Quick reference
SAT Math Formula Sheet.
Every formula tested on the Digital SAT Math section — with worked examples, SAT tips, and common mistakes. Click any card to expand.
Algebra
12 formulasSlope of a line
m = (y₂ − y₁) / (x₂ − x₁)
Rise over run between two points.
Tap for example →Slope-intercept form
y = mx + b
m = slope, b = y-intercept.
Tap for example →Point-slope form
y − y₁ = m(x − x₁)
Line through (x₁, y₁) with slope m.
Tap for example →Standard form of a line
Ax + By = C
A, B, C are integers; A ≥ 0.
Tap for example →Parallel & perpendicular slopes
Parallel: m₁ = m₂ Perpendicular: m₁ × m₂ = −1
Parallel lines share the same slope. Perpendicular slopes are negative reciprocals.
Tap for example →Linear percent change
New = Original × (1 ± r)
r is the decimal rate; + for increase, − for decrease.
Tap for example →Percent change
% change = (New − Old) / Old × 100
Measures relative change between two values.
Tap for example →Direct variation
y = kx (k = y/x)
y varies directly with x; k is the constant of variation.
Tap for example →Inverse variation
y = k/x (xy = k)
y varies inversely with x; product xy is constant.
Tap for example →Solving linear systems
Substitution: solve one, plug into other Elimination: add/subtract to cancel a variable
Two equations, two unknowns.
Tap for example →No solution / Infinite solutions
No solution: same slope, different intercepts Infinite: identical equations
Parallel lines never meet; coincident lines overlap.
Tap for example →Absolute value equation
|ax + b| = c → ax + b = c or ax + b = −c
Split into two cases when the expression is isolated.
Tap for example →Advanced Math
12 formulasQuadratic formula
x = (−b ± √(b² − 4ac)) / 2a
Solves ax² + bx + c = 0 for any quadratic.
Tap for example →Vertex form
y = a(x − h)² + k
Vertex at (h, k). Opens up if a > 0, down if a < 0.
Tap for example →Standard form → vertex (h, k)
h = −b/(2a)
x-coordinate of vertex for y = ax² + bx + c.
Tap for example →Discriminant
D = b² − 4ac
D > 0: two real roots. D = 0: one root. D < 0: no real roots.
Tap for example →Difference of squares
a² − b² = (a + b)(a − b)
Used to factor or simplify expressions quickly.
Tap for example →Perfect square trinomial
(a ± b)² = a² ± 2ab + b²
The square of a binomial expands predictably.
Tap for example →Sum and product of roots
Sum = −b/a Product = c/a
For ax² + bx + c = 0 with roots r₁ and r₂.
Tap for example →Exponential growth/decay
f(t) = a · bᵗ Growth: b > 1 Decay: 0 < b < 1
a = initial value, b = growth factor, t = time periods.
Tap for example →Laws of exponents
aᵐ · aⁿ = aᵐ⁺ⁿ aᵐ / aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1/aⁿ
Rules for multiplying, dividing, and raising powers.
Tap for example →Rational exponents & radicals
aᵐ/ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Fractional exponents equal roots.
Tap for example →Polynomial remainder theorem
If f(x) ÷ (x − k), remainder = f(k)
Evaluate the polynomial at the root to get the remainder.
Tap for example →Systems with quadratics
Substitute the linear equation into the quadratic, then solve.
Find intersections of a line and a parabola.
Tap for example →Functions
8 formulasFunction notation
f(x) means "f evaluated at x"
f(3) means substitute x = 3 into the rule for f.
Tap for example →Average rate of change
ARC = [f(b) − f(a)] / (b − a)
Slope of the secant line between two points on a curve.
Tap for example →Vertical & horizontal shifts
f(x) + k → up k f(x) − k → down k f(x − h) → right h f(x + h) → left h
Transformations shift the graph without changing shape.
Tap for example →Reflections & stretches
−f(x) → reflect over x-axis f(−x) → reflect over y-axis a·f(x) → vertical stretch by |a|
Multiply output to stretch/reflect vertically; multiply input to stretch horizontally.
Tap for example →Composition of functions
(f ∘ g)(x) = f(g(x))
Apply g first, then apply f to the result.
Tap for example →Inverse function
Swap x and y, then solve for y.
f⁻¹(x) undoes f(x). Graphically: reflect over y = x.
Tap for example →Linear vs. exponential growth
Linear: f(x) = mx + b (constant rate) Exponential: f(x) = abˣ (constant ratio)
Distinguish by checking differences (linear) vs. ratios (exponential).
Tap for example →Domain and range
Domain: all valid x inputs Range: all resulting y outputs
Exclude values that cause division by zero or negative square roots.
Tap for example →Geometry & Trig
17 formulasArea of a triangle
A = ½bh
b = base, h = perpendicular height (not slant).
Tap for example →Pythagorean theorem
a² + b² = c²
c is the hypotenuse (longest side) of a right triangle.
Tap for example →Special right triangles
30-60-90: sides x, x√3, 2x 45-45-90: sides x, x, x√2
These ratios are fixed. Memorize them — they appear on nearly every test.
Tap for example →Circumference of a circle
C = 2πr = πd
r = radius, d = diameter.
Tap for example →Area of a circle
A = πr²
r = radius.
Tap for example →Arc length
L = (θ/360) × 2πr
θ is the central angle in degrees.
Tap for example →Area of a sector
A = (θ/360) × πr²
Sector = "pie slice." θ is the central angle in degrees.
Tap for example →Volume of a cylinder
V = πr²h
r = base radius, h = height.
Tap for example →Volume of a cone
V = (1/3)πr²h
One-third the volume of a cylinder with the same base and height.
Tap for example →Volume of a sphere
V = (4/3)πr³
r = radius.
Tap for example →Area of a rectangle
A = l × w
Length times width.
Tap for example →Area of a trapezoid
A = ½(b₁ + b₂)h
Average of the parallel bases times height.
Tap for example →Distance formula
d = √[(x₂−x₁)² + (y₂−y₁)²]
Pythagorean theorem applied to coordinate points.
Tap for example →Midpoint formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Average the x-coordinates and average the y-coordinates.
Tap for example →Equation of a circle
(x − h)² + (y − k)² = r²
Center (h, k), radius r.
Tap for example →SOH-CAH-TOA
sin θ = opp/hyp cos θ = adj/hyp tan θ = opp/adj
Trigonometric ratios for right triangles.
Tap for example →Complementary angle trig
sin θ = cos(90° − θ) cos θ = sin(90° − θ)
Sine and cosine are co-functions; they swap at complementary angles.
Tap for example →Data & Statistics
12 formulasMean (average)
x̄ = Σx / n
Sum of all values divided by the count of values.
Tap for example →Median
Middle value of an ordered list Even count: average of two middle values
Resistant to outliers unlike the mean.
Tap for example →Mode
The value that appears most frequently
A data set can have multiple modes or no mode.
Tap for example →Range
Range = Maximum − Minimum
Spread from the smallest to the largest value.
Tap for example →Weighted average
x̄ = Σ(value × weight) / Σweights
Used when values contribute unequally to the average.
Tap for example →Percentage
Part = (Percent / 100) × Whole
Converts a fraction to a percent.
Tap for example →Probability
P(A) = favorable outcomes / total outcomes
For equally likely outcomes. P ranges from 0 to 1.
Tap for example →Compound probability
P(A and B) = P(A) × P(B) [if independent] P(A or B) = P(A) + P(B) − P(A and B)
Multiply for "and" (independent events), add-then-subtract for "or."
Tap for example →Compound interest
A = P(1 + r/n)^(nt)
P = principal, r = annual rate (decimal), n = compounds/yr, t = years.
Tap for example →Simple interest
I = Prt A = P(1 + rt)
P = principal, r = rate (decimal), t = time in years.
Tap for example →Standard deviation (concept)
σ = measure of how spread out data is from the mean
Higher σ = more spread. SAT never asks you to calculate it — just compare.
Tap for example →Line of best fit
y = mx + b (interpreted in context)
Slope = rate of change. y-intercept = starting value.
Tap for example →