ACT Math – Complete Memorization Sheet & Concept Notes

Core Angle & Basic Geometry Facts

• Vertical angles (opposite each other when two lines cross) are congruent: e.g. $x = x$.
• Adjacent angles formed by two intersecting lines are supplementary (sum $=180^{\circ}$).
• Parallel-line angle pairs (corresponding, alternate interior, etc.) are congruent; heavily tested.
• Sum of interior angles of an $n$-gon: $180(n-2)$.
  • Example: pentagon ($n=5$) ⇒ $180(5-2)=540^{\circ}$.

Plane & Solid Shapes: Formulas to Memorize

• Triangle area: $A=\tfrac12 bh$ (works with altitude even if triangle is not right-angled).
• Special right triangles
  • 45^{\circ}40^{\circ}90^{\circ} ⇒ sides $1:1:\sqrt2$.
  • 30^{\circ}60^{\circ}90^{\circ} ⇒ sides $1:\sqrt3:2$.
• Circle
  • Circumference $=2\pi r$.
  • Area $=\pi r^2$.
• Trapezoid area: $A=\tfrac12(b<em>1+b</em>2)h$.
• Kite area: $A=\tfrac12 d<em>1d</em>2$ (diagonals $d<em>1,d</em>2$).
• Right circular cylinder
  • Volume $=\pi r^2 h$.
  • Lateral area $=2\pi r h$; total surface area $=2\pi r h+2\pi r^2$.
• Cube
  • Volume $=s^3$; surface area $=6s^2$.
• Rectangular prism
  • Volume $=lwh$; surface area $=2(lw+lh+wh)$.

Triangles & The Pythagorean Toolkit

• Pythagorean theorem (right triangles): $a^2+b^2=c^2$.
  • Key triples: $3-4-5, 5-12-13$ (+ multiples).
• Third-side rule: for any triangle with sides $a,b,c$ (where $c$ is longest)
  $|a-b|<c<a+b$.

Coordinate-Geometry Formulas

• Slope: $m=\tfrac{y<em>2-y</em>1}{x<em>2-x</em>1}$.
• Midpoint: $\bigl(\tfrac{x<em>1+x</em>2}{2},\tfrac{y<em>1+y</em>2}{2}\bigr)$.
• Distance: $\sqrt{(x<em>2-x</em>1)^2+(y<em>2-y</em>1)^2}$.
• Point-slope: $y-y<em>1=m(x-x</em>1)$.
• Slope-intercept: $y=mx+b$.

Number Theory Essentials

• Integers: …, $-2,-1,0,1,2,…$.
• Primes: only divisors $1$ & itself; $2$ is the sole even prime.
• Rational numbers: expressible as $\tfrac{p}{q}$ where $p,q$ integers, $q\neq0$.
• Irrational: cannot be expressed as a fraction (e.g. $\sqrt5,\pi$).
• Undefined: division by $0$.
• Imaginary/complex: involve $i$ where $i=\sqrt{-1}$.
• GCF: largest integer dividing every member of a set (ex. $\text{GCF}(12,18,30)=6$).
• LCM: smallest positive integer divisible by each number (ex. $\text{LCM}(2,3)=6$).

Linear-Equation Solution Cases

• One solution: slopes different (e.g. $6x-5=8x+2$).
• Zero solutions (parallel lines): slopes equal, intercepts different.
• Infinite solutions (identical lines): both sides identical.

Percent, Ratio & Average Toolkit

• Basic percent proportion: $\tfrac{\text{is}}{\text{of}}=\tfrac{\%}{100}$.
• Decimal method: convert percent → decimal (40 % ⇒ 0.4). Example: 40 % of 80 $=0.4\times80=32$.
• Percent change: $\tfrac{\text{new}-\text{old}}{\text{old}}\times100\%$.
• Direct variation: $y=kx$. Inverse variation: $y=\tfrac{k}{x}$.
• Mean (average): $\bar{x}=\tfrac{\sum x}{n}$.
• Weighted mean: $\bar{x}<em>w=\tfrac{\sum w</em>ix<em>i}{\sum w</em>i}$.
  • Example (running speeds): $\bar{v}=\tfrac{10\,(7)+8\,(11)}{10+8}=\tfrac{70+88}{18}\approx8.78$ mph.

Exponent & Root Laws (ACT Favourites)

• Product: $a^x a^y=a^{x+y}$.
• Quotient: $\tfrac{a^x}{a^y}=a^{x-y}$.
• Zero power: $a^0=1$ ($a\neq0$).
• Power of a power: $(a^x)^y=a^{xy}$.
• Fractional exponents: $a^{\tfrac{x}{y}}=\sqrt[y]{a^{x}}$.
• Roots
  • $\sqrt[n]{ab}=\sqrt[n]{a}\,\sqrt[n]{b}$.
  • $\sqrt[n]{\tfrac{a}{b}}=\tfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$.

Logarithms & Natural Logs

• Definition: $\log_a b=c\;\Longleftrightarrow\;a^{c}=b$.
• Change of base: $\log_b a=\tfrac{\log a}{\log b}$ (works with common or natural log buttons).
• Advanced (for 33-36 scorers)
  • Product: $\log<em>a(MN)=\log</em>a M+\log<em>a N$. • Quotient: $\log</em>a\tfrac{M}{N}=\log<em>a M-\log</em>a N$.
  • Power: $\log<em>a M^{k}=k\log</em>a M$.
• Natural log $\ln x$ : base $e$ (≈2.718 28).

Quadratics

• Quadratic formula: $x=\tfrac{-b\pm\sqrt{b^2-4ac}}{2a}$.
• Vertex form: $y=a(x-h)^2+k$ ⇒ vertex $(h,k)$.
• Discriminant $\Delta=b^2-4ac$
  • $\Delta>0$ ⇒ 2 real roots.
  • $\Delta=0$ ⇒ 1 real root.
  • $\Delta<0$ ⇒ 2 complex roots.

Trigonometry Must-Knows

• SOH-CAH-TOA
  • $\sin\theta=\tfrac{\text{opp}}{\text{hyp}}$, $\cos\theta=\tfrac{\text{adj}}{\text{hyp}}$, $\tan\theta=\tfrac{\text{opp}}{\text{adj}}$.
• Reciprocal & inverse trig functions appear; know calculator usage.
• Pythagorean identity: $\sin^{2}\theta+\cos^{2}\theta=1$.
• Law of sines: $\tfrac{\sin A}{a}=\tfrac{\sin B}{b}=\tfrac{\sin C}{c}$.
• Law of cosines: $c^{2}=a^{2}+b^{2}-2ab\cos C$.
• Periods of trig graphs
  • $y=\sin(bx),\cos(bx),\sec(bx),\csc(bx)$ ⇒ period $\tfrac{2\pi}{b}$.
  • $y=\tan(bx),\cot(bx)$ ⇒ period $\tfrac{\pi}{b}$.

Matrices

• Addition/Subtraction: combine corresponding entries.
• Multiplication: row × column dot products (be sure inner dimensions match).
• Determinant (2×2): (\begin{vmatrix}a&b\c&d\end{vmatrix}=ad-bc).
• ACT treats determinants as “hard” → easy points if memorized.

Conic Sections

• Circle: $(x-h)^2+(y-k)^2=r^{2}$.
• Ellipse (horizontal major axis): $\tfrac{(x-h)^2}{a^{2}}+\tfrac{(y-k)^2}{b^{2}}=1\;(a>b)$.
• Hyperbola
  • Horizontal: $\tfrac{(x-h)^2}{a^{2}}-\tfrac{(y-k)^2}{b^{2}}=1$.
  • Vertical: $\tfrac{(y-k)^2}{a^{2}}-\tfrac{(x-h)^2}{b^{2}}=1$.

Probability & Counting

• Probability: $P=\tfrac{\text{desired}}{\text{total}}$.
• All probabilities in a sample space sum to 1.
• "OR" (mutually exclusive) ⇒ add: $P(A\text{ or }B)=P(A)+P(B)$.
• Independent "AND" ⇒ multiply: $P(A \cap B)=P(A)P(B)$.
• Expected value: $E=\sum x<em>ip</em>i$ (advanced).
• Factorial: $n! = n(n-1)(n-2)\dots1$.
• Permutations (order matters): $^{n}P_{r}=\tfrac{n!}{(n-r)!}$.
• Combinations (order unimportant): $^{n}C_{r}=\tfrac{n!}{r!(n-r)!}$.

Sequences & Series

• Arithmetic nth term: $a<em>n=a</em>1+(n-1)d$.
• Geometric nth term: $a<em>n=a</em>1 r^{n-1}$.

Complex Numbers

• Definition: $i=\sqrt{-1}$ ⇒ $i^{2}=-1$.
• Powers cycle every 4: $i, -1, -i, 1, …$.
• Complex conjugate of $a+bi$ is $a-bi$ (used in rationalizing denominators).
• Absolute value (modulus): $|a+bi|=\sqrt{a^{2}+b^{2}}$ (advanced).

Inequalities & Graphing

• Multiplying/dividing by a negative reverses the inequality sign.
• Graph:
  • ">" or "

Exponential Growth & Decay

• Discrete growth: $y=a(1+r)^{t}$.
• Discrete decay: $y=a(1-r)^{t}$.
• Continuous form (advanced): $y=ae^{kt}$ ((k>0) growth, (k<0) decay).

Circles: Arcs, Sectors & Radians

• Arc length: $\tfrac{\text{arc}}{2\pi r}=\tfrac{\theta}{360}$ ⇒ $\text{arc}=\tfrac{\theta}{360}(2\pi r)$.
• Sector area: $\tfrac{\text{sector}}{\pi r^{2}}=\tfrac{\theta}{360}$ ⇒ $A_{\text{sector}}=\tfrac{\theta}{360}(\pi r^{2})$.

Vectors

• Component form: $\langle x,y\rangle$ or $xi+yj$.
• Addition/Subtraction: add/subtract corresponding components.
• Magnitude: $|\vec{v}|=\sqrt{x^{2}+y^{2}}$ (links to Pythagorean theorem).

Descriptive Statistics

• Standard deviation measures spread; larger spread ⇒ larger SD; tight cluster ⇒ smaller SD.
• Empirical rule (normal curve, advanced):
  • 68 % within $1\sigma$, 95 % within $2\sigma$, 99.7 % within $3\sigma$ of the mean.

Miscellaneous Advanced Topics

• Binomial theorem: $(a+b)^{n}=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^{k}$; coefficients via Pascal’s triangle.
• Vertical asymptote: set denominator of rational function to $0$, solve for $x$.
• Horizontal asymptote quick rules (not often tested): compare degrees of numerator & denominator.

Study Strategy & Resources Mentioned

• Equation sheet downloadable via free trial of Prep Pros “Ultimate ACT Math Course”.
• Book + 30-hour video course uses 4-level system:
  • Level 1: core concepts for any score.
  • Levels 2-4: progressively advanced (aiming for 27-36).
• Free trial grants three full chapters, full equation sheet, video lessons, practice sets & solutions.
• Matrices chapter in trial—covers add/subtract/multiply/determinant; easy points despite high ACT difficulty label.
• Study plan: take a practice ACT, identify level, follow matching study guide.
• Matt (Prep Pros) publishes weekly ACT content; comments welcomed for questions.