Complete K-12 Math Formulas

📚 Complete K-12 Math Formulas 📚

Your Ultimate Mathematical Reference Guide

🔍 Quick Search

Find any formula instantly by typing keywords...

📑 Table of Contents

1. Basic Arithmetic 2. Algebra 3. Geometry 4. Trigonometry 5. Coordinate Geometry 6. Sequences & Series 7. Exponentials & Logs 8. Statistics & Probability 9. Pre-Calculus & Calculus

1. Basic Arithmetic

Order of Operations

PEMDAS / BODMAS
Parentheses/Brackets → Exponents/Orders → MD Multiplication & Division (left to right) → AS Addition & Subtraction (left to right)

Basic Operations

Addition

a + b = c

Commutative: a + b = b + a

Subtraction

a − b = c

a = c + b

Multiplication

a × b = c

Commutative: a × b = b × a

Division

ab = c

a = b × c (b ≠ 0)

Properties

a + 0 = a (Identity)

a × 1 = a (Identity)

a × 0 = 0 (Zero Property)

(a + b) + c = a + (b + c) (Associative)

a(b + c) = ab + ac (Distributive)

Fractions

ab + cd = ad + bcbd

ab × cd = acbd

ab ÷ cd = ab × dc = adbc

Percentages

Percentage = PartWhole × 100

Part = Percentage × Whole100

2. Algebra

Exponent Rules

a^m × aⁿ = a^m+n

a^maⁿ = a^m−n

(a^m)ⁿ = a^mn

(ab)ⁿ = aⁿbⁿ

a⁰ = 1 (a ≠ 0)

a⁻ⁿ = 1aⁿ

a^1/n = aⁿ

Algebraic Identities

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

a² − b² = (a + b)(a − b)

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a − b)³ = a³ − 3a²b + 3ab² − b³

a³ + b³ = (a + b)(a² − ab + b²)

a³ − b³ = (a − b)(a² + ab + b²)

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

Quadratic Formula

For equation: ax² + bx + c = 0

x = −b ± b² − 4ac 2a

Discriminant: Δ = b² − 4ac

Linear Equations

Slope-intercept form: y = mx + b

Point-slope form: y − y₁ = m(x − x₁)

Standard form: Ax + By = C

Slope: m = y₂ − y₁x₂ − x₁

Systems of Equations

Cramer's Rule: x = D_xD, y = D_yD

Polynomials

Sum of roots: r₁ + r₂ = −ba

Product of roots: r₁ × r₂ = ca

Remainder Theorem: f(a) = remainder when f(x) ÷ (x − a)

Factor Theorem: If f(a) = 0, then (x − a) is a factor

3. Geometry

2D Shapes - Area

Shape	Area Formula	Perimeter
Square	A = s²	P = 4s
Rectangle	A = l × w	P = 2(l + w)
Triangle	A = ½bh	P = a + b + c
Circle	A = πr²	C = 2πr = πd
Parallelogram	A = bh	P = 2(a + b)
Trapezoid	A = ½(a + b)h	P = a + b + c + d
Rhombus	A = ½d₁d₂	P = 4s
Ellipse	A = πab	C ≈ π(a + b)

Triangles - Special Formulas

Pythagorean Theorem: a² + b² = c²

Heron's Formula: A = s(s−a)(s−b)(s−c)

where s = (a + b + c)/2 (semi-perimeter)

Triangle Area = 12ab sin C

Angle Sum: A + B + C = 180°

3D Solids - Volume & Surface Area

Solid	Volume	Surface Area
Cube	V = s³	SA = 6s²
Rectangular Prism	V = lwh	SA = 2(lw + lh + wh)
Sphere	V = (4/3)πr³	SA = 4πr²
Cylinder	V = πr²h	SA = 2πr² + 2πrh
Cone	V = (1/3)πr²h	SA = πr² + πrl
Pyramid	V = (1/3)Bh	SA = B + ½Pl

Circle Properties

Arc Length = rθ (θ in radians)

Sector Area = 12r²θ

Chord Length = 2r sin(θ2)

Polygon Formulas

Interior Angle Sum = (n − 2) × 180°

Each Interior Angle = (n − 2) × 180°n

Exterior Angle Sum = 360°

Number of Diagonals = n(n − 3)2

4. Trigonometry

Basic Trig Ratios

sin θ = oppositehypotenuse

cos θ = adjacenthypotenuse

tan θ = oppositeadjacent

csc θ = 1sin θ

sec θ = 1cos θ

cot θ = 1tan θ

Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Angle Sum & Difference

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = tan A ± tan B1 ∓ tan A tan B

Double Angle Formulas

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

tan 2θ = 2 tan θ1 − tan²θ

Half Angle Formulas

sin θ2 = ±1 − cos θ2

cos θ2 = ±1 + cos θ2

Law of Sines

asin A = bsin B = csin C

Law of Cosines

c² = a² + b² − 2ab cos C

b² = a² + c² − 2ac cos B

a² = b² + c² − 2bc cos A

Special Angles

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

5. Coordinate Geometry

Distance & Midpoint

Distance: d = (x₂−x₁)² + (y₂−y₁)²

Midpoint: M = (x₁+x₂2, y₁+y₂2)

Slope & Lines

Slope: m = y₂ − y₁x₂ − x₁ = riserun

Parallel lines: m₁ = m₂

Perpendicular lines: m₁ × m₂ = −1

Conic Sections

Circle

(x − h)² + (y − k)² = r²

Center: (h, k), Radius: r

Parabola

y = a(x − h)² + k

x = a(y − k)² + h

Vertex: (h, k)

Ellipse

(x−h)²a² + (y−k)²b² = 1

Center: (h, k)

Hyperbola

(x−h)²a² − (y−k)²b² = 1

Center: (h, k)

Vectors

Magnitude: |v| = x² + y²

Dot Product: a · b = a₁b₁ + a₂b₂

a · b = |a| |b| cos θ

6. Sequences & Series

Arithmetic Sequences

a_n = a₁ + (n − 1)d

S_n = n2(a₁ + a_n)

S_n = n2[2a₁ + (n−1)d]

d = common difference

Geometric Sequences

a_n = a₁ × rⁿ⁻¹

S_n = a₁ 1 − rⁿ1 − r

S_∞ = a₁1 − r (|r| < 1)

r = common ratio

Special Series

1 + 2 + 3 + ... + n = n(n+1)2

1² + 2² + 3² + ... + n² = n(n+1)(2n+1)6

1³ + 2³ + 3³ + ... + n³ = [n(n+1)2]²

7. Exponentials & Logarithms

Logarithm Rules

log_b(xy) = log_bx + log_by

log_b(xy) = log_bx − log_by

log_b(xⁿ) = n log_bx

log_bb = 1

log_b1 = 0

b^log_bx = x

Change of Base

log_bx = log_axlog_ab

log_bx = ln xln b

Natural Logarithm

ln(e) = 1

e^{ln x} = x

ln(e^x) = x

e ≈ 2.71828...

Exponential Growth & Decay

A = P(1 + r)^t

A = Pe^rt

A = P(1 + rn)^nt

Compound Interest Formula

8. Statistics & Probability

Measures of Central Tendency

Mean: μ = Σxn

Median: Middle value when ordered

Mode: Most frequent value

Measures of Spread

Range = Maximum − Minimum

Variance: σ² = Σ(x − μ)²n

Standard Deviation: σ = σ²

Probability

P(A) = Number of favorable outcomesTotal number of outcomes

P(A or B) = P(A) + P(B) − P(A and B)

P(A and B) = P(A) × P(B|A)

P(not A) = 1 − P(A)

Combinations & Permutations

Permutations: P(n,r) = n!(n−r)!

Combinations: C(n,r) = n!r!(n−r)!

n! = n × (n−1) × (n−2) × ... × 2 × 1

Normal Distribution

Z-score: z = x − μσ

Empirical Rule:
68% within 1σ
95% within 2σ
99.7% within 3σ

9. Pre-Calculus & Calculus Basics

Limits

lim_x→a [f(x) + g(x)] = lim_x→a f(x) + lim_x→a g(x)

lim_x→a [f(x) × g(x)] = lim_x→a f(x) × lim_x→a g(x)

lim_x→a c × f(x) = c × lim_x→a f(x)

Derivatives - Basic Rules

ddx(c) = 0

ddx(xⁿ) = nxⁿ⁻¹

ddx[f(x) + g(x)] = f'(x) + g'(x)

ddx[cf(x)] = c × f'(x)

Product & Quotient Rules

(fg)' = f'g + fg'

(fg)' = f'g − fg'g²

Chain Rule

ddx[f(g(x))] = f'(g(x)) × g'(x)

Common Derivatives

ddx(sin x) = cos x

ddx(cos x) = −sin x

ddx(e^x) = e^x

ddx(ln x) = 1x

Integrals - Basic Rules

∫ xⁿ dx = xⁿ⁺¹n+1 + C (n ≠ −1)

∫ e^x dx = e^x + C

∫ 1x dx = ln|x| + C

∫ sin x dx = −cos x + C

∫ cos x dx = sin x + C

Fundamental Theorem of Calculus

∫_a^b f(x) dx = F(b) − F(a)

where F'(x) = f(x)

↑