SAT Math Essentials: Formulas, Concepts & Key Terms for All 4 Domains

Domain 1 · ~35% of the section · largest domain

Algebra

The biggest domain on the section. These questions test linear equations and inequalities, systems of equations, slope and line equations, and using linear models to describe real-world relationships. Fluent algebra is the single highest-leverage SAT Math skill.

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Slope

Rise over run — how steep a line is.

m = (y₂ − y₁) / (x₂ − x₁)

Memorize

Slope-Intercept Form

Write a line as y equals slope times x plus the y-intercept.

y = mx + b

Memorize

Point-Slope Form

Useful when you know one point and the slope.

y − y₁ = m(x − x₁)

Memorize

Standard Form of a Line

Lines can also be written with both variables on the same side.

Ax + By = C

Memorize

Parallel Lines

Parallel lines have equal slopes.

m₁ = m₂

Memorize

Perpendicular Lines

Perpendicular slopes are negative reciprocals; their product is −1.

m₁ × m₂ = −1

Memorize

Distance Between Two Points

The straight-line distance between any two points in the plane.

d = √[(x₂−x₁)² + (y₂−y₁)²]

Memorize

Midpoint

The point exactly halfway between two given points.

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Memorize

Solving an Inequality

Treat it like an equation, but flip the sign whenever you multiply or divide by a negative.

Flip < or > when × or ÷ by −

Rule

System of Equations

Two equations solved together by substitution or elimination.

Sub or add/subtract to cancel

Strategy

Variable

A letter that stands for an unknown number.

Coefficient

The number multiplied by a variable (the 3 in 3x).

Constant

A fixed number with no variable attached.

Y-Intercept

The point where a line crosses the y-axis — the value of y when x = 0.

X-Intercept

The point where a line crosses the x-axis — the value of x when y = 0.

Substitution

Solving a system by replacing one variable using another equation.

Elimination

Solving a system by adding or subtracting equations to cancel a variable.

Linear Model

A real-world situation described by a linear equation.

Big Idea 1

Slope is rate of change

Every linear equation describes a constant rate of change. In word problems, the coefficient of x is the rate (per item, per hour, per dollar) and the constant is the starting value or fixed cost. Match the slope to the real-world rate and the intercept to the real-world starting value.

LinesModeling

Big Idea 2

Systems describe two relationships at once

When two equations are solved together, the solution satisfies both at once. Use substitution when one variable is already isolated, and use elimination when adding or subtracting the equations cancels a variable cleanly. Practice both so you can pick the faster method.

SystemsStrategy

Big Idea 3

Inequalities flip when you multiply by a negative

The most common algebra mistake on the SAT. Multiplying or dividing both sides of an inequality by a negative number flips the direction of the inequality sign. Mark this clearly when solving so you do not forget.

InequalitiesMistakes

Domain 2 · ~35% of the section · largest domain

Advanced Math

The other big half of the section. Tests quadratic equations and parabolas, function notation, exponent rules, exponential growth and decay, and basic polynomial manipulation. Strong here usually means a strong overall Math score.

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Quadratic Formula

Solves any quadratic equation in the form ax² + bx + c = 0.

x = [−b ± √(b² − 4ac)] / (2a)

Memorize

Discriminant

The part under the square root. Tells you how many real solutions a quadratic has.

b² − 4ac

Memorize

Vertex Form

A parabola written so the vertex (h, k) is visible.

y = a(x − h)² + k

Memorize

Vertex X-Value

For y = ax² + bx + c, the vertex is at this x-value.

x = −b / (2a)

Memorize

Difference of Squares

A common factoring pattern.

a² − b² = (a − b)(a + b)

Memorize

Exponent Rule (Multiply)

When multiplying like bases, add the exponents.

x^a · x^b = x^(a+b)

Memorize

Exponent Rule (Divide)

When dividing like bases, subtract the exponents.

x^a / x^b = x^(a−b)

Memorize

Exponent Rule (Power)

A power raised to another power: multiply the exponents.

(x^a)^b = x^(a·b)

Memorize

Negative Exponent

A negative exponent means a reciprocal.

x^(−a) = 1 / x^a

Memorize

Exponential Growth

A quantity increases by the same percentage each step.

A = A₀ · (1 + r)^t

Memorize

Exponential Decay

A quantity decreases by the same percentage each step.

A = A₀ · (1 − r)^t

Memorize

Quadratic Equation

An equation with a squared variable; its graph is a parabola.

Parabola

The U-shaped curve produced by a quadratic equation.

Vertex

The highest or lowest point of a parabola.

Factoring

Rewriting an expression as a product of simpler parts.

Function

A rule that assigns exactly one output to each input.

Function Notation

Writing a function as f(x), read "f of x."

Root / Zero

An input value that makes a function equal zero.

Polynomial

An expression of variables and constants using only addition, subtraction, and whole-number powers.

Big Idea 1

Parabolas open up or down based on the leading coefficient

If a is positive, the parabola opens upward and the vertex is the minimum. If a is negative, it opens downward and the vertex is the maximum. Knowing this tells you which side of the vertex the function increases or decreases on.

QuadraticsGraphs

Big Idea 2

The discriminant tells you the number of solutions

If b² − 4ac is positive, there are two real solutions. If it equals zero, there is exactly one. If it is negative, there are no real solutions. Many SAT problems test this without making you actually solve the equation.

QuadraticsDiscriminant

Big Idea 3

Exponential is not the same as quadratic growth

A linear function grows by adding the same amount; an exponential function grows by multiplying by the same factor. When a word problem says "doubles every year" or "increases by 5% each month," it is always exponential, never linear or quadratic.

ExponentialModeling

Domain 3 · ~15% of the section

Problem Solving & Data Analysis

Real-world math with numbers. Ratios, proportions, rates, percentages, unit conversions, basic statistics, scatterplots, and probability. These questions usually come as word problems, so reading carefully and setting up the right relationship matters more than fancy techniques.

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Percent

A percent is just a part out of one hundred.

% = (part / whole) × 100

Memorize

Percent Change

How much a value increased or decreased relative to the original.

% change = (new − old) / old × 100

Memorize

Percent Increase Result

After increasing by a percent, multiply by 1 plus the decimal.

new = old × (1 + r)

Memorize

Percent Decrease Result

After decreasing by a percent, multiply by 1 minus the decimal.

new = old × (1 − r)

Memorize

Mean (Average)

Sum the values and divide by how many there are.

mean = sum / count

Memorize

Proportion

Set two ratios equal and cross-multiply.

a/b = c/d ⇒ a·d = b·c

Memorize

Probability

The chance of an event, written as a fraction between 0 and 1.

P = favorable / total

Memorize

Speed, Distance, Time

The classic motion relationship.

distance = rate × time

Memorize

Ratio

A comparison of two quantities by division.

Rate

A ratio comparing quantities with different units, like miles per hour.

Unit Conversion

Changing a measurement from one unit to another by multiplying by conversion factors.

Median

The middle value when data is ordered from smallest to largest.

Mode

The value that appears most often in a data set.

Range

The difference between the largest and smallest values.

Standard Deviation

A measure of how spread out the data values are.

Scatterplot

A graph showing the relationship between two variables as points.

Line of Best Fit

A straight line drawn to model the trend in a scatterplot.

Big Idea 1

Percent change is asymmetric

A 20% increase followed by a 20% decrease does not return you to the original. Multiply 1.2 by 0.8 and you get 0.96, a 4% decrease overall. Many SAT questions test this asymmetry. Always multiply through; never just add or subtract percents.

PercentTrap

Big Idea 2

Setting up the proportion correctly is the whole question

Ratio and rate problems hinge on lining up matching units on each side of the equals sign. If "cats per dog" is on one side, it must be on the other. Once it is set up correctly, the arithmetic is usually easy.

RatiosSetup

Big Idea 3

Mean is sensitive to outliers, median is not

When a question asks how an extreme value would affect a statistic, remember: a single outlier pulls the mean strongly but barely moves the median. The SAT loves to test this exact distinction in statistics word problems.

StatisticsComparison

Domain 4 · ~15% of the section

Geometry & Trigonometry

Shapes, sizes, and angles. Area, perimeter, volume, the Pythagorean theorem, similar and congruent figures, circles, and right-triangle trigonometry. The Bluebook app provides a reference sheet with the most common formulas, but knowing them by heart saves precious time.

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Area of a Rectangle

Length times width.

A = l × w

Memorize

Area of a Triangle

One-half base times height.

A = ½ b × h

Memorize

Area of a Circle

Pi times the radius squared.

A = π r²

Given on test

Circumference of a Circle

Two pi times the radius, or pi times the diameter.

C = 2π r = π d

Given on test

Volume of a Box

Length times width times height.

V = l × w × h

Given on test

Volume of a Cylinder

Pi times radius squared times height.

V = π r² h

Given on test

Pythagorean Theorem

In a right triangle, the two legs squared and summed equal the hypotenuse squared.

a² + b² = c²

Given on test

Angles in a Triangle

Always add up to 180 degrees.

∑ angles = 180°

Memorize

Angles on a Straight Line

Always add up to 180 degrees.

∑ = 180°

Memorize

Angles Around a Point

Always add up to 360 degrees.

∑ = 360°

Memorize

Sine (SOH)

Opposite over hypotenuse.

sinθ = opposite / hypotenuse

Memorize

Cosine (CAH)

Adjacent over hypotenuse.

cosθ = adjacent / hypotenuse

Memorize

Tangent (TOA)

Opposite over adjacent.

tanθ = opposite / adjacent

Memorize

Hypotenuse

The longest side of a right triangle, opposite the right angle.

Right Triangle

A triangle with one 90-degree angle.

Complementary Angles

Two angles that add up to 90 degrees.

Supplementary Angles

Two angles that add up to 180 degrees.

Similar Figures

Same shape, different size; matching sides are proportional.

Congruent Figures

Identical in both shape and size.

Radius

The distance from a circle's center to its edge.

Diameter

Twice the radius; the distance across a circle through its center.

SOH-CAH-TOA

The memory phrase for the three trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Big Idea 1

Some formulas are given, some are not

The Bluebook reference sheet gives the most common geometry formulas (circle area, cylinder volume, Pythagorean theorem). But algebra formulas — slope, the quadratic formula, exponent rules — are never provided. Memorize the ones not given; double-check the ones that are.

StrategyFormulas

Big Idea 2

Similar figures share ratios, not sizes

When two figures are similar, every pair of matching sides shares the same ratio (the "scale factor"). If one triangle's side is 1.5 times longer than the matching side on the other, every other matching side is 1.5 times longer too. Areas scale by the square of this factor; volumes by the cube.

Similar FiguresScale

Big Idea 3

Trig ratios depend only on the angle

Sine, cosine, and tangent of an angle stay the same no matter how large the right triangle is. This is why SOH-CAH-TOA works: the ratio of "opposite over hypotenuse" stays constant for the same angle even as the triangle is scaled. Use this to find missing sides when you know one side and one angle.

TrigRight Triangles

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