shape: Square

formula: A = l²

symbol: l

description: length of side


shape: Rectangle

formula: A = w × h

symbol: w

description: width

symbol: h

description: height


shape: Triangle

symbol: b

description: base

description: height


shape: Rhombus

symbol: D

description: large diagonal

symbol: d

description: small diagonal


shape: Trapezoid

symbol: B

description: large side

description: small side

description: height


shape: Regular polygon

symbol: P

description: perimeter

symbol: a

description: apothem


shape: Circle

Area

formula: A = πr²

Perimeter

formula: P = 2πr

symbol: r

description: radius


shape: Cone (lateral surface)

formula: A = πr × s

description: radius

symbol: s

description: slant height


shape: Sphere (surface area)

formula: A = 4πr²

description: radius


category: Volumes

topic: Cube

formula: V = s³

description: side


topic: Parallelepiped

formula: V = l × w × h

description: length

description: width

description: height


topic: Regular prism

formula: V = b × h

description: base

description: height


topic: Cylinder

formula: V = πr² × h

description: radius

description: height


topic: Cone (or pyramid)

formula: V = ⅓b × h

description: base

description: height


topic: Sphere

formula: V = ⁴⁄₃πr³

description: radius


category: Functions

topic: Directly Proportional

expression: y = kx

expression: k = y/x


topic: Inversely Proportional

expression: y = k/x

expression: k = yx


topic: Concavity

result: Concave up

result: Concave down


topic: Zero-product property

rule: A×B=0 ⇔ A=0 ∨ B=0


topic: Binomial theorem

formula: (x+y)ⁿ = Σ[nCk·xⁿ⁻ᵏ·yᵏ]


category: Exponents

rule: Product (same base)

formula: aᵐ × aⁿ = aᵐ⁺ⁿ

example: 3⁵ × 3² = 3⁷


rule: Product (same exponent)

formula: aᵐ × bᵐ = (a×b)ᵐ

example: 3⁵ × 2⁵ = 6⁵


rule: Quotient (same base)

formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

example: 3⁷ ÷ 3² = 3⁵


rule: Quotient (same exponent)

formula: aᵐ ÷ bᵐ = (a÷b)ᵐ


rule: Power of power

formula: (aᵐ)ᵖ = aᵐˣᵖ


rule: Negative exponents

formula: a⁻ⁿ = (1/a)ⁿ

examples:
3⁻² = (⅓)²
(⅔)⁻⁴ = (³⁄₂)⁴


rule: Fractional exponents

formula: aᵖ/ᵃ = ᵃ√(aᵖ)

example: 2⁴⁄₃ = ³√(2⁴)


category: Radicals

operation: Multiplication

rule: ⁿ√x × ⁿ√y = ⁿ√(x×y)

example: ³√2 × ³√5 = ³√10


operation: Exponents

rule: (ⁿ√x)ᵖ = ⁿ√xᵖ

example: (³√2)³ = ³√8


operation: Nested Radicals

rule: ⁿ√(ᵐ√x) = ᵐⁿ√x

example: ³√5 = ³√5 (identity)


operation: Exponentiation

form: Radical to Exponent

rule: ⁿ√aᵐ = aᵐ⁄ₙ

example: ³√4⁵ = 4⁵⁄₃

form: Radical Power

rule: (ⁿ√a)ⁿ = a


operation: Simplification

rule: (ⁿ√a)ᵐ = ⁿ√aᵐ

example: (³√4)⁵ = ³√4⁵


category: Geometry

theorem: Euler's Polyhedral Formula

symbol: F

description: Faces

symbol: V

description: Vertices

symbol: E

description: Edges


theorem: Interior Angles

symbol: n

description: Number of sides


theorem: Pythagorean

formula: H² = C₁² + C₂²

symbol: H

description: Hypotenuse

description: Legs

note: Right triangles only


operation: Distance

formula: AB = √[(x₁-x₂)² + (y₁-y₂)²]

solution: √[(8-4)² + (2+1)²] = 5


form: Slope-Intercept

equation: y = mx + b

symbol: m

description: Slope

description: Y-intercept


form: Vector Equation

equation: (x,y,z) = (x₀,y₀,z₀) + k(u₁,u₂,u₃)

components:
Point: (x₀,y₀,z₀)
Direction vector: u(u₁,u₂,u₃)


form: Cartesian Line

equation: (x-x₀)/u₁ = (y-y₀)/u₂ = (z-z₀)/u₃


form: Parametric Line

equations:
x = x₀ + ku₁
y = y₀ + ku₂
z = z₀ + ku₃


form: Plane Equation

standard: n₁(x-x₀) + n₂(y-y₀) + n₃(z-z₀) = 0

scalar: n₁x + n₂y + n₃z + d = 0


equation: (x-x₀)² + (y-y₀)² = r²

parameters:
Center: (x₀,y₀)
Radius: r


shape: Sphere

equation: (x-x₀)² + (y-y₀)² + (z-z₀)² = r²


shape: Ellipse

parameters:
Center: (h,k)
Semi-axes: a, b


category: Logic

property: Associativity

rules:
(p∧q)∧r ⇔ p∧(q∧r)
(p∨q)∨r ⇔ p∨(q∨r)


property: Identity

rules:
p∧T ⇔ p
p∨F ⇔ p
p∧F ⇔ F
p∨T ⇔ T


property: Distributivity

rules:
p∧(q∨r) ⇔ (p∧q)∨(p∧r)
p∨(q∧r) ⇔ (p∨q)∧(p∨r)


rule: Implication

equivalences:
p⇒q ⇔ ¬p∨q
¬(p⇒q) ⇔ p∧¬q


rule: De Morgan's

laws:
¬(p∧q) ⇔ ¬p∨¬q
¬(p∨q) ⇔ ¬p∧¬q


category: Vectors

concept: Notation

definition: AB→ = B - A = (b₁-a₁, b₂-a₂)


concept: Magnitude

formula: ‖u→‖ = √(u₁² + u₂²)


concept: Magnitude Square

identity: (u→)² = ‖u→‖²


operation: Point-Vector Addition

formula: A + u→ = (a₁+u₁, a₂+u₂)


operation: Vector Addition

formula: u→ + v→ = (u₁+v₁, u₂+v₂)


operation: Scalar Multiplication

formula: k×u→ = (k×u₁, k×u₂)


operation: Dot Product

formulas:
u→·v→ = u₁v₁ + u₂v₂
‖u→‖×‖v→‖×cos(θ)


concept: Angle Between Vectors

formula: cosα = |u→·v→|/(‖u→‖×‖v→‖)

note: Extends to 3D with third coordinate


category: Statistics

rule: Summation

formulas:
∑ᵢ₌ₚⁿλ = (n-p+1)λ
∑λxᵢ = λ∑xᵢ
∑(xᵢ+yᵢ) = ∑xᵢ + ∑yᵢ
∑ᵢ₌₁ⁿxᵢ = ∑ᵢ₌₁ᵖxᵢ + ∑ᵢ₌ₚ₊₁ⁿxᵢ


concept: Data Description

terms:
Sample: x = (x₁,x₂,...,xₙ)
Size: N
Abs. Freq: nᵢ
Rel. Freq: fᵢ = nᵢ/N
Cum. Freq: Nᵢ, Fᵢ


calculation: Mean

formulas:
Ungrouped: x̄ = (∑xᵢ)/N
Weighted: x̄ = (∑nᵢxᵢ)/N
Grouped: x̄ = ∑fᵢxᵢ


calculation: Median

rules:
Odd N: Me = xₖ where k=(N+1)/2
Even N: Me = (xₖ+xₖ₊₁)/2 where k=N/2


property: Sum of Deviations

theorem: ∑(xᵢ - x̄) = 0


calculation: Sum of Squares

formulas:
SSₓ = ∑(xᵢ - x̄)²
Ungrouped: SSₓ = ∑xᵢ² - kx̄²
Grouped: SSₓ = ∑(xᵢ - x̄)²nᵢ


category: Sequences

type: Arithmetic

properties: Monotonicity: Increasing if r>0, Decreasing if r<0


type: Geometric

properties: Monotonicity: Increasing if u₁>0∧r>1, Decreasing if u₁<0∧r>1, Non-monotonic if r<0


application: Simple Interest

formula: FV = P×(1 + r×t)

symbol: FV

description: Future Value

description: Principal

description: Interest rate

symbol: t

description: Time


application: Compound Interest

formula: FV = P×(1 + r)ᵗ


category: Calculus

concept: Average Rate of Change

description: Slope of secant line between points [a,b]


rule: Exponential

formulas:
(aᵘ)' = u'×aᵘ×ln(a)
(eᵘ)' = u'×eᵘ

examples:
(7ⁿ²)' = 3×7ⁿ²×ln7
(e²ˣ)' = 2e²ˣ


rule: Trigonometric

formulas:
(sin u)' = u'cos u
(cos u)' = -u'sin u
(tan u)' = u'/cos² u


rule: Logarithmic

formulas:
(logₐu)' = u'/(u ln a)
(ln u)' = u'/u


category: Probability

law: Commutative

rules:
A∪B = B∪A
A∩B = B∩A


law: Associative

rules:
A∪(B∪C) = (A∪B)∪C
A∩(B∩C) = (A∩B)∩C


rules:
A∪∅ = A
A∩E = A
A∪E = E
A∩∅ = ∅


law: Distributive

rules: A∪(B∩C)=(A∪B)∩(A∪C) A∩(B∪C)=(A∩B)∪(A∩C)


law: De Morgan's

rules:
A̅∩B̅ = A̅∪B̅
A̅∪B̅ = A̅∩B̅


rule: Laplace

formula: P(A) = (# favorable outcomes)/(# total outcomes)


rule: Union

formula: P(A∪B) = P(A)+P(B)-P(A∩B)


rule: Conditional

formula: P(A|B) = P(A∩B)/P(B)


rule: Independence

conditions: P(A|B)=P(A) P(A∩B)=P(A)×P(B)


distribution: Properties

Mean

formula: μ = Σxᵢpᵢ

Standard Deviation

formula: σ = √[Σpᵢ(xᵢ-μ)²]


category: Logarithms

definition: logₐb=x ⇔ b=aˣ

example: 3ˣ=15 ⇔ x=log₃15


rule: Product

formula: logₐ(u×v)=logₐu + logₐv

example: log₆10 + log₆2 = log₆20


rule: Quotient

formula: logₐ(u/v)=logₐu - logₐv

example: log₄9 - log₄3 = log₄3


rule: Power

formula: logₐuᵛ = v×logₐu

example: log₄36 = 2×log₄6


rule: Change of Base

formula: logₐu = logᵦu/logᵦa

application: log₄5 × log₅6 = log₄6


type: Limit

expression: lim(x→∞) logₐx/x = 0

conditions: a>1, a∈ℝ


expression: lim(uₙ→∞) (1 + k/uₙ)ᵘⁿ = eᵏ


type: Power Rule

formula: ∫u(x)ᵃ·u'(x) dx = u(x)ᵃ⁺¹/(a+1) + c


type: Logarithmic

formula: ∫u'(x)/u(x) dx = ln|u(x)| + c

constraint: c∈ℝ


type: Exponential

formula: ∫eᵘ⁽ˣ⁾·u'(x) dx = eᵘ⁽ˣ⁾ + c


type: Trigonometric

formulas:
∫sin(u(x))·u'(x) dx = -cos(u(x)) + c
∫cos(u(x))·u'(x) dx = sin(u(x)) + c


type: Linearity

rules:
∫(f+g) dx = ∫f dx + ∫g dx
∫k·f(x) dx = k∫f(x) dx


type: Definite Integral

properties:
∫ₐᵇ f(x) dx = -∫ₐᵇ f(x) dx
∫ₐᵃ f(x) dx = 0
∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx


type: Barrow's Rule

formula: ∫ₐᵇ f(x) dx = F(b)-F(a)

description: F is primitive of f on [a,b]


category: Complex Numbers

form: Algebraic

definition: z = a + bi

operations:
Conjugate: z̄ = a - bi
Negation: -z = -a - bi
Equality: z₁=z₂ ⇔ a=c ∧ b=d


formula: (a+bi)×(c+di) = (ac-bd)+(ad+bc)i


operation: Division

formula: (a+bi)/(c+di) = (ac+bd)/(c²+d²) + (bc-ad)i/(c²+d²)


operation: Inverse

formulas:
z⁻¹ = 1/z
z⁻¹ = z̄/|z|²


property: Magnitude

formulas:
|z| = √(a²+b²)
|z|² = z·z̄


form: Polar

conversion:
|z| = √(a²+b²)
θ = arctan(b/a)
z = |z|(cosθ + isinθ)


form: Exponential

operations:
Conjugate: z̄ = |z|e^(-iθ)
Negation: -z = |z|e^(i(θ+π))


operation: Polar Multiplication

formula: z₁×z₂ = |z₁||z₂|e^(i(θ₁+θ₂))


formula: zⁿ = |z|ⁿe^(inθ)


operation: Roots

formula: ⁿ√z = ⁿ√|z|e^(i(θ+2πk)/n


category: Algebra

expression_type: Polynomial

definition: Multi-term expression with non-negative exponents


formula_type: Basic Identities

formulas:
(a+b)² = a² + 2ab + b²
(a-b)² = a² - 2ab + b²
a² - b² = (a-b)(a+b)
a² + b² = (a-b)² + 2ab


formula_type: Trinomial Expansion

formulas:
(a+b+c)² = a²+b²+c²+2ab+2ac+2bc
(a-b-c)² = a²+b²+c²-2ab-2ac+2bc


formula_type: Cubic Identities

formulas:
(a+b)³ = a³ + 3a²b + 3ab² + b³
(a-b)³ = a³ - b³ + 3ab² - 3a²b
a³-b³ = (a²+ab+b²)(a-b)
a³+b³ = (a²-ab+b²)(a+b)


formula_type: Higher Degree

formulas:
a⁴ - b⁴ = (a²+b²)(a²-b²)
a⁵ - b⁵ = (a-b)(a⁴+a³b+a²b²+ab³+b⁴)
a⁵ + b⁵ = (a+b)(a⁴-a³b+a²b²-ab³+b⁴)


formula_type: Quadratic Formula

application: Solves ax²+bx+c=0

note: Discriminant Δ = b²-4ac determines nature of roots


formula_type: Distance Formula

formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

application: Distance between points (x₁,y₁) and (x₂,y₂)


formula_type: Slope Formula

formula: m = (y₂-y₁)/(x₂-x₁)

application: Slope of line through (x₁,y₁) and (x₂,y₂)


formula_type: Exponential Growth/Decay

formulas:
Growth: A = A₀e^(rt)
Decay: A = A₀e^(-rt)

variables:
A₀=initial amount
r=rate
t=time


formula_type: Logarithmic

formula: log_b(y) = x ⇔ y = b^x

common_bases:
Base 10
Base e (natural log)


formula_type: Pythagorean Theorem

formula: c² = a² + b²

application: Right triangle hypotenuse calculation


formula_type: Laws of Exponents

rules:
aᵐ×aⁿ = aᵐ⁺ⁿ
(ab)ᵐ = aᵐbᵐ
(aᵐ)ⁿ = aᵐⁿ


formula_type: Fractional Exponents

rules:
a⁰ = 1
aᵐ/aⁿ = aᵐ⁻ⁿ
a⁻ᵐ = 1/aᵐ
aᵐ = 1/a⁻ᵐ


category: Trigonometry

type: Function Relationships

formulas:
sinθ = 1/cscθ
cosθ = 1/secθ
tanθ = 1/cotθ
cscθ = 1/sinθ
secθ = 1/cosθ
cotθ = 1/tanθ
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ


type: Pythagorean Identities

formulas:
sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
cot²θ + 1 = csc²θ


type: Double Angle

formulas: cos2θ = cos²θ - sin²θ = 1-2sin²θ = 2cos²θ-1


type: Triple Angle

formulas:
sin3θ = 3sinθ - 4sin³θ
cos3θ = 4cos³θ - 3cosθ
tan3θ = (3tanθ - tan³θ)/(1-3tan²θ)


type: Opposite Angle

formulas:
sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
csc(-θ) = -cscθ


type: Half Angle

formulas: tan(θ/2) = ±√[(1-cosθ)/(1+cosθ)] = (1-cosθ)/sinθ = sinθ/(1+cosθ)


type: Power Reducing

formulas:
sin²θ = (1-cos2θ)/2
cos²θ = (1+cos2θ)/2
tan²θ = (1-cos2θ)/(1+cos2θ)


type: Law of Cosines

formulas:
a² = b² + c² - 2bc·cosA
b² = a² + c² - 2ac·cosB
c² = a² + b² - 2ab·cosC


type: Cofunction

formulas:
sinθ = cos(π/2-θ)
cosθ = sin(π/2-θ)
tanθ = cot(π/2-θ)
cotθ = tan(π/2-θ)
secθ = csc(π/2-θ)
cscθ = sec(π/2-θ)

note: Valid in Quadrant I


type: Angle Addition

formulas:
sin(A±B) = sinAcosB ± cosAsinB
cos(A±B) = cosAcosB ∓ sinAsinB
tan(A+B) = (tanA+tanB)/(1-tanAtanB)
tan(A-B) = (tanA-tanB)/(1+tanAtanB)


type: Product-to-Sum

formulas:
sinA·sinB = ½[cos(A-B)-cos(A+B)]
cosA·cosB = ½[cos(A-B)+cos(A+B)]
sinA·cosB = ½[sin(A+B)+sin(A-B)]
cosA·sinB = ½[sin(A+B)-sin(A-B)]


type: Sum-to-Product

formulas:
sinA±sinB = 2sin[(A±B)/2]cos[(A∓B)/2]
cosA+cosB = 2cos[(A+B)/2]cos[(A-B)/2]
cosA-cosB = -2sin[(A+B)/2]sin[(A-B)/2]