📋 Formula sheet
All formulas with illustrations — print and hang on the wall!
📐 Geometry
Area — 2D
A = l · w
Rectangle
A = s²
Square
A = (b · h) / 2
Triangle
A = π · r²
Circle
A = (a+b)·h / 2
Trapezoid
Volume — 3D
V = l · w · h
Rectangular prism
V = s³
Cube
V = π · r² · h
Cylinder
V = (4/3) · π · r³
Sphere
V = (1/3)·π·r²·h
Cone
Pythagoras
a² + b² = c²
Pythagorean theorem
c = √(a² + b²)
Find hypotenuse
Trigonometry
sin(θ) = opposite / hypotenuse
Sine
cos(θ) = adjacent / hypotenuse
Cosine
tan(θ) = opposite / adjacent
Tangent
sin²(θ) + cos²(θ) = 1
Pythagorean identity
🔢 Numbers and algebra
Algebraic identities
(a+b)² = a² + 2ab + b²
Square of sum
(a-b)² = a² - 2ab + b²
Square of difference
(a+b)(a-b) = a² - b²
Difference of squares
Exponent rules
aᵐ · aⁿ = aᵐ⁺ⁿ
Product rule
aᵐ / aⁿ = aᵐ⁻ⁿ
Quotient rule
(aᵐ)ⁿ = aᵐⁿ
Power of a power
a⁰ = 1
Zero exponent
a⁻ⁿ = 1/aⁿ
Negative exponent
a^(1/n) = ⁿ√a
Fractional exponent
Logarithms
log(a·b) = log(a) + log(b)
Product rule
log(a/b) = log(a) - log(b)
Quotient rule
log(aⁿ) = n · log(a)
Power rule
ln(e) = 1
Natural log
Absolute value
|x| = x if x ≥ 0
Positive value
|x| = -x if x < 0
Negative value
〰️ Functions
Linear function
f(x) = ax + b
a=slope, b=y-intercept
a = (y₂-y₁) / (x₂-x₁)
Slope formula
Quadratic function
f(x) = ax² + bx + c
Parabola
x = -b / (2a)
Vertex x
x = (-b ± √(b²-4ac)) / (2a)
Quadratic formula
D = b² - 4ac
Discriminant
Exponential function
f(x) = a · bˣ
Growth/decay
e ≈ 2.71828
Euler's number
∫ Calculus
Differentiation rules
(xⁿ)' = n·xⁿ⁻¹
Power rule
(c)' = 0
Constant
(eˣ)' = eˣ
Exponential
(ln x)' = 1/x
Natural log
(sin x)' = cos x
Sine
(cos x)' = -sin x
Cosine
(u·v)' = u'v + uv'
Product rule
(u/v)' = (u'v-uv')/v²
Quotient rule
Integration rules
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
Power rule
∫ k dx = kx + C
Constant
∫ eˣ dx = eˣ + C
Exponential
∫ₐᵇ f(x) dx = F(b)-F(a)
Definite integral
📊 Statistics
Measures of centre
x̄ = Σxᵢ / n
Mean
σ = √(Σ(xᵢ-x̄)² / n)
Std deviation
z = (x - μ) / σ
Z-score
68-95-99.7 rule
μ ± 1σ → 68%
Within 1σ
μ ± 2σ → 95%
Within 2σ
μ ± 3σ → 99.7%
Within 3σ
Probability
P(A) = favourable / total
Probability
P(not A) = 1 - P(A)
Complement
0 ≤ P(A) ≤ 1
Always 0-1
🔷 Vectors
Vector operations
|⃗v| = √(x² + y²)
Magnitude
⃗a + ⃗b = (a₁+b₁, a₂+b₂)
Addition
k · ⃗v = (kx, ky)
Scalar multiplication
⃗a · ⃗b = a₁b₁ + a₂b₂
Dot product
cos θ = (⃗a·⃗b)/(|⃗a||⃗b|)
Angle between vectors