Math Formula

Basic Arithmetic

Number Properties

Commutative Property of Addition

  • Formula: a + b = b + a
  • Example: 5 + 3 = 3 + 5 = 8

Commutative Property of Multiplication

  • Formula: a × b = b × a
  • Example: 4 × 6 = 6 × 4 = 24

Associative Property of Addition

  • Formula: (a + b) + c = a + (b + c)
  • Example: (2 + 3) + 4 = 2 + (3 + 4) = 9

Associative Property of Multiplication

  • Formula: (a × b) × c = a × (b × c)
  • Example: (2 × 3) × 4 = 2 × (3 × 4) = 24

Distributive Property

  • Formula: a × (b + c) = (a × b) + (a × c)
  • Example: 3 × (4 + 2) = (3 × 4) + (3 × 2) = 18

Basic Geometry

Perimeter Formulas

Rectangle

  • Formula: P = 2(l + w)
  • Example: For length = 5 cm, width = 3 cm
  • P = 2(5 + 3) = 16 cm

Square

  • Formula: P = 4s
  • Example: For side = 4 cm
  • P = 4 × 4 = 16 cm

Triangle

  • Formula: P = a + b + c
  • Example: For sides 3 cm, 4 cm, 5 cm
  • P = 3 + 4 + 5 = 12 cm

Area Formulas

Rectangle

  • Formula: A = l × w
  • Example: For length = 6 cm, width = 4 cm
  • A = 6 × 4 = 24 cm²

Square

  • Formula: A = s²
  • Example: For side = 5 cm
  • A = 5² = 25 cm²

Triangle

  • Formula: A = (b × h) ÷ 2
  • Example: For base = 6 cm, height = 4 cm
  • A = (6 × 4) ÷ 2 = 12 cm²

Circle

  • Formula: A = πr²
  • Example: For radius = 3 cm
  • A = π × 3² ≈ 28.27 cm²

Algebra

Linear Equations

Slope Formula

  • Formula: m = (y₂ – y₁)/(x₂ – x₁)
  • Example: Points (2,3) and (4,7)
  • m = (7 – 3)/(4 – 2) = 2

Point-Slope Form

  • Formula: y – y₁ = m(x – x₁)
  • Example: Point (2,3), slope = 2
  • y – 3 = 2(x – 2)

Slope-Intercept Form

  • Formula: y = mx + b
  • Example: Slope = 2, y-intercept = 1
  • y = 2x + 1

Exponent Rules

Product Rule

  • Formula: aᵐ × aⁿ = aᵐ⁺ⁿ
  • Example: 2³ × 2⁴ = 2⁷ = 128

Quotient Rule

  • Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
  • Example: 2⁵ ÷ 2³ = 2² = 4

Power Rule

  • Formula: (aᵐ)ⁿ = aᵐⁿ
  • Example: (2³)⁴ = 2¹² = 4,096

Geometry

Pythagorean Theorem

  • Formula: a² + b² = c²
  • Example: Right triangle with legs 3 and 4
  • 3² + 4² = c²
  • 9 + 16 = c²
  • c = 5

Circle Formulas

Circumference

  • Formula: C = 2πr
  • Example: For radius = 5 cm
  • C = 2π × 5 ≈ 31.42 cm

Arc Length

  • Formula: L = (θ/360°) × 2πr
  • Example: For radius = 6 cm, angle = 90°
  • L = (90/360) × 2π × 6 ≈ 9.42 cm

Advanced Algebra

Quadratic Formula

  • Formula: x = [-b ± √(b² – 4ac)] ÷ 2a
  • Example: For 2x² + 5x – 3 = 0
  • a = 2, b = 5, c = -3
  • x = [-5 ± √(25 + 24)] ÷ 4
  • x = (-5 ± 7) ÷ 4
  • x = 0.5 or -3

Polynomial Factoring

Difference of Squares

  • Formula: a² – b² = (a + b)(a – b)
  • Example: x² – 16 = (x + 4)(x – 4)

Perfect Square Trinomial

  • Formula: a² + 2ab + b² = (a + b)²
  • Example: x² + 6x + 9 = (x + 3)²

Trigonometry

Basic Ratios

Sine

  • Formula: sin θ = opposite/hypotenuse
  • Example: In 30-60-90 triangle
  • sin 30° = 1/2

Cosine

  • Formula: cos θ = adjacent/hypotenuse
  • Example: In 30-60-90 triangle
  • cos 30° = √3/2

Tangent

  • Formula: tan θ = opposite/adjacent
  • Example: In 45-45-90 triangle
  • tan 45° = 1

Trigonometric Identities

Pythagorean Identity

  • Formula: sin²θ + cos²θ = 1
  • Example: For θ = 30°
  • sin²30° + cos²30° = (1/2)² + (√3/2)² = 1

Double Angle Formulas

  • Formula: sin(2θ) = 2sin θ cos θ
  • Example: sin(60°) = 2sin(30°)cos(30°)
  • = 2(1/2)(√3/2) = √3/2

Calculus Basics

Basic Derivatives

Power Rule

  • Formula: d/dx(xⁿ) = nxⁿ⁻¹
  • Example: d/dx(x³) = 3x²

Product Rule

  • Formula: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
  • Example: d/dx(x²sin x) = 2x sin x + x² cos x

Chain Rule

  • Formula: d/dx[f(g(x))] = f'(g(x)) × g'(x)
  • Example: d/dx(sin(x²))= 2x cos(x²)

Conic Sections

Parabola

  • Formula: y = ax² + bx + c
  • Example: y = 2x² – 4x + 1
  • Vertex form: y = 2(x – 1)² – 1
  • Vertex: (1, -1)

Ellipse

  • Formula: (x²/a²) + (y²/b²) = 1
  • Example: (x²/16) + (y²/9) = 1
  • Center: (0,0)
  • Vertices: (±4, 0)
  • Co-vertices: (0, ±3)

Hyperbola

  • Formula: (x²/a²) – (y²/b²) = 1
  • Example: (x²/16) – (y²/9) = 1
  • Asymptotes: y = ±(3/4)x
  • Vertices: (±4, 0)