Math Reference Sheet- Essential Formulas and Rules

Why You Need This Math Reference Sheet

Math tests don't wait for you to "figure it out." You either know the formula or you don't. This reference sheet gives you the formulas and rules you actually need, organized so you can find them fast.

Bookmark this page. Use it during homework. Drill these into your memory for exams.

Basic Arithmetic Formulas

These are the foundation. If these feel shaky, fix that now before anything else.

Order of Operations (PEMDAS/BODMAS)

Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

Calculate left to right for multiplication/division and addition/subtraction. Nothing else.

Percentage Calculations

• Find X% of Y: (X ÷ 100) × Y
• What percent is X of Y: (X ÷ Y) × 100
• Percentage increase: ((New − Old) ÷ Old) × 100
• Percentage decrease: ((Old − New) ÷ Old) × 100

Exponent Rules

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^m×n
• a⁰ = 1 (when a ≠ 0)
• a^-n = 1 ÷ aⁿ

Square Root Basics

√(a × b) = √a × √b

√(a ÷ b) = √a ÷ √b

√a² = |a| (the absolute value)

Algebra Essentials

Factoring Formulas

• Difference of squares: a² − b² = (a + b)(a − b)
• Perfect square trinomials: a² + 2ab + b² = (a + b)²
• a² − 2ab + b² = (a − b)²
• Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)
• Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)

Quadratic Formula

For ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) ÷ 2a

The discriminant (b² − 4ac) tells you what you're dealing with:

• Positive = two real solutions
• Zero = one repeated solution
• Negative = no real solutions

Slope and Linear Equations

Slope formula: m = (y₂ − y₁) ÷ (x₂ − x₁)

Point-slope form: y − y₁ = m(x − x₁)

Slope-intercept form: y = mx + b (m = slope, b = y-intercept)

Standard form: Ax + By = C

Logarithm Rules

• log_b(xy) = log_bx + log_by
• log_b(x ÷ y) = log_bx − log_by
• log_b(xⁿ) = n × log_bx
• log_b(b) = 1
• log_b(1) = 0
• log_b(bⁿ) = n

The change of base formula: log_bx = (log_ax) ÷ (log_ab)

Geometry Formulas

2D Shapes

3D Shapes

Pythagorean Theorem

a² + b² = c²

c is always the hypotenuse (longest side, opposite the right angle). This only works for right triangles.

Trigonometry Basics

Sine, Cosine, Tangent

For a right triangle with angle θ:

• sin(θ) = opposite ÷ hypotenuse
• cos(θ) = adjacent ÷ hypotenuse
• tan(θ) = opposite ÷ adjacent

Remember it as SOH CAH TOA.

Reciprocal Trig Functions

• csc(θ) = 1 ÷ sin(θ) = hypotenuse ÷ opposite
• sec(θ) = 1 ÷ cos(θ) = hypotenuse ÷ adjacent
• cot(θ) = 1 ÷ tan(θ) = adjacent ÷ opposite

Pythagorean Identities

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

Special Angles (Radians)

Law of Sines and Cosines

Law of Sines: a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C)

Use this when you know two angles and one side, or two sides and an angle opposite one of them.

Law of Cosines: c² = a² + b² − 2ab·cos(C)

Use this when you know all three sides and want to find an angle, or two sides and the angle between them.

Calculus Fundamentals

Derivative Rules

• Power rule: d÷dx(xⁿ) = n·x^n-1
• Constant rule: d÷dx(c) = 0
• Constant multiple: d÷dx(c·f(x)) = c·f'(x)
• Sum rule: d÷dx(f + g) = f' + g'
• Product rule: d÷dx(f·g) = f'g + fg'
• Quotient rule: d÷dx(f÷g) = (f'g − fg') ÷ g²
• Chain rule: d÷dx(f(g(x))) = f'(g(x))·g'(x)

Derivatives of Common Functions

• d÷dx(sin x) = cos x
• d÷dx(cos x) = −sin x
• d÷dx(tan x) = sec² x
• d÷dx(eˣ) = eˣ
• d÷dx(ln x) = 1 ÷ x
• d÷dx(aˣ) = aˣ · ln a

Integral Rules

• Power rule: ∫xⁿ dx = (xⁿ⁺¹ ÷ (n + 1)) + C (n ≠ −1)
• Constant: ∫c dx = cx + C
• Sum rule: ∫(f + g) dx = ∫f dx + ∫g dx
• u-substitution: Reverse of chain rule
• Integration by parts: ∫u dv = uv − ∫v du

Common Integrals

• ∫sin x dx = −cos x + C
• ∫cos x dx = sin x + C
• ∫eˣ dx = eˣ + C
• ∫(1 ÷ x) dx = ln|x| + C
• ∫sec² x dx = tan x + C

Statistics Basics

Measures of Central Tendency

• Mean: Sum of all values ÷ number of values
• Median: Middle value when data is ordered (or average of two middle values)
• Mode: Most frequent value

Variance and Standard Deviation

Population variance (σ²): Σ(x − μ)² ÷ N

Sample variance (s²): Σ(x − x̄)² ÷ (n − 1)

Standard deviation: √variance

Use sample variance (÷ n−1) when working with sample data. Use population variance (÷ N) when you have every data point.

Z-Scores

z = (x − μ) ÷ σ

A z-score tells you how many standard deviations a value is from the mean. Positive = above mean. Negative = below mean.

Probability Rules

• Complement: P(not A) = 1 − P(A)
• Addition (OR): P(A or B) = P(A) + P(B) − P(A and B)
• Multiplication (AND): P(A and B) = P(A) × P(B|A)
• Independence: If A and B are independent, P(A and B) = P(A) × P(B)

How to Use This Reference Sheet Effectively

Having this list doesn't help if you can't access it when you need it.

Step 1: Identify Your Gaps

Go through each section. Mark what you know cold. Mark what you sort of remember. Mark what looks completely foreign. Focus your study time on the foreign and sort-of sections.

Step 2: Practice the Derivations

Don't just memorize. Understand why the quadratic formula works. Derive the Pythagorean identities from sin² + cos² = 1. The formulas stick longer when you understand them.

Step 3: Drill Until It's Automatic

For exams, you need these accessible without thinking. Flashcards work. Repetition works. Practice problems work. Pick your method and put in the reps.

Step 4: Check Your Work

Plug answers back into the original equation. Use the discriminant to verify quadratic solutions. Check that your trig answer is in the right quadrant. Small checks catch big mistakes.

Quick Reference Comparison

This covers the formulas you'll encounter most often in algebra through introductory calculus and statistics. Work through the sections relevant to your current coursework. The rest will be here when you need it.