The Ultimate Formula Sheet for ACT Math

1.Number Definitions 2.Fractions, Decimals & Percentages 3.Rates, Ratios & Proportions 4.Exponents, Roots & Polynomials 5.Parabolas 6.Graphing Lines 7.Parent Graphs & Transformations 8.Data, Probability & Combinatorics 9.Angles 10.Triangles 11.Circles 12.Polygons 13.Solids 14.Trigonometry 15.Sequences & Series 16.Logarithms 17.Vectors 18.Matrices 19.Conic Sections

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Number Definitions

Natural numbers1, 2, 3, 4, … (positive integers; also called counting numbers)

Whole numbers0, 1, 2, 3, … (natural numbers plus zero)

IntegersAll whole numbers and their negatives; no fractions or decimals

Rational numbersAny number that can be expressed as a ratio of two integers; includes all integers, fractions, and terminating or repeating decimals

Irrational numbersCannot be expressed as a ratio of two integers; non-terminating, non-repeating decimals (ex: π, √2)

Real numbersAll rational and irrational numbers

Complex numbersAll real numbers plus imaginary numbers; form: a + bi

Prime numbers to memorize2, 3, 5, 7, 11, 13, 17, 19, 23 — 2 is the only even prime

Fractions, Decimals & Percentages (r = percent in decimal form)

Fractionpartwhole

Percentpart100

Percent increase or decrease|old − new|old × 100%

Increase by a percentmultiply by (1 + r)

Decrease by a percentmultiply by (1 − r)

Simple InterestA = P(1 + rt)

Compounded annuallyA = P(1 + r)ᵗ

Compounded n times/yearA = P(1 + rn)ⁿᵗ

Rates, Ratios & Proportions

Distance, Rate, TimeDistance = Rate × Time

Conversion factor (general form)ending unitsstarting units

Conversion factor (example)10 feet × 12 inches1 foot = 120 inches

Mixture formula(Concentration of A × Volume of A) + (Concentration of B × Volume of B) = Final concentration × (Volume of A + Volume of B)

Direct variationy = kx (y varies directly as x)

Inverse variationy = kx (y varies inversely as x)

Exponents, Roots & Polynomials

Multiplication rulea^b · a^c = a^b+c

Division rulea^ba^c = a^b−c

Power rule(a^b)^c = a^bc

Negative exponentsa^−b = 1a^b

Fractional exponentsa^b/c = c√a^b = (c√a)^b

Zero exponenta⁰ = 1

Imaginary uniti = √−1; i² = −1; i³ = −i; i⁴ = 1

Imaginary cycle patterni⁴ⁿ = 1; i⁴ⁿ⁺¹ = i; i⁴ⁿ⁺² = −1; i⁴ⁿ⁺³ = −i
To find iⁿ: divide n by 4 and use the remainder (0→1, 1→i, 2→−1, 3→−i)

Complex conjugates(a + bi)(a − bi) = a² + b²

Difference of squaresa² − b² = (a + b)(a − b)

Perfect square trinomiala² ± 2ab + b² = (a ± b)²

Sum of cubesa³ + b³ = (a + b)(a² − ab + b²)

Difference of cubesa³ − b³ = (a − b)(a² + ab + b²)

Completing the squarex² + bx + b²4 = (x + b2)²

Parabolas

Standard Form: f(x) = ax² + bx + c

Vertex(−b2a, f(−b2a))

y-interceptc

x-intercepts (quadratic formula)x = −b ± √b² − 4ac2a

Discriminant b² − 4acPositive → 2 real roots
Zero → 1 real root
Negative → 2 imaginary roots

Sum of solutions−ba

Product of solutionsca

Factored Form: f(x) = a(x − m)(x − n)

x-interceptsm and n

x-coordinate of vertexm + n2

Vertex Form: f(x) = a(x − h)² + k

Vertex(h, k)

Directiona > 0 opens up; a < 0 opens down

Graphing Lines

Slope formulam = y₂ − y₁x₂ − x₁

Slope-intercept formy = mx + b

Standard formAx + By = C

Point-slope formy − y₁ = m(x − x₁)

Horizontal line slopem = 0

Vertical line slopeundefined

Parallel linesEqual slopes

Perpendicular linesSlopes are opposite reciprocals

Distance formulad = √(x₂ − x₁)² + (y₂ − y₁)²

Midpoint formulaM = (x₁ + x₂2, y₁ + y₂2)

Quadrants

Parent Graphs & Transformations

Transformation	Visual effect
f(x) + k	Shift up by k units
f(x) − k	Shift down by k units
f(x + h)	Shift left by h units
f(x − h)	Shift right by h units
−f(x)	Reflect over the x-axis (flip upside down)
f(−x)	Reflect over the y-axis (flip left-right)
c · f(x)	Stretch vertically by factor of c (becomes skinnier)
1c · f(x)	Shrink vertically by factor of c (becomes fatter)

Data, Probability & Combinatorics

Averagesum of itemsnumber of items

Probabilitydesired outcomespossible outcomes

MedianCenter data point (when ordered)

ModeMost frequent data point

RangeMaximum − minimum

P(A and B) — independent eventsP(A) × P(B)

P(A or B)P(A) + P(B) − P(A and B)

Expected ValueE(x) = Σ xᵢ · P(xᵢ)

Factorialn! = n(n−1)(n−2)…1 0! = 1

Fundamental Counting PrincipleIf one event can occur in m ways and a second independent event can occur in n ways, then the two events can occur in m×n total ways (ex: 3 shirts × 2 pants = 6 outfits)

Combinations — order doesn't matter nCrn!r!(n−r)!

Permutations — order matters nPrn!(n−r)!

Angles

Vertical anglesVertical angles are congruent

Complementary anglesAdd up to 90°

Linear pairAngles that form a linear pair are supplementary (add up to 180°)

Angles around a pointAdd up to 360°

Parallel lines cut by a transversalAll acute angles are congruent; all obtuse angles are congruent

Triangles

AreaA = ½bh

Area (non-right triangle)A = ½ab sin C

AnglesThree angles of a △ add up to 180°

Exterior angle= sum of two remote interior angles

Pythagorean Theorema² + b² = c²

Pythagorean Triples3-4-5, 5-12-13, 7-24-25

Law of Sinesasin A = bsin B = csin C

Law of Cosinesa² = b² + c² − 2bc·cos(A)

Special Right Triangles

Law of Sines / Cosines

Triangle Inequality TheoremThe third side x must satisfy:
|a − b| < x < a + b

Circles

AreaA = πr²

CircumferenceC = 2πr

Arc lengthx360 = arccircumference (x = central angle)

Sector areax360 = sectorarea of circle (x = central angle)

Radius and tangentA radius and tangent line meet at a right angle

Central vs. inscribed angleA central angle is double the inscribed angle that subtends the same arc

Radius ⊥ tangent

Central angle = 2 × inscribed angle

Polygons (n = number of sides)

Area of a rectangleA = lw

Area of a trapezoidA = 12(b₁ + b₂)h

Sum of exterior angles360°

Sum of interior angles180(n − 2)

One interior angle (regular polygon)180(n − 2)n

Number of diagonals (convex only)n(n − 3)2

Regular hexagonCan be divided into 6 equilateral triangles; all sides and all interior angles are equal (each interior angle = 120°)

Properties of Parallelograms

SidesOpposite sides are parallel and congruent

AnglesOpposite angles are congruent; consecutive angles are supplementary

DiagonalsBisect each other and each forms a pair of congruent triangles; if congruent → rectangle; if perpendicular → rhombus

AreaA = base × height

Solids

Volume

Rectangular prism (box)V = lwh

CylinderV = πr²h

SphereV = 43πr³

ConeV = 13πr²h

PyramidV = 13lwh

Triangular prismV = 12bh × length

Surface Area

Rectangular prism (box)SA = 2(lw + lh + wh)

CylinderSA = 2πr² + 2πrh

CubeSA = 6s²

SphereSA = 4πr²

Trigonometry

SOHCAHTOA

Sinesin = opphyp

Cosinecos = adjhyp

Tangenttan = oppadj

Cosecantcsc(x) = 1sin(x)

Secantsec(x) = 1cos(x)

Cotangentcot(x) = 1tan(x)

Tangent identitytan(x) = sin(x)cos(x)

Pythagorean identitysin²(x) + cos²(x) = 1

Cofunction identitysin(x) = cos(90° − x)

Radians360° = 2π radians

Sinusoidal Functions: y = A sin(Bx − C) + D (also for cos, csc, sec)

Amplitude|A|

Period2πB

Phase shiftCB

Vertical shiftD

Tangent Functions: y = A tan(Bx − C) + D (also for cot)

AmplitudeNone

PeriodπB

Phase shiftCB

Vertical shiftD

Sequences & Series (a₁ = first term, n = number of terms, d = common difference, r = common ratio)

Arithmetic — add a constant each time (ex: 5, 8, 11, 14, 17…)

nth term (closed form)a_n = a₁ + (n − 1)d

nth term (recursive)a_n = a_n-1 + d

Sum of n termsS_n = n2(a₁ + a_n)

Geometric — multiply by a constant each time (ex: 3, 6, 12, 24, 48…)

nth term (closed form)a_n = a₁ · rⁿ⁻¹

nth term (recursive)a_n = a_n-1 · r

Sum of n termsS_n = a₁(rⁿ − 1)r − 1

Logarithms

DefinitionIf log_b(a) = x, then b^x = a

Change of Baselog_b(a) = log alog b

Product Propertylog_b(x) + log_b(y) = log_b(xy)

Quotient Propertylog_b(x) − log_b(y) = log_b(xy)

Power Propertylog(x^a) = a · log(x)

Natural Logln x = log_e(x), e ≈ 2.718

Vectors

Component formv = <x, y>

Magnitude|v| = √x² + y²

Addition<x₁, y₁> + <x₂, y₂> = <x₁+x₂, y₁+y₂>

Subtraction<x₁, y₁> − <x₂, y₂> = <x₁−x₂, y₁−y₂>

Scalar multiplicationa<x, y> = <ax, ay>

Unit vector (i + j form)<x, y> = xi + yj

Matrices

Addition — only possible when rows of first = rows of second AND columns of first = columns of second Matrix addition

Scalar multiplication

Multiplication — only possible when columns of first = rows of second Matrix multiplication

Conic Sections

Circle — center (h, k), radius r(x − h)² + (y − k)² = r²

Ellipse — center (h, k); 2a = horizontal axis, 2b = vertical axis; c = distance from center to focus(x − h)²a² + (y − k)²b² = 1

Horizontal ellipsea² = b² + c²

Vertical ellipseb² = a² + c²

Horizontal hyperbola(x − h)²a² − (y − k)²b² = 1

Vertical hyperbola(y − k)²a² − (x − h)²b² = 1

Circle

Ellipse

Hyperbola

The Ultimate Formula Sheet for ACT Math

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