + Perpendicular²).
   
4. Daily Practice: Apply maths formulas for class 9 to problems daily. Even just 15 minutes makes a difference!
   
5. Concept Mapping: Draw diagrams showing how maths formulas for class 9 relate to each other.

Class 9 Maths Formulas Practice Questions & Answers

1: If the area of a triangle is 30 square units and its base is 10 units, find its height.
Answer: Using the formula Area = (1/2) × base × height 30 = (1/2) × 10 × height height = 30 × 2 ÷ 10 = 6 units

2: Find the volume of a sphere with radius 7 cm.
Answer: Using V = (4/3)πr³ V = (4/3) × 22/7 × 7³ V = (4/3) × 22/7 × 343 V = (4/3) × 22 × 49 V = (4 × 22 × 49)/3 V = 1437.33… cm³

3: Find the probability of getting a prime number when rolling a fair six-sided die.
Answer: Prime numbers on a die are 2, 3, and 5. Total possible outcomes = 6. P(prime) = Number of favourable outcomes/Total outcomes = 3/6 = 1/2

4: The coordinates of the midpoint of a line segment are (6, 4). If one endpoint is (8, 7), find the other endpoint.
Answer: Using the midpoint formula (x₁+x₂)/2 = 6 and (y₁+y₂)/2 = 4: (8+x₂)/2 = 6, so x₂ = 12 – 8 = 4 (7+y₂)/2 = 4, so y₂ = 8 – 7 = 1 Therefore, the other endpoint is (4, 1).

5: Using Heron’s formula, find the area of a triangle with sides 5 cm, 12 cm, and 13 cm.
Answer: Semi-perimeter s = (5+12+13)/2 = 15 Area = √[s(s-a)(s-b)(s-c)] = √[15(15-5)(15-12)(15-13)] = √[15 × 10 × 3 × 2] = √[900] = 30 square cm

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