The SAT gives you 12 geometry formulas on a reference sheet. The ACT gives you nothing. Everything else — and there's a lot — you're expected to have memorized before you walk into test day.

This page covers every formula that regularly appears on the Digital SAT and ACT, organized by topic. We've tagged each one with the test it appears on, how often it's tested, and a note on when to use it. Whether you're prepping for one test or both, this is the only formula page you need. Not sure which test to take? See our breakdown of ACT vs. SAT in 2026 to find out which one gives your child the best advantage.

📐
119 Formulas Every one you need
📚
8 Topics Algebra to matrices
🎯
SAT + ACT Both tests covered
⭐
Rated Frequency for each
Show:

📋 What the SAT Reference Sheet Provides

These 12 formulas are printed at the start of every SAT Math section. You still need to know when to apply them — but you don't need to memorize them.

✓ A = πr² (circle area)
✓ C = 2πr (circumference)
✓ A = lw (rectangle area)
✓ A = ½bh (triangle area)
✓ a² + b² = c² (Pythagorean)
✓ 30-60-90 ratios
✓ 45-45-90 ratios
✓ V = lwh (rect. prism)
✓ V = πr²h (cylinder)
✓ V = ⅓πr²h (cone)
✓ V = ⁴⁄₃πr³ (sphere)
✓ V = ⅓lwh (pyramid)
SAT: 12 formulas provided
ACT: No formula sheet — memorize everything
How to Use This Sheet

Don't just read through this list passively. For every formula, cover the formula with your hand, try to recall it, then check. Active recall beats re-reading by 3x. Work through each section until you can write every formula from memory in under 3 seconds.

📊 Algebra Essentials

Linear equations and systems account for roughly 35–40% of all math questions on both the SAT and ACT. This is the highest-yield section to master.

Linear Equations

Slope Formula
SAT + ACT ★★★
m = y₂ − y₁x₂ − x₁
Use when given two points and asked for slope, rate of change, or to find a parallel/perpendicular line.
Slope-Intercept Form
SAT + ACT ★★★
y = mx + b
m = slope, b = y-intercept. Convert every linear equation to this form to quickly identify slope and intercept.
Point-Slope Form
SAT + ACT ★★★
y − y₁ = m(x − x₁)
Fastest way to write a line's equation when you know the slope and one point. No need to find the y-intercept first.
Standard Form of a Line
SAT + ACT ★★
Ax + By = C
Use for finding intercepts quickly. x-intercept: set y = 0. y-intercept: set x = 0. A, B, C are integers.
Parallel Lines
SAT + ACT ★★★
m₁ = m₂ (same slope)
Parallel lines never intersect. They have identical slopes but different y-intercepts.
Perpendicular Lines
SAT + ACT ★★★
m₁ · m₂ = −1
Slopes are negative reciprocals. If one slope is ²⁄₃, the perpendicular slope is −³⁄₂.
Horizontal & Vertical Line Slopes
SAT + ACT ★★
Horizontal line: slope = 0   |   Vertical line: slope = undefined
Horizontal lines have equation y = k. Vertical lines have equation x = k. Vertical lines have no slope (division by zero).

Systems of Equations

Every SAT and ACT includes at least 2–3 systems questions. Know when to substitute vs. eliminate.

Substitution Method
SAT + ACT ★★★
Solve one equation for a variable, plug into the other
Best when one variable is already isolated (y = … or x = …). On the digital SAT, Desmos can also graph both equations.
Elimination Method
SAT + ACT ★★★
Add or subtract equations to cancel one variable
Best when coefficients line up for easy cancellation. Multiply one or both equations to make coefficients match if needed.
No Solution (Parallel)
SAT + ACT ★★★
Same slope, different y-intercept → 0 solutions
Lines are parallel and never cross. On the SAT, this often shows up as "for what value of k does the system have no solution?"
Infinite Solutions (Same Line)
SAT + ACT ★★
Same slope, same y-intercept → ∞ solutions
The equations describe the same line. One equation is a multiple of the other. Every point on the line is a solution.
💡 VPT Pro Tip

On the Digital SAT, graph every system in Desmos first. You'll see the intersection (or lack of one) immediately. Then confirm algebraically only if the answer choices are close. This saves 30–60 seconds per question.

📈 Quadratics & Polynomials

Quadratics are the second-highest-frequency topic on the SAT (and top 3 on the ACT). Master all three forms and the quadratic formula — they appear in roughly 20% of math questions.

Quadratic Forms

Standard Form
SAT + ACT ★★★
y = ax² + bx + c
c is the y-intercept. Use when you need to apply the quadratic formula or find the discriminant. a > 0 opens up, a < 0 opens down.
Vertex Form
SAT + ACT ★★★
y = a(x − h)² + k
Vertex at (h, k). Use when asked for min/max value, axis of symmetry, or vertex coordinates. Watch the signs carefully.
Factored Form
SAT + ACT ★★★
y = a(x − r₁)(x − r₂)
r₁ and r₂ are the x-intercepts (roots/zeros). Use when asked where the parabola crosses the x-axis.
Quadratic Formula
SAT + ACT ★★★
x = −b ± √(b² − 4ac)2a
Use when ax² + bx + c = 0 can't be easily factored. The expression b² − 4ac (discriminant) tells you how many real solutions exist.
Vertex x-coordinate
SAT + ACT ★★★
x = −b2a
Fastest way to find the vertex from standard form. Also gives you the axis of symmetry. Plug back in for the y-coordinate.
Sum of Solutions
SAT ★★
r₁ + r₂ = −ba
Sum of roots without solving. SAT loves asking "what is the sum of the solutions?" — use this shortcut instead of solving.
Product of Solutions
SAT ★★
r₁ · r₂ = ca
Product of roots without solving. Useful for SAT questions that ask about the product of solutions in a quadratic.
Discriminant
SAT ★★★
D = b² − 4ac
D > 0 → 2 real solutions. D = 0 → 1 real solution (touches x-axis). D < 0 → 0 real solutions (no x-intercepts).

Factoring Patterns

Recognize these instantly — they're tested directly and used in dozens of problems on both exams.

Difference of Squares
SAT + ACT ★★★
a² − b² = (a + b)(a − b)
Use whenever you see a perfect square minus a perfect square. Example: x² − 25 = (x + 5)(x − 5).
Perfect Square Trinomial (Sum)
SAT + ACT ★★★
(a + b)² = a² + 2ab + b²
Common trap: students forget the 2ab middle term. This pattern is key for "completing the square" problems.
Perfect Square Trinomial (Diff)
SAT + ACT ★★
(a − b)² = a² − 2ab + b²
Same pattern with subtraction. Note: (a − b)² is always ≥ 0 — squaring can never produce a negative result.
Sum of Cubes
ACT ★
a³ + b³ = (a + b)(a² − ab + b²)
Rare on the ACT, but it does appear. Remember: "Same, Opposite, Always Positive" (SOAP) for the signs.
Difference of Cubes
ACT ★
a³ − b³ = (a − b)(a² + ab + b²)
Same SOAP pattern. The first binomial matches the sign; the trinomial flips the middle sign.

⚡ Exponents, Radicals & Growth

Exponent rules are tested on every single SAT and ACT. Exponential growth/decay and compound interest appear regularly, especially on the SAT's Problem Solving section.

Exponent Rules

Product Rule
SAT + ACT ★★★
am · an = am+n
Same base? Add the exponents. Only works when the bases are identical.
Quotient Rule
SAT + ACT ★★★
am ÷ an = am−n
Same base? Subtract the exponents. Also written as am/an.
Power Rule
SAT + ACT ★★★
(am)n = amn
Raising a power to a power? Multiply the exponents. These three rules handle nearly every exponent question.
Zero Exponent
SAT + ACT ★★
a⁰ = 1 (a ≠ 0)
Anything (except 0) raised to the zero power equals 1. Tricky concept but simple rule.
Negative Exponent
SAT + ACT ★★★
a−n = 1an
A negative exponent flips the base to the denominator. 2⁻³ = 1/8. Does NOT make the result negative.
Fractional Exponent
SAT + ACT ★★★
am/n = (n√a)m = n√(am)
The denominator of the fraction is the root; the numerator is the power. 8^(2/3) = (³√8)² = 2² = 4.

Exponential Growth & Decay

Exponential Function
SAT + ACT ★★★
y = a · bx
a = initial value, b = growth/decay factor. b > 1 is growth, 0 < b < 1 is decay. x is the time period.
Compound Interest
SAT + ACT ★★
A = P(1 + rn)nt
P = principal, r = annual rate (decimal), n = times compounded per year, t = years. SAT and ACT both test this directly.
Continuous Growth/Decay
SAT + ACT ★
A = Pert
For continuous compounding. e ≈ 2.718. Rare on both tests but does occasionally appear. r positive = growth, r negative = decay.

Imaginary & Complex Numbers

Powers of i (Cycle of 4)
SAT ★★
i¹ = i   i² = −1   i³ = −i   i⁴ = 1
The pattern repeats every 4. For in, divide n by 4 and use the remainder: 0→1, 1→i, 2→−1, 3→−i.
Complex Conjugate Product
SAT ★★
(a + bi)(a − bi) = a² + b²
Multiplying conjugates eliminates the imaginary part. Use to rationalize denominators containing i.
💡 VPT Pro Tip

For compound interest problems, read the question carefully for the compounding frequency. "Annually" means n = 1. "Quarterly" means n = 4. "Monthly" means n = 12. Students lose easy points by using the wrong n value.

📐 Geometry

The ACT is heavier on geometry than the SAT (roughly 30% vs. 20%), but both tests cover area, volume, triangles, and angle relationships extensively.

Area & Perimeter

Rectangle
SAT + ACT ★★★
A = lw    P = 2l + 2w
SAT provides the area formula. l = length, w = width. Also applies to squares: A = s², P = 4s.
Square
SAT + ACT ★★
A = s²    P = 4s
A special rectangle where all sides are equal. Diagonal of a square = s√2 (from the 45-45-90 triangle).
Triangle
SAT + ACT ★★★
A = 12bh
h must be the perpendicular height — not the slant side. Given on the SAT reference sheet.
Circle
SAT + ACT ★★★
A = πr²    C = 2πr    d = 2r
Given on SAT reference sheet. Know which formula to use: area for "how much space" and circumference for "how far around."
Trapezoid
SAT + ACT ★★
A = 12(b₁ + b₂)h
b₁ and b₂ are the two parallel bases, h is the perpendicular height between them. NOT given on SAT reference sheet.
Parallelogram
SAT + ACT ★★
A = bh
h is the perpendicular height, not the slant side. Opposite sides are parallel and equal in length.

Volume Formulas

Rectangular Prism (Box)
SAT + ACT ★★
V = lwh
Length × width × height. Given on SAT reference sheet. Use for boxes, rooms, tanks, and rectangular containers.
Cylinder
SAT + ACT ★★
V = πr²h
Base area (πr²) × height. Given on SAT reference sheet. Think: "area of circle times how tall."
Cone
SAT + ACT ★
V = 13πr²h
Exactly ⅓ of a cylinder with the same base and height. Given on SAT reference sheet.
Sphere Volume
SAT + ACT ★
V = 43πr³
Given on SAT reference sheet. The ACT does NOT provide this — memorize it for the ACT.
Pyramid
SAT + ACT ★
V = 13lwh
Given on SAT reference sheet. Like a cone — it's ⅓ of the rectangular prism with the same base and height.

Surface Area

Sphere Surface Area
SAT + ACT ★
SA = 4πr²
NOT on the SAT reference sheet — must be memorized. More common on ACT than SAT.
Cylinder Surface Area
SAT + ACT ★
SA = 2πrh + 2πr²
Lateral area (2πrh) + two circular bases (2πr²). Think of unrolling the cylinder into a rectangle plus two circles.

Pythagorean Theorem & Triangles

Pythagorean Theorem
SAT + ACT ★★★
a² + b² = c²
a, b = legs, c = hypotenuse (longest side, opposite the 90° angle). Right triangles only. Given on SAT reference sheet.
Common Pythagorean Triples
SAT + ACT ★★★
3-4-5    5-12-13    8-15-17    7-24-25
Plus their multiples: 6-8-10, 9-12-15, 10-24-26, etc. Recognizing these saves significant calculation time on test day.
30-60-90 Triangle
SAT + ACT ★★★
x : x√3 : 2x
Short leg (opp 30°) = x. Long leg (opp 60°) = x√3. Hypotenuse (opp 90°) = 2x. Given on SAT reference sheet.
45-45-90 Triangle
SAT + ACT ★★★
x : x : x√2
Both legs equal x. Hypotenuse = x√2. Appears when a square is cut diagonally. Given on SAT reference sheet.
Triangle Inequality
SAT + ACT ★★
a + b > c (any two sides > third)
The sum of any two sides of a triangle must be greater than the third side. Use to determine if three lengths can form a triangle.
Triangle Angle Sum
SAT + ACT ★★★
Angles sum to 180°
If you know two angles, subtract their sum from 180° to find the third. Noted on the SAT reference sheet as a fact.

Angles & Polygons

Supplementary & Complementary
SAT + ACT ★★★
Supplementary = 180°    Complementary = 90°
Supplementary angles form a straight line. Complementary angles form a right angle. These appear in almost every geometry section.
Vertical Angles
SAT + ACT ★★
Vertical angles are equal
When two lines cross, the opposite angles are always equal. Use together with supplementary angles to find all four angles.
Interior Angles of n-gon
SAT + ACT ★★
Sum = (n − 2) × 180°
Pentagon (n=5): 540°. Hexagon (n=6): 720°. Use when given a polygon and asked about angle sums.
Each Angle of Regular n-gon
SAT + ACT ★★
(n − 2) × 180°n
Only for regular polygons (all sides and angles equal). Regular hexagon: each angle = 120°.

⭕ Circles & Coordinate Geometry

Circle equations and arc/sector problems are high-frequency SAT topics. The distance and midpoint formulas appear on both tests regularly.

Circles & Arcs

Arc Length
SAT + ACT ★★
Arc = θ360° × 2πr
θ = central angle in degrees. Proportion: arc length / circumference = central angle / 360°.
Sector Area
SAT + ACT ★★
Sector = θ360° × πr²
"Slice of pie" area. Same proportional logic as arc length but applied to the full circle area.
Central Angle = Arc
SAT + ACT ★★
Central angle = intercepted arc measure
A central angle (vertex at center) has the same degree measure as its intercepted arc.
Inscribed Angle
SAT + ACT ★★
Inscribed angle = 12 × arc measure
An inscribed angle (vertex on circle) is half its intercepted arc. Inscribed in a semicircle = 90°.
Standard Equation of a Circle
SAT ★★★
(x − h)² + (y − k)² = r²
Center at (h, k), radius = r. Watch the signs: (x − 3)² means center x = 3, not −3. High-frequency SAT question type.

Coordinate Geometry

Distance Formula
SAT + ACT ★★★
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Derived from the Pythagorean theorem. Use when finding the length between two coordinate points.
Midpoint Formula
SAT + ACT ★★
M = (x₁ + x₂2 , y₁ + y₂2)
Average the x's, average the y's. Use to find the center of a segment or verify a midpoint.
💡 VPT Pro Tip

When the SAT asks you to find a circle's equation from the general form x² + y² + Dx + Ey + F = 0, complete the square for both x and y terms. Group x terms and y terms, add (D/2)² and (E/2)² to both sides, then factor. This converts directly to standard form.

📐 Trigonometry

The ACT is heavy on trig (roughly 7–10 questions). The SAT usually has 2–4 trig questions. SOH-CAH-TOA and the Pythagorean identity are must-knows for both tests.

Basic Trig Ratios

Sine (SOH)
SAT + ACT ★★★
sin θ = OppositeHypotenuse
The "O" and "H" from SOH-CAH-TOA. Always identify the angle first, then label sides relative to that angle.
Cosine (CAH)
SAT + ACT ★★★
cos θ = AdjacentHypotenuse
The "A" and "H" from CAH. "Adjacent" means the leg next to the angle (not the hypotenuse).
Tangent (TOA)
SAT + ACT ★★★
tan θ = OppositeAdjacent
Also equals sin θ / cos θ. Use when a problem gives you opposite and adjacent sides (no hypotenuse).
Pythagorean Identity
SAT + ACT ★★★
sin²θ + cos²θ = 1
The most important trig identity. If you know sin θ, find cos θ (and vice versa). Rearranges to sin²θ = 1 − cos²θ.
Complementary Angle Relationship
SAT + ACT ★★
sin(x) = cos(90° − x)
Sine and cosine are "co-functions." sin 30° = cos 60°, sin 45° = cos 45°. The SAT tests this relationship directly.

Unit Circle Key Values

Memorize these exact values — they're tested directly on both exams (more heavily on the ACT).

0° (0 radians)
SAT + ACT ★★
sin 0° = 0   cos 0° = 1   tan 0° = 0
Starting position on the unit circle — no vertical displacement, maximum horizontal displacement.
30° (π/6 radians)
SAT + ACT ★★
sin = 12   cos = √32   tan = √33
From the 30-60-90 triangle ratios. sin 30° = ½ is the most commonly tested single trig value.
45° (π/4 radians)
SAT + ACT ★★
sin = √22   cos = √22   tan = 1
From the 45-45-90 triangle. At 45°, sine and cosine are equal. tan 45° = 1.
60° (π/3 radians)
SAT + ACT ★★
sin = √32   cos = 12   tan = √3
Mirror of 30° values: sine and cosine are swapped. sin 60° = cos 30° = √3/2.
90° (π/2 radians)
SAT + ACT ★★
sin 90° = 1   cos 90° = 0   tan 90° = undef.
Maximum vertical, zero horizontal. Tangent is undefined because cos 90° = 0 (division by zero).

Radian Conversion

Degrees to Radians
SAT + ACT ★★
Radians = Degrees × π180
Multiply degrees by π/180 to convert to radians. 90° = π/2, 180° = π, 360° = 2π.
Key Radian Values
SAT + ACT ★★
180° = π    360° = 2π    90° = π/2
Both tests use radians. If a question gives angles in radians and you're confused, convert to degrees first.

📊 Statistics & Probability

Statistics is one of the four main SAT math domains (roughly 25% of the test). Probability and counting show up on both tests. Master the basics — they're straightforward point-getters.

Ratios & Percentages

Percent Formula
SAT + ACT ★★★
Percent = PartWhole × 100
The core percent relationship. "What percent of 80 is 20?" → 20/80 × 100 = 25%.
Percent Change
SAT + ACT ★★★
% Change = New − OldOld × 100
Always divide by the ORIGINAL value. Positive = increase, negative = decrease. One of the most-tested concepts.
Increase by r%
SAT + ACT ★★★
New = Original × (1 + r)
r as a decimal. Increase by 15% → multiply by 1.15. This is the basis for growth models and compound interest.
Decrease by r%
SAT + ACT ★★★
New = Original × (1 − r)
r as a decimal. Decrease by 20% → multiply by 0.80. This is the basis for decay and depreciation problems.

Statistics

Mean (Average)
SAT + ACT ★★★
Mean = SumCount    ↔    Sum = Mean × Count
"Sum = Mean × Count" is often MORE useful on tests. When they give you the average and number of items, use it to find the total.
Median
SAT + ACT ★★★
Middle value when data is sorted
Even count: average the two middle values. Always sort first. Median is resistant to outliers — mean is not.
Mode
SAT + ACT ★
Most frequently occurring value
A data set can have no mode, one mode, or multiple modes. Rarely tested directly but good to know the definition.
Range
SAT + ACT ★★
Range = Max − Min
Simplest measure of spread. Tells you the total span of the data set. Not the same as standard deviation.
Standard Deviation (Concept)
SAT + ACT ★★
Measures average distance from the mean
You won't calculate this on the test. Just know: more spread out = higher SD, more clustered = lower SD. Compare data sets visually.

Probability

Basic Probability
SAT + ACT ★★★
P(event) = Favorable outcomesTotal outcomes
Always between 0 and 1 (or 0% and 100%). The foundation for every probability question on both tests.
P(A and B) — Independent
SAT + ACT ★★
P(A and B) = P(A) × P(B)
For independent events (one doesn't affect the other). Flipping heads twice: ½ × ½ = ¼.
P(A or B) — Mutually Exclusive
SAT + ACT ★★
P(A or B) = P(A) + P(B)
For mutually exclusive events (can't both happen). Rolling a 2 or a 5: 1/6 + 1/6 = 2/6.
General Addition Rule
SAT + ACT ★★
P(A or B) = P(A) + P(B) − P(A and B)
Subtract the overlap to avoid double-counting. Use for Venn diagram and two-way table problems.
Complement Rule
SAT + ACT ★★★
P(not A) = 1 − P(A)
Sometimes it's easier to find the probability of what you DON'T want and subtract from 1. Huge time-saver.

Counting & Distance/Rate

Combinations (nCr)
SAT + ACT ★★
nCr = n!r!(n − r)!
Order does NOT matter. "How many ways to choose 3 from 10?" = 10C3. ACT tests this more than SAT.
Permutations (nPr)
SAT + ACT ★
nPr = n!(n − r)!
Order DOES matter. "How many ways to arrange 3 from 10?" = 10P3. Always larger than nCr for the same values.
Fundamental Counting Principle
SAT + ACT ★★
Total = choices₁ × choices₂ × choices₃ × ...
Multiply the number of options at each step. 3 shirts × 4 pants × 2 shoes = 24 outfits.
Distance / Rate / Time
SAT + ACT ★★★
d = r × t
Also: r = d/t and t = d/r. The basis for all rate, speed, and work-rate problems on both tests.
💡 VPT Pro Tip

On SAT data analysis questions, read the table/graph labels first before reading the question. Students lose 30+ seconds re-reading because they didn't understand the data format upfront. Circle the units — misreading "thousands" vs. "millions" is a classic trap.

🎯 ACT-Only Advanced Topics

These formulas appear on the ACT but are rarely (or never) tested on the Digital SAT. If you're only taking the SAT, you can skip this section. Targeting 30+ on the ACT? Memorize these.

Advanced Trigonometry

Reciprocal Trig Functions
ACT ★★
csc θ = 1sin θ   sec θ = 1cos θ   cot θ = 1tan θ
Cosecant, secant, cotangent are the reciprocals of sine, cosine, tangent. The ACT tests these directly — the SAT does not.
Law of Sines
ACT ★★
asin A = bsin B = csin C
Use with non-right triangles when you know AAS, ASA, or SSA. Lowercase = sides, uppercase = opposite angles.
Law of Cosines
ACT ★★
c² = a² + b² − 2ab · cos C
Use for SAS or SSS non-right triangles. When C = 90°, this simplifies to the Pythagorean theorem (cos 90° = 0).
Triangle Area (Non-Right)
ACT ★
A = 12ab · sin C
Use when you know two sides and the included angle of any triangle (not just right triangles).
Trig Graph Properties
ACT ★★
y = A · sin(Bx + C) + D
Amplitude = |A|    Period = 2π/B    Phase Shift = −C/B    Vertical Shift = D
The ACT frequently asks about amplitude and period. Remember: larger B = shorter period (more compressed horizontally).

Logarithms

Logarithm Definition
ACT ★★
logb(x) = y   ↔   by = x
A log is the inverse of an exponent. log₁₀(1000) = 3 because 10³ = 1000. Convert between log and exponential form.
Log of a Product
ACT ★★
log(ab) = log a + log b
Multiplication inside the log becomes addition outside. Works for any base.
Log of a Quotient
ACT ★★
log(ab) = log a − log b
Division inside becomes subtraction outside. These three log properties cover all ACT log questions.
Log of a Power
ACT ★★
log(an) = n · log a
The exponent comes down as a coefficient. This is the "power rule" for logarithms.

Sequences & Series

Arithmetic Sequence (nth term)
ACT ★★
an = a₁ + (n − 1)d
a₁ = first term, d = common difference, n = term number. Each term increases (or decreases) by d.
Arithmetic Series Sum
ACT ★
Sn = n(a₁ + an)2
Sum of the first n terms. Also written as S = n × (average of first and last term).
Geometric Sequence (nth term)
ACT ★★
an = a₁ · rn−1
a₁ = first term, r = common ratio. Each term is multiplied by r. If |r| < 1, the terms shrink.
Geometric Series Sum
ACT ★
Sn = a₁(1 − rn)1 − r
Sum of the first n terms of a geometric sequence. r ≠ 1. Less commonly tested but appears on harder ACT sections.

Matrices

The ACT occasionally includes basic matrix operations. These never appear on the Digital SAT.

Matrix Addition/Subtraction
ACT ★
Add/subtract corresponding elements
Matrices must have the same dimensions. Add each element in position (i,j) to the corresponding element.
Scalar Multiplication
ACT ★
k × [a b; c d] = [ka kb; kc kd]
Multiply every element in the matrix by the scalar. The matrix dimensions don't change.
Matrix Multiplication
ACT ★
Row × Column for each element
# columns of first = # rows of second. Result dimensions: (rows of first) × (columns of second).
Matrix Multiplication is Non-Commutative
ACT ★
AB ≠ BA
Order matters in matrix multiplication (unlike regular multiplication). The ACT may test whether you know this property.

Function Transformations

Both tests cover transformations, but the ACT tests them more frequently and in more complex ways.

Vertical Shift Up
SAT + ACT ★★
f(x) + k → shift up k units
Adding to the output moves the graph up. Every y-value increases by k.
Vertical Shift Down
SAT + ACT ★★
f(x) − k → shift down k units
Subtracting from the output moves the graph down. Every y-value decreases by k.
Horizontal Shift Left
SAT + ACT ★★
f(x + h) → shift left h units
Adding inside the function moves left (counterintuitive). The change is OPPOSITE to the sign inside.
Horizontal Shift Right
SAT + ACT ★★
f(x − h) → shift right h units
Subtracting inside the function moves right. This is the same reason (x − 3)² centers the parabola at x = 3.
Reflect Over x-axis
SAT + ACT ★★
−f(x) → reflect over x-axis
Negating the output flips the graph upside down. All y-values become their opposites.
Reflect Over y-axis
SAT + ACT ★★
f(−x) → reflect over y-axis
Negating the input flips the graph left-to-right. The shape is mirrored across the y-axis.
Vertical Stretch
SAT + ACT ★★
c · f(x) → vertical stretch by c (c > 1)
Multiplying the output by c > 1 stretches vertically. 0 < c < 1 compresses vertically. Affects amplitude in trig.
💡 VPT Pro Tip

For transformations, remember this: changes INSIDE the function (to x) are horizontal and counterintuitive (opposite direction). Changes OUTSIDE the function (to y) are vertical and intuitive (same direction as the sign). This single rule handles every transformation question.

⭐ If You Only Memorize 15 Formulas

These are the 15 highest-frequency formulas across both the SAT and ACT. If you're short on time, start here — these cover the vast majority of formula-dependent questions.

1 Slope Formula m = (y₂−y₁)/(x₂−x₁)
2 Slope-Intercept Form y = mx + b
3 Quadratic Formula x = (−b ± √(b²−4ac)) / 2a
4 Vertex Form / x = −b/2a y = a(x−h)² + k
5 Difference of Squares a²−b² = (a+b)(a−b)
6 Exponent Rules aᵐ·aⁿ = aᵐ⁺ⁿ
7 Pythagorean Theorem a² + b² = c²
8 Circle Equation (x−h)²+(y−k)² = r²
9 SOH-CAH-TOA sin = O/H, cos = A/H
10 Pythagorean Identity sin²θ + cos²θ = 1
11 Mean (Average) Mean = Sum / Count
12 Probability P = Favorable / Total
13 Percent Change (New−Old)/Old × 100
14 Distance Formula d = √((x₂−x₁)²+(y₂−y₁)²)
15 Distance/Rate/Time d = rt

That's all 119 formulas. Every single one of them has appeared on a real SAT or ACT — most of them multiple times. The difference between a 600 and a 750 on SAT Math (or a 25 and a 34 on ACT Math) often comes down to formula fluency: can you recall and apply these under timed pressure without hesitation? Knowing the formulas is only half the battle — avoid the other mistakes that cost students points by reading 5 SAT Math Mistakes That Cost Students 100+ Points.

Use this page as your study checklist. Go through each section, cover the formulas, and try to write them from memory. Once you can nail every formula in under 3 seconds, shift to timed practice problems. That's where scores actually go up. For context on where these formulas fit into the full Digital SAT math section, see our guide to The Digital SAT in 2026.

If you're preparing for the SAT or ACT and want expert guidance on how to prioritize your formula study for your specific score goals, our 1-on-1 SAT tutors and ACT tutors at Victory Prep have been building personalized math prep plans for students since 2014. Our students average a 210-point improvement on the SAT — and math is almost always where the biggest gains come from.

Need help mastering these formulas? Our tutors break down every concept step by step until it clicks — and build the timed-practice habits that lock them in for test day.

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