SAT Math Formula Sheet (Detailed)
Jump links at the top, clean print layout, and “what it’s for” notes so students don’t just memorize symbols.
Print tip: choose “Save as PDF” to make a one-click downloadable sheet.
1) Algebra Basics (core tools)exponents, radicals, factoring, fractions
2) Lines & Systemsslope, forms, intersections, inequalities
3) Functions & Graph Featuresdomain/range, transformations, rate of change
4) Quadratics & Polynomialsvertex, roots, discriminant, remainder
5) Exponential & Rational Modelsgrowth/decay, asymptotes, domain restrictions
6) Percent, Ratio, Unitspercent change, proportionality, unit conversion
7) Data, Statistics, Probabilitymean/median, regression, probability rules
8) Geometry & Coordinate Geometrytriangles, circles, distance, area/volume
9) Trigonometry (SAT level)SOHCAHTOA, special triangles, radians
10) Quick Checklist“If you only memorize one screen…”
2) Lines & Systemsslope, forms, intersections, inequalities
3) Functions & Graph Featuresdomain/range, transformations, rate of change
4) Quadratics & Polynomialsvertex, roots, discriminant, remainder
5) Exponential & Rational Modelsgrowth/decay, asymptotes, domain restrictions
6) Percent, Ratio, Unitspercent change, proportionality, unit conversion
7) Data, Statistics, Probabilitymean/median, regression, probability rules
8) Geometry & Coordinate Geometrytriangles, circles, distance, area/volume
9) Trigonometry (SAT level)SOHCAHTOA, special triangles, radians
10) Quick Checklist“If you only memorize one screen…”
How to use this: When you miss a question, tag it to a section below and re-do it using the formulas listed there.
The goal is not “memorize everything,” it’s “know what tool matches what situation.”
The goal is not “memorize everything,” it’s “know what tool matches what situation.”
1) Algebra Basics (core tools)
A. Exponent Laws
a^m · a^n = a^(m+n)
a^m / a^n = a^(m−n) (a ≠ 0)
(a^m)^n = a^(mn)
(ab)^n = a^n b^n
(a/b)^n = a^n / b^n (b ≠ 0)
a^0 = 1 (a ≠ 0)
a^(−n) = 1/a^n
a^m / a^n = a^(m−n) (a ≠ 0)
(a^m)^n = a^(mn)
(ab)^n = a^n b^n
(a/b)^n = a^n / b^n (b ≠ 0)
a^0 = 1 (a ≠ 0)
a^(−n) = 1/a^n
Roots as exponents
ⁿ√a = a^(1/n)
a^(m/n) = ⁿ√(a^m)
a^(m/n) = ⁿ√(a^m)
B. Radicals & Simplifying
√(ab) = √a · √b (a,b ≥ 0)
√(a/b) = √a / √b (b > 0)
√(k^2) = |k|
√(x^2) = |x| (important!)
√(a/b) = √a / √b (b > 0)
√(k^2) = |k|
√(x^2) = |x| (important!)
Common trap: √(x²) is not “x” automatically. It’s |x|. SAT sometimes tests this.
C. Factoring Patterns
GCF: ax + ay = a(x + y)
Difference of squares: a² − b² = (a − b)(a + b)
Perfect square trinomials:
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Trinomial: x² + (p+q)x + pq = (x+p)(x+q)
Difference of squares: a² − b² = (a − b)(a + b)
Perfect square trinomials:
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Trinomial: x² + (p+q)x + pq = (x+p)(x+q)
Grouping (4 terms)
ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y)
D. Fractions & Rational Expressions
a/b + c/d = (ad + bc)/bd
a/b − c/d = (ad − bc)/bd
(a/b)·(c/d) = ac/bd
(a/b)÷(c/d) = (a/b)·(d/c)
a/b − c/d = (ad − bc)/bd
(a/b)·(c/d) = ac/bd
(a/b)÷(c/d) = (a/b)·(d/c)
Domain rule: If an expression has a denominator, that denominator cannot be 0.
Even if something cancels later, the original restriction still matters.
Even if something cancels later, the original restriction still matters.
2) Lines & Systems
A. Slope + Line Forms
Slope between points: m = (y₂ − y₁)/(x₂ − x₁)
Slope-intercept: y = mx + b
Point-slope: y − y₁ = m(x − x₁)
Standard: Ax + By = C
Slope-intercept: y = mx + b
Point-slope: y − y₁ = m(x − x₁)
Standard: Ax + By = C
Intercepts
x-intercept: set y = 0, solve for x
y-intercept: set x = 0, solve for y
y-intercept: set x = 0, solve for y
Fast recognition: In y=mx+b, the slope is the coefficient of x. b is the y-intercept.
B. Parallel & Perpendicular
Parallel lines: same slope
Perpendicular (non-vertical): m₁m₂ = −1
Negative reciprocal: if m = a/b, perp slope = −b/a
Perpendicular (non-vertical): m₁m₂ = −1
Negative reciprocal: if m = a/b, perp slope = −b/a
Vertical / Horizontal
Vertical line: x = constant (slope undefined)
Horizontal line: y = constant (slope 0)
Horizontal line: y = constant (slope 0)
C. Systems (2 equations)
One solution: lines intersect once
No solution: parallel distinct lines
Infinite solutions: same line (equivalent equations)
No solution: parallel distinct lines
Infinite solutions: same line (equivalent equations)
Quick check for “same line”
If you can multiply one equation by a constant to get the other, they’re the same line.
D. Inequalities
If you multiply/divide by a negative, flip the inequality sign.
Example: −2x > 6 → x < −3
Example: −2x > 6 → x < −3
Compound inequalities
−a < x < a means x is between −a and a
x < −a OR x > a means outside that range
x < −a OR x > a means outside that range
3) Functions & Graph Features
A. Function notation
f(x) means “output when input is x”
If f(x)=2x+3, then f(5)=2(5)+3=13
If f(x)=2x+3, then f(5)=2(5)+3=13
Domain & Range (SAT level)
Domain: allowed x-values (inputs)
Range: possible y-values (outputs)
Restrictions often come from: denominator=0, even root of negative
Range: possible y-values (outputs)
Restrictions often come from: denominator=0, even root of negative
B. Average rate of change
Average rate of change on [a,b]: (f(b) − f(a)) / (b − a)
Think of this as the slope of the secant line between (a,f(a)) and (b,f(b)).
C. Transformations
f(x) + k : shift up k
f(x) − k : shift down k
f(x − h) : shift right h
f(x + h) : shift left h
a·f(x) : vertical stretch by |a| (flip if a<0)
f(ax) : horizontal shrink by factor 1/|a| (flip if a<0)
f(x) − k : shift down k
f(x − h) : shift right h
f(x + h) : shift left h
a·f(x) : vertical stretch by |a| (flip if a<0)
f(ax) : horizontal shrink by factor 1/|a| (flip if a<0)
Common confusion: inside changes go “opposite.”
f(x−3) shifts RIGHT 3, not left.
f(x−3) shifts RIGHT 3, not left.
D. Composition (sometimes)
(f ∘ g)(x) = f(g(x))
Inverse idea (when asked)
Swap x and y, solve for y, then rename y as f⁻¹(x).
Check: f(f⁻¹(x)) = x
Check: f(f⁻¹(x)) = x
4) Quadratics & Polynomials
A. Quadratic forms
Standard: y = ax² + bx + c
Vertex: y = a(x − h)² + k (vertex = (h,k))
Factored: y = a(x − r₁)(x − r₂) (roots r₁, r₂)
Vertex: y = a(x − h)² + k (vertex = (h,k))
Factored: y = a(x − r₁)(x − r₂) (roots r₁, r₂)
Axis + vertex x-value
Axis of symmetry: x = −b/(2a)
B. Quadratic formula + discriminant
x = (−b ± √(b² − 4ac)) / (2a)
Δ = b² − 4ac
Δ = b² − 4ac
Discriminant meaning:
Δ > 0 → 2 real roots
Δ = 0 → 1 real root (double root)
Δ < 0 → no real roots
Δ > 0 → 2 real roots
Δ = 0 → 1 real root (double root)
Δ < 0 → no real roots
C. Completing the square (fast version)
x² + bx = (x + b/2)² − (b/2)²
Example: x² + 6x = (x+3)² − 9
D. Polynomial extras
Remainder theorem: remainder when f(x) ÷ (x − a) is f(a)
Factor theorem: if f(a)=0, then (x − a) is a factor
End behavior (rough): leading term controls big-x behavior
Factor theorem: if f(a)=0, then (x − a) is a factor
End behavior (rough): leading term controls big-x behavior
5) Exponential & Rational Models
A. Exponential basics
y = a·b^x (b>1 growth, 0<b<1 decay)
Percent growth/decay model
y = a(1 + r)^t (growth)
y = a(1 − r)^t (decay)
y = a(1 − r)^t (decay)
B. Exponential interpretation
a = initial value (when x=0)
b = growth factor per 1 unit of x
b = growth factor per 1 unit of x
If something grows 8% each period, b = 1.08.
If it decays 12% each period, b = 0.88.
If it decays 12% each period, b = 0.88.
C. Rational functions (SAT level)
f(x) = (polynomial)/(polynomial)
Domain excludes values that make denominator 0
Domain excludes values that make denominator 0
Asymptotes (common cases)
Vertical asymptote: denominator = 0 (that doesn’t cancel)
Horizontal asymptote (degree rule):
deg(top) < deg(bottom) → y=0
deg(top) = deg(bottom) → y = (leading coeffs ratio)
Horizontal asymptote (degree rule):
deg(top) < deg(bottom) → y=0
deg(top) = deg(bottom) → y = (leading coeffs ratio)
D. Simplifying rational expressions
Factor numerator and denominator, cancel common factors.
But keep original domain restrictions.
But keep original domain restrictions.
6) Percent, Ratio, Units
A. Percent basics
percent = (part/whole) × 100%
part = (percent as decimal) × whole
whole = part / (percent as decimal)
part = (percent as decimal) × whole
whole = part / (percent as decimal)
Percent change
Percent change = (new − old)/old × 100%
B. Multipliers (fastest way)
Increase by r% → multiply by (1 + r/100)
Decrease by r% → multiply by (1 − r/100)
Decrease by r% → multiply by (1 − r/100)
Two changes compound. Example: +20% then −20% is not back to original.
It’s ×1.2×0.8 = ×0.96 (a 4% net decrease).
It’s ×1.2×0.8 = ×0.96 (a 4% net decrease).
C. Ratios & proportions
a:b = a/b
a/b = c/d → ad = bc
a/b = c/d → ad = bc
Direct variation
y = kx (k is constant of proportionality)
D. Units & rates
rate = distance/time
distance = rate·time
Work problems: combined rate = sum of individual rates
distance = rate·time
Work problems: combined rate = sum of individual rates
7) Data, Statistics, Probability
A. Center & spread
Mean = (sum of values)/(number of values)
Median = middle value (sorted) or avg of 2 middle values
Range = max − min
Median = middle value (sorted) or avg of 2 middle values
Range = max − min
Boxplot pieces (concept)
Min, Q1, Median, Q3, Max
IQR = Q3 − Q1
IQR = Q3 − Q1
B. Standard deviation (what SAT expects)
Bigger SD → more spread out
Add constant to all values → SD unchanged
Multiply all values by k → SD multiplied by |k|
Add constant to all values → SD unchanged
Multiply all values by k → SD multiplied by |k|
SAT usually tests SD conceptually, not with the full calculation.
C. Scatterplots & lines of best fit
Positive association: as x increases, y tends to increase
Negative association: as x increases, y tends to decrease
Stronger association: points closer to a line
Negative association: as x increases, y tends to decrease
Stronger association: points closer to a line
Residual
Residual = actual y − predicted y
D. Probability rules
P(A) = favorable/total
P(not A) = 1 − P(A)
P(A or B) = P(A) + P(B) − P(A and B)
Independent: P(A and B) = P(A)P(B)
Conditional: P(A|B) = P(A and B)/P(B)
P(not A) = 1 − P(A)
P(A or B) = P(A) + P(B) − P(A and B)
Independent: P(A and B) = P(A)P(B)
Conditional: P(A|B) = P(A and B)/P(B)
Two-way tables
Joint: cell/total
Conditional: cell/(row total or column total)
Conditional: cell/(row total or column total)
E. Counting (when it shows up)
n! = n·(n−1)·…·2·1
Permutations: nP r = n!/(n−r)!
Combinations: nC r = n!/[r!(n−r)!]
Permutations: nP r = n!/(n−r)!
Combinations: nC r = n!/[r!(n−r)!]
8) Geometry & Coordinate Geometry
A. Triangles
Angle sum: A + B + C = 180°
Area: A = (1/2)bh
Pythagorean theorem: a² + b² = c²
45-45-90: legs x, hyp x√2
30-60-90: short x, long x√3, hyp 2x
Area: A = (1/2)bh
Pythagorean theorem: a² + b² = c²
45-45-90: legs x, hyp x√2
30-60-90: short x, long x√3, hyp 2x
Similar triangles
Scale factor = k
Perimeter scales by k
Area scales by k²
Perimeter scales by k
Area scales by k²
B. Circles
Circumference: C = 2πr
Area: A = πr²
Arc length: (θ/360)·2πr
Sector area: (θ/360)·πr²
Area: A = πr²
Arc length: (θ/360)·2πr
Sector area: (θ/360)·πr²
Circle equations
Center (0,0): x² + y² = r²
Center (h,k): (x − h)² + (y − k)² = r²
Center (h,k): (x − h)² + (y − k)² = r²
C. Polygons & area formulas
Rectangle: A=lw, P=2l+2w
Parallelogram: A=bh
Trapezoid: A=(1/2)(b₁+b₂)h
Interior angle sum: (n−2)·180°
Parallelogram: A=bh
Trapezoid: A=(1/2)(b₁+b₂)h
Interior angle sum: (n−2)·180°
D. Coordinate geometry
Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Slope reminder
m = (y₂ − y₁)/(x₂ − x₁)
E. 3D solids (common)
Rectangular prism: V=lwh, SA=2(lw+lh+wh)
Cylinder: V=πr²h, SA=2πr²+2πrh
Cone (sometimes): V=(1/3)πr²h
Sphere (sometimes): V=(4/3)πr³, SA=4πr²
Cylinder: V=πr²h, SA=2πr²+2πrh
Cone (sometimes): V=(1/3)πr²h
Sphere (sometimes): V=(4/3)πr³, SA=4πr²
SAT sometimes includes cone/sphere. If you want, I can add a “SAT-provided vs must-memorize” label.
9) Trigonometry (SAT level)
A. SOH-CAH-TOA
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
One identity you might see
sin²(θ) + cos²(θ) = 1
B. Degrees ↔ Radians
π radians = 180°
degrees → radians: × (π/180)
radians → degrees: × (180/π)
degrees → radians: × (π/180)
radians → degrees: × (180/π)
Arc length (radians version)
s = rθ (θ in radians)
C. Special values
sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3
sin 45°=√2/2, cos 45°=√2/2, tan 45°=1
sin 60°=√3/2, cos 60°=1/2, tan 60°=√3
sin 45°=√2/2, cos 45°=√2/2, tan 45°=1
sin 60°=√3/2, cos 60°=1/2, tan 60°=√3
Special triangles are the fastest way to remember these, not memorizing a table.
10) Quick Checklist (high-frequency formulas)
If you only want the “core SAT page,” this is it. Everything else above is the expanded version with notes.
- Slope: (y₂−y₁)/(x₂−x₁)
- Line forms: y=mx+b, y−y₁=m(x−x₁)
- Distance: √[(x₂−x₁)²+(y₂−y₁)²] • Midpoint: ((x₁+x₂)/2,(y₁+y₂)/2)
- Quadratic vertex x: −b/(2a) • Quadratic formula: (−b±√(b²−4ac))/(2a)
- Discriminant: b²−4ac
- Pythagorean: a²+b²=c² • Triangle area: (1/2)bh
- Circle: C=2πr, A=πr²
- Percent change: (new−old)/old
- Growth/decay: a(1±r)^t
- Probability: P(A or B)=P(A)+P(B)−P(A and B), P(A|B)=P(A and B)/P(B)
- Trig: sin=opp/hyp, cos=adj/hyp, tan=opp/adj