Top set · End-of-year exam

GCSE Maths Formulas

A complete revision sheet for Ease's class. Every formula you're expected to know — and the ones the exam won't give you.

Number & Percentages

  • Percentage change
    % change = (new − old) / old × 100
  • Percentage of amount
    P% of A = (P / 100) × A
  • Reverse percentage
    original = new / (1 ± %/100)
  • Compound interest
    A = P(1 + r/100)ⁿ
  • Simple interest
    I = P × r × t / 100
  • Speed
    speed = distance ÷ time
  • Density
    density = mass ÷ volume
  • Pressure
    pressure = force ÷ area

Algebra

  • Quadratic formula
    x = (−b ± √(b² − 4ac)) / 2a
    for ax² + bx + c = 0
  • Discriminant
    Δ = b² − 4ac
    >0 two roots, =0 one, <0 none
  • Difference of two squares
    a² − b² = (a − b)(a + b)
  • Completing the square
    x² + bx + c = (x + b/2)² − (b/2)² + c
  • nth term (linear)
    aₙ = a + (n − 1)d
  • nth term (geometric)
    aₙ = a·rⁿ⁻¹
  • Laws of indices
    aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
  • Negative / fractional index
    a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a

Coordinate Geometry

  • Gradient
    m = (y₂ − y₁) / (x₂ − x₁)
  • Straight line
    y = mx + c
  • Point-slope form
    y − y₁ = m(x − x₁)
  • Distance between points
    d = √((x₂ − x₁)² + (y₂ − y₁)²)
  • Midpoint
    M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
  • Parallel / perpendicular
    parallel: m₁ = m₂, perpendicular: m₁·m₂ = −1

Shape, Area & Volume

  • Circle circumference
    C = 2πr = πd
  • Circle area
    A = πr²
  • Arc length
    L = (θ/360) × 2πr
  • Sector area
    A = (θ/360) × πr²
  • Triangle area
    A = ½ × base × height
  • Trapezium area
    A = ½(a + b)h
  • Parallelogram area
    A = base × height
  • Cuboid volume
    V = l × w × h
  • Prism volume
    V = area of cross-section × length
  • Cylinder volume / surface
    V = πr²h, SA = 2πr² + 2πrh
  • Pyramid volume
    V = ⅓ × base area × height
  • Cone volume / curved SA
    V = ⅓πr²h, CSA = πrl
  • Sphere volume / surface
    V = 4⁄3 πr³, SA = 4πr²

Pythagoras & Trigonometry

  • Pythagoras' theorem
    a² + b² = c²
    right-angled triangle
  • SOH CAH TOA
    sin θ = O/H, cos θ = A/H, tan θ = O/A
  • Sine rule
    a / sin A = b / sin B = c / sin C
  • Cosine rule
    a² = b² + c² − 2bc·cos A
  • Area of any triangle
    A = ½ab·sin C
  • Exact values
    sin30 = ½, cos60 = ½, tan45 = 1, sin60 = cos30 = √3/2

Circle Theorems (key facts)

  • Angle at centre
    = 2 × angle at circumference (same arc)
  • Angle in semicircle
    = 90°
  • Angles in same segment
    are equal
  • Cyclic quadrilateral
    opposite angles sum to 180°
  • Tangent & radius
    meet at 90°
  • Two tangents from a point
    are equal in length
  • Alternate segment theorem
    angle between tangent & chord = angle in alternate segment

Probability & Statistics

  • Probability
    P(event) = favourable ÷ total
  • Not event
    P(not A) = 1 − P(A)
  • Independent events (AND)
    P(A ∩ B) = P(A) × P(B)
  • Mutually exclusive (OR)
    P(A ∪ B) = P(A) + P(B)
  • Mean
    mean = Σx / n
  • Mean from frequency table
    mean = Σfx / Σf
  • Range
    range = max − min
  • Interquartile range
    IQR = Q₃ − Q₁

Vectors & Transformations

  • Vector magnitude
    |a| = √(x² + y²)
  • Column vector addition
    (a, b) + (c, d) = (a+c, b+d)
  • Scalar multiple
    k(a, b) = (ka, kb)
  • Enlargement (area)
    area scale factor = (linear SF)²
  • Enlargement (volume)
    volume scale factor = (linear SF)³

Good luck to Ease and the class — you've got this.