Math Formulas Collection

Search and browse essential math formulas from elementary to high school

27 formulas

Triangle Area

Elem

S = (1/2) × a × h

Area of triangle with base a and height h

Example: a=6, h=4 → S = (1/2)×6×4 = 12

Rectangle Area

Elem

S = a × b

Area of rectangle with width a and height b

Example: a=5, b=3 → S = 15

Circle Area

Elem

S = π × r²

Area of circle with radius r

Example: r=3 → S = π×9 ≈ 28.27

Circle Circumference

Elem

C = 2 × π × r

Circumference of circle with radius r

Example: r=5 → C = 2π×5 ≈ 31.42

Mean (Average)

Elem

평균 = (a₁+a₂+...+aₙ) / n

Sum of n values divided by n

Example: (3+5+7)/3 = 5

Percentage

Elem

% = (부분/전체) × 100

Ratio of part to whole as percentage

Example: 30/120 × 100 = 25%

Distributive Law

Elem

a × (b + c) = a×b + a×c

Distributive property of multiplication

Example: 3×(4+5) = 3×4+3×5 = 27

Pythagorean Theorem

Mid

a² + b² = c²

In a right triangle, hypotenuse² = sum of other sides²

Example: 3²+4² = 9+16 = 25 = 5²

Linear Equation

Mid

ax + b = 0 → x = -b/a

Solution of linear equation

Example: 2x+6=0 → x=-3

Quadratic Formula

Mid

x = (-b ± √(b²-4ac)) / 2a

Roots of ax²+bx+c=0

Example: x²-5x+6=0 → x=(5±1)/2 → x=3,2

Difference of Squares

Mid

a² - b² = (a+b)(a-b)

Factor difference of two squares

Example: x²-9 = (x+3)(x-3)

Perfect Square

Mid

(a±b)² = a² ± 2ab + b²

Perfect square expansion

Example: (x+3)² = x²+6x+9

System of Equations

Mid

a₁x+b₁y=c₁, a₂x+b₂y=c₂

Solve by elimination or substitution

Example: x+y=5, x-y=1 → x=3, y=2

Variance & Std Dev

Mid

σ² = Σ(xᵢ-x̄)²/n, σ = √σ²

Measures of data spread

Example: Data {2,4,6} → x̄=4, σ²=8/3≈2.67

Probability

Mid

P(A) = n(A) / n(S)

Probability = favorable outcomes / total outcomes

Example: Even dice: P = 3/6 = 1/2

Trig Functions

High

sinθ, cosθ, tanθ = sinθ/cosθ

sin²θ + cos²θ = 1

Example: sin30°=1/2, cos60°=1/2, tan45°=1

Trig Addition

High

sin(α±β) = sinα·cosβ ± cosα·sinβ

cos(α±β) = cosα·cosβ ∓ sinα·sinβ

Example: sin75°=sin(45°+30°)=(√6+√2)/4

Basic Derivatives

High

(xⁿ)' = n·xⁿ⁻¹

(const)' = 0, (kf)' = kf', (f±g)' = f'±g'

Example: (x³)' = 3x², (5x²)' = 10x

Product Rule

High

(fg)' = f'g + fg'

Derivative of product of two functions

Example: (x·sinx)' = sinx + x·cosx

Basic Integrals

High

∫xⁿdx = xⁿ⁺¹/(n+1) + C

When n≠-1 (indefinite integral)

Example: ∫x²dx = x³/3 + C

Arithmetic Series

High

Sₙ = n(a₁+aₙ)/2 = n(2a₁+(n-1)d)/2

Sum of arithmetic sequence with first term a₁, common difference d

Example: 1+2+...+100 = 100×101/2 = 5050

Geometric Series

High

Sₙ = a₁(rⁿ-1)/(r-1)

Sum of geometric sequence (r≠1)

Example: 1+2+4+8 = 1×(2⁴-1)/(2-1) = 15

Logarithm Laws

High

log(ab) = loga + logb

log(a/b) = loga - logb, log(aⁿ) = n·loga

Example: log100 = log(10²) = 2

Combination (nCr)

High

ₙCᵣ = n! / (r!(n-r)!)

Number of ways to choose r from n (unordered)

Example: ₅C₂ = 5!/(2!3!) = 10

Permutation (nPr)

High

ₙPᵣ = n! / (n-r)!

Number of ways to arrange r from n (ordered)

Example: ₅P₂ = 5!/3! = 20

Binomial Theorem

High

(a+b)ⁿ = Σ ₙCₖ · aⁿ⁻ᵏ · bᵏ

Expansion of binomial power

Example: (a+b)³ = a³+3a²b+3ab²+b³

Limits

High

lim(x→a) f(x) = L

Value of f(x) as x approaches a

Example: lim(x→0) sinx/x = 1

FAQ

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30+ essential formulas from elementary (area, perimeter), middle school (factoring, Pythagorean), to high school (trigonometry, calculus, sequences, probability).

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