SAT Math Cheatsheet: Complete Formula Reference for Test Day Success
Every point matters on the SAT Math section. This comprehensive cheatsheet from SATHELP24x7.com compiles every essential formula, theorem, and mathematical relationship you need to know for the Digital SAT. While the test provides a basic reference sheet during the exam, relying on it wastes precious time—memorizing these formulas beforehand allows instant application.
This cheatsheet organizes formulas by mathematical topic, includes memory aids for complex equations, and highlights high-frequency formulas that appear most often on actual SAT exams. Whether you're in final preparation or creating your study materials, this complete reference ensures you have every mathematical tool needed for success.
Quick Note: This cheatsheet includes formulas NOT provided on the SAT reference sheet. The test provides basic geometry formulas, but algebra, advanced math, statistics, and most critical formulas must be memorized. Print this guide, review it daily, and practice applying each formula through problem-solving—application creates durable memory.
Algebra Formulas (35% of SAT Math)
Linear Equations & Functions
Core Linear Formulas
Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Calculates steepness of line between two points \((x_1, y_1)\) and \((x_2, y_2)\)
Slope-Intercept Form
\( y = mx + b \)
Where \(m\) = slope and \(b\) = y-intercept
Point-Slope Form
\( y - y_1 = m(x - x_1) \)
Used when you know slope \(m\) and one point \((x_1, y_1)\)
Standard Form
\( Ax + By = C \)
General form of linear equation where \(A, B, C\) are constants
Parallel Lines
\( m_1 = m_2 \)
Parallel lines have equal slopes
Perpendicular Lines
\( m_1 \cdot m_2 = -1 \) or \( m_1 = -\frac{1}{m_2} \)
Perpendicular lines have negative reciprocal slopes
Systems of Equations
Solution Methods
Substitution Method
Solve one equation for one variable, substitute into other equation
Elimination Method
Add or subtract equations to eliminate one variable
Graphical Solution
Solution is intersection point of the two lines (use Desmos on Digital SAT)
Quadratic Equations & Functions
Essential Quadratic Formulas
Quadratic Formula
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Solves \( ax^2 + bx + c = 0 \) — MUST MEMORIZE, not on reference sheet
Discriminant
\( \Delta = b^2 - 4ac \)
If \(\Delta > 0\): two real solutions; \(\Delta = 0\): one solution; \(\Delta < 0\): no real solutions
Vertex Form
\( y = a(x - h)^2 + k \)
Vertex is at point \((h, k)\); \(a\) determines opening direction and width
Standard Form
\( y = ax^2 + bx + c \)
Vertex x-coordinate: \( x = -\frac{b}{2a} \)
Factored Form
\( y = a(x - r_1)(x - r_2) \)
Where \(r_1\) and \(r_2\) are the roots (x-intercepts)
Difference of Squares
\( a^2 - b^2 = (a + b)(a - b) \)
High-frequency factoring pattern on SAT
Perfect Square Trinomial
\( a^2 + 2ab + b^2 = (a + b)^2 \)
\( a^2 - 2ab + b^2 = (a - b)^2 \)
Quadratic Formula Memory Trick: "Negative boy, couldn't decide (±), went to the radical party, then went to the square dance (b²), saw 4 awesome chicks (4ac), all over 2 awesome guys (2a)." Silly, but effective: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Exponential & Radical Expressions
Exponent Rules
Product Rule
\( x^a \cdot x^b = x^{a+b} \)
Quotient Rule
\( \frac{x^a}{x^b} = x^{a-b} \)
Power Rule
\( (x^a)^b = x^{ab} \)
Negative Exponent
\( x^{-a} = \frac{1}{x^a} \)
Zero Exponent
\( x^0 = 1 \) (where \( x \neq 0 \))
Fractional Exponent
\( x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m \)
Exponential Growth/Decay
\( y = a(1 + r)^t \) (growth) or \( y = a(1 - r)^t \) (decay)
Where \(a\) = initial amount, \(r\) = rate, \(t\) = time
Compound Interest
\( A = P\left(1 + \frac{r}{n}\right)^{nt} \)
\(P\) = principal, \(r\) = rate, \(n\) = compounds per year, \(t\) = years
Coordinate Geometry
Distance & Midpoint
Distance Formula
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Distance between points \((x_1, y_1)\) and \((x_2, y_2)\) — NOT on reference sheet
Midpoint Formula
\( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)
Midpoint between two points — NOT on reference sheet
Advanced Math Formulas (35% of SAT Math)
Polynomials
Polynomial Operations
Sum of Cubes
\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
Difference of Cubes
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
Remainder Theorem
When polynomial \(P(x)\) is divided by \((x - a)\), remainder is \(P(a)\)
Function Transformations
Transformation Rules
Vertical Shift
\( f(x) + k \): shifts graph UP \(k\) units
\( f(x) - k \): shifts graph DOWN \(k\) units
Horizontal Shift
\( f(x + k) \): shifts graph LEFT \(k\) units (opposite!)
\( f(x - k) \): shifts graph RIGHT \(k\) units
Vertical Stretch/Compression
\( kf(x) \): if \(|k| > 1\), stretches vertically; if \(0 < |k| < 1\), compresses
Horizontal Stretch/Compression
\( f(kx) \): if \(|k| > 1\), compresses horizontally; if \(0 < |k| < 1\), stretches
Reflections
\( -f(x) \): reflects over x-axis (flips upside down)
\( f(-x) \): reflects over y-axis (flips left-right)
Horizontal Shift Memory Aid: For horizontal shifts, "do the opposite." \( f(x+3) \) moves LEFT 3 units, not right. Think of it as: "What x-value do I need to input to get back to the original function?"
Rational Expressions & Equations
Rational Expression Rules
Simplification
Factor numerator and denominator, cancel common factors
Multiplication
\( \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \)
Division
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc} \)
Addition/Subtraction
\( \frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c} \) (same denominator)
Different denominators: find LCD, convert both fractions, then add/subtract
Geometry & Trigonometry Formulas (15% of SAT Math)
Triangles
Triangle Formulas (Some on reference sheet)
Area of Triangle
\( A = \frac{1}{2}bh \)
Where \(b\) = base and \(h\) = height — PROVIDED on reference sheet
Pythagorean Theorem
\( a^2 + b^2 = c^2 \)
For right triangles: \(a, b\) are legs, \(c\) is hypotenuse — PROVIDED on reference sheet
Pythagorean Triples
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples)
Triangle Angle Sum
Sum of interior angles = \( 180° \)
Special Right Triangles
Critical Triangle Ratios (PROVIDED on reference sheet but MUST memorize for speed)
45-45-90 Triangle
Side ratios: \( x : x : x\sqrt{2} \)
Legs are equal, hypotenuse = leg × √2
30-60-90 Triangle
Side ratios: \( x : x\sqrt{3} : 2x \)
Short leg : long leg : hypotenuse (opposite 30° : opposite 60° : opposite 90°)
Special Triangle Memory: 45-45-90 is easier: "square root of 2 for hypotenuse." For 30-60-90: "1, 2, root 3" or sing to a tune. Short leg is 1, hypotenuse is 2, medium leg is √3.
Circles
Circle Formulas
Circle Area
\( A = \pi r^2 \)
Where \(r\) = radius — PROVIDED on reference sheet
Circle Circumference
\( C = 2\pi r \) or \( C = \pi d \)
Where \(r\) = radius, \(d\) = diameter — PROVIDED on reference sheet
Arc Length
\( L = \frac{\theta}{360°} \times 2\pi r \)
Where \(\theta\) = central angle in degrees — NOT on reference sheet
Sector Area
\( A = \frac{\theta}{360°} \times \pi r^2 \)
Area of "pizza slice" with central angle \(\theta\) — NOT on reference sheet
Equation of Circle
\( (x - h)^2 + (y - k)^2 = r^2 \)
Circle with center \((h, k)\) and radius \(r\) — NOT on reference sheet
Other Polygons
Quadrilaterals & Polygons
Rectangle Area
\( A = lw \)
Length × width — PROVIDED on reference sheet
Parallelogram Area
\( A = bh \)
Base × height
Trapezoid Area
\( A = \frac{1}{2}(b_1 + b_2)h \)
Average of parallel bases × height
Sum of Interior Angles (n-sided polygon)
\( S = (n - 2) \times 180° \)
3D Geometry (Volume & Surface Area)
Solid Geometry Formulas (PROVIDED on reference sheet)
Rectangular Solid (Box) Volume
\( V = lwh \)
Length × width × height
Cylinder Volume
\( V = \pi r^2 h \)
Where \(r\) = radius of base, \(h\) = height
Sphere Volume
\( V = \frac{4}{3}\pi r^3 \)
Sphere Surface Area
\( SA = 4\pi r^2 \)
Cone Volume
\( V = \frac{1}{3}\pi r^2 h \)
Pyramid Volume
\( V = \frac{1}{3}Bh \)
Where \(B\) = area of base, \(h\) = height
Trigonometry
Basic Trig Formulas
SOHCAHTOA
\( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
\( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
\( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \)
Pythagorean Identity
\( \sin^2\theta + \cos^2\theta = 1 \)
Complementary Angle Relationship
\( \sin(90° - \theta) = \cos\theta \)
\( \cos(90° - \theta) = \sin\theta \)
Reciprocal Identities
\( \csc\theta = \frac{1}{\sin\theta} \)
\( \sec\theta = \frac{1}{\cos\theta} \)
\( \cot\theta = \frac{1}{\tan\theta} \)
SOHCAHTOA Memory: "Some Old Hippie Caught Another Hippie Trippin' On Acid" — crude but unforgettable. Or the cleaner: "Silly Old Harry Caught A Herring Trawling Off America."
Problem-Solving & Data Analysis (15% of SAT Math)
Ratios, Proportions & Percentages
Essential Formulas
Proportion
\( \frac{a}{b} = \frac{c}{d} \) leads to \( ad = bc \)
Cross-multiply to solve
Percent
\( \text{Percent} = \frac{\text{part}}{\text{whole}} \times 100\% \)
Percent Change
\( \text{Percent Change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\% \)
Positive = increase, negative = decrease
Percent of a Number
\( n\% \text{ of } x = \frac{n}{100} \times x = 0.0n \times x \)
Example: 15% of 80 = 0.15 × 80 = 12
Simple Interest
\( I = Prt \)
\(P\) = principal, \(r\) = rate (as decimal), \(t\) = time in years
Rate, Work & Distance
Rate Formulas
Distance Formula
\( d = rt \)
Distance = rate × time (also written as \( r = \frac{d}{t} \) or \( t = \frac{d}{r} \))
Average Speed
\( \text{Average Speed} = \frac{\text{total distance}}{\text{total time}} \)
Work Rate
\( \text{Work} = \text{rate} \times \text{time} \)
If person A takes \(a\) hours alone, rate = \(\frac{1}{a}\) jobs/hour
Combined Work Rate
\( \frac{1}{t_{\text{together}}} = \frac{1}{t_1} + \frac{1}{t_2} \)
Statistics
Statistical Measures
Mean (Average)
\( \bar{x} = \frac{\sum x}{n} = \frac{x_1 + x_2 + \ldots + x_n}{n} \)
Sum of all values divided by number of values
Median
Middle value when data is arranged in order (average of two middle values if even number of data points)
Mode
Most frequently occurring value in dataset
Range
\( \text{Range} = \text{Maximum} - \text{Minimum} \)
Standard Deviation Concept
Measures spread of data around mean (larger SD = more spread out data)
Probability
Probability Formulas
Basic Probability
\( P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \)
Probability of A or B (mutually exclusive)
\( P(A \text{ or } B) = P(A) + P(B) \)
Probability of A and B (independent events)
\( P(A \text{ and } B) = P(A) \times P(B) \)
Probability of NOT A
\( P(\text{not } A) = 1 - P(A) \)
Quick Reference Tables
Common Values to Memorize
Perfect Squares
\(1^2 = 1\) \(2^2 = 4\) \(3^2 = 9\)
\(4^2 = 16\) \(5^2 = 25\) \(6^2 = 36\)
\(7^2 = 49\) \(8^2 = 64\) \(9^2 = 81\)
\(10^2 = 100\) \(11^2 = 121\) \(12^2 = 144\)
\(13^2 = 169\) \(14^2 = 196\) \(15^2 = 225\)
Perfect Cubes
\(1^3 = 1\) \(2^3 = 8\) \(3^3 = 27\)
\(4^3 = 64\) \(5^3 = 125\)
\(6^3 = 216\) \(10^3 = 1000\)
Common Square Roots
\(\sqrt{2} \approx 1.414\)
\(\sqrt{3} \approx 1.732\)
\(\sqrt{5} \approx 2.236\)
Common Fractions to Decimals
\(\frac{1}{2} = 0.5\) \(\frac{1}{3} \approx 0.333\)
\(\frac{1}{4} = 0.25\) \(\frac{1}{5} = 0.2\)
\(\frac{2}{3} \approx 0.667\) \(\frac{3}{4} = 0.75\)
Critical Memorization Priorities
Must Memorize (NOT on SAT Reference Sheet): Quadratic formula, distance formula, midpoint formula, slope formula, all linear equation forms, exponential growth/decay, circle equation, arc length, sector area, percent change formula, all exponent rules, function transformations, special triangle ratios (for speed), and all statistics formulas.
On SAT Reference Sheet (but memorize anyway for speed): Basic area formulas (triangle, rectangle), circle area and circumference, volume formulas (cylinder, sphere, cone, pyramid), Pythagorean theorem, special right triangle ratios (45-45-90 and 30-60-90).
Study Strategies for Formula Mastery
Active Memorization
- Write each formula 5 times by hand
- Create flashcards with formula on one side, application on other
- Practice applying formulas through problem-solving
- Quiz yourself daily for 10-15 minutes
Application Practice
- Solve 10-15 problems using each formula
- Application creates stronger memory than repetition
- Focus on recognizing when to use which formula
- Time yourself to build speed
Visual Memory
- Create a one-page cheatsheet in your handwriting
- Use colors to group related formulas
- Post it where you'll see it daily
- Visual familiarity speeds recall
Test Day Prep
- Review cheatsheet morning of test
- Don't try to memorize new formulas last-minute
- Focus on formulas you've struggled with
- Trust your preparation
Download & Print This Cheatsheet
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Master These Formulas with Expert Guidance
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