This page is a complete ACT Math playbook for students. You’ll find formulas, fast strategies, topic tips, timing rules, and two study plans.
Official scope: Read the ACT Math overview to confirm domains and test format. (act.org).
Memorize the 31 formulas and when to use each.
Write what’s given, what’s asked, and the link (formula or strategy).
Plug in easy numbers on abstract algebra; back-solve from choices on multi-step.
Draw and label for geometry and word problems.
Skip, star, and return. Guess one letter at the end - no penalty.
Practice like the real thing: 60 questions in 60 minutes.
ACT Math at a glance
The section mixes algebra, functions, geometry, trigonometry, statistics, and probability. Most questions are one or two steps; a few are multi-step with traps.
Calculator: You may use one. Check the official Calculator Policy before test day (act.org).
Internal help: see ACT Test Dates & Deadlines 2025–26.
Download the one-page sheet and keep it visible this week.
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Line: y = m x + b
Point-slope: y − y₁ = m(x − x₁)
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
Distance: √((x₂ − x₁)² + (y₂ − y₁)²)
Parallels: slopes equal; Perpendicular: m₁·m₂ = −1
Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a)
Discriminant: b² − 4ac
Vertex (for ax²+bx+c): x = −b / (2a)
Zeros from factors: a(x − r₁)(x − r₂) → r₁, r₂
Circle: (x − h)² + (y − k)² = r²
Pythagorean: a² + b² = c²
45-45-90: 1 : 1 : √2; 30-60-90: 1 : √3 : 2
Triangle area: A = ½ b h
Parallelogram area: A = b h
Trapezoid area: A = ½ (b₁ + b₂) h
Rectangle/Square: A = l w; A = s²
Circle: A = π r²; C = 2π r
Sector/Arc: A = (θ/360) π r²; L = (θ/360)·2π r
Prism/Cylinder: V = B h; V = π r² h
Pyramid/Cone: V = ⅓ B h; V = ⅓ π r² h
Sphere: V = ⁴⁄₃ π r³; SA = 4π r²
Exponents: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n
Logs: log(ab) = log a + log b; log(a^k) = k·log a
Stats: mean, median, mode, range
Probability: P = favorable/total; independent: P(A∩B) = P(A)P(B)
Permutations/Combinations: nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]
Arithmetic seq: aₙ = a₁ + (n−1)d; Sₙ = (n/2)(a₁ + aₙ)
Geometric seq: aₙ = a₁ r^(n−1); Sₙ = a₁(1 − r^n)/(1 − r)
Degree–radian: 180° = π rad
Right-triangle trig: sin = opp/hyp; cos = adj/hyp; tan = opp/adj
How to study formulas: make flashcards with “name → trigger → one example.” Test yourself with 3-minute “formula sprints” each day.
Plug-in (turn variables into easy numbers)
If a question uses letters only, try x=2x=2 or x=3x=3.
Compare choices using your test value to find what matches.
Back-solve (work from answers)
For multi-step algebra, plug each choice into the equation.
Often C or D hits first; stop once one choice fits.
Draw-and-label (words → picture)
Sketch a quick diagram. Mark lengths, angles, and parallel lines.
Right triangles unlock many geometry items.
Units-first (avoid nonsense answers)
Write units beside each number. Cancel units like factors.
If units don’t match the ask, you’re on the wrong path.
Algebra & Equations
Systems: Start with elimination if coefficients line up.
Quadratics: Connect zeros ↔ factors ↔ vertex.
Example: If 2x + 3 = 13 → x = 5. If the ask is 2x − 12x − 1, answer 99 directly—don’t stop at x.
Functions & Graphs
Lines: Identify slope and intercept fast.
Transformations: y = f(x ± a) ± b → shifts left/right and up/down.
Example: y = 2f(x − 3) + 4 means right 3, vertical stretch by 2, then up 4.
Geometry
Similar triangles: Set side ratios; solve in one step.
Angles: Parallel lines create equal or supplementary angles.
Example: If two triangles are similar with scale 2, all corresponding sides double; areas scale by 4.
Trigonometry
Defaults: SOH-CAH-TOA on right triangles.
Special triangles: Know 30°-60°-90° and 45°-45°-90° cold.
Example: In a 30°-60°-90° with hypotenuse 10, short leg = 5, long leg = 5√3.
Statistics & Probability
Tables/trees: Organize totals to avoid double counts.
Complements: For “at least one,” use 1 − P(none).
Example: Two independent events 0.3 and 0.5: both = 0.15; at least one = 1 − (0.7 × 0.5) = 1 − 0.35 = 0.65.
2-week sprint (45–60 minutes/day)
Days 1-2: Learn and quiz the 31 formulas.
Days 3-4: Algebra drills, 40 timed Qs across lines, systems, quadratics.
Days 5-6: Geometry + trig drills, 40 timed Qs with sketches
Day 7: Full 60-question set under real timing.
Days 8–9: Functions + stats/probability, 40 timed Qs.
Day 10: Hard-only set (plug-in, back-solve, multi-step).
Day 11: Full test.
Day 12: Deep review; redo starred misses.
Day 13: Speed set: 30 Qs in 25 minutes.
Day 14: Final full test + light formula refresh.
4-week steady plan (5 days/week)
Week 1: Formulas + algebra basics; 30-Q checkpoint.
Week 2: Geometry + trig; 30-Q checkpoint.
Week 3: Functions, stats, probability; one 60-Q test.
Week 4: Two full tests; fix your two weakest strands.
You have 60 minutes for 60 questions.
Some items take 20–30 seconds; bank that time for tougher ones.
How to fill it: Record set/topic, Q#, what you did, the correct approach, and a short fix. Aim to kill repeat mistakes within two practice sets.
Check which calculators are allowed and which functions are restricted. Practice with the same device you’ll use on test day.
Dropping negative signs when substituting.
Mixing units (feet vs inches) in the same problem.
Solving for xx when the ask is an expression in xx.
Rounding too early and compounding error.
Ignoring non-real roots when the discriminant is negative.
Graph a random line y=mx+by=mx+b. Mark slope, intercept, and one more point.
Build two similar triangles; write three true side ratios.
Write a quadratic with zeros 22 and -3-3; expand; find vertex.
Draw a probability tree for two draws without replacement.
Convert 150∘150^\circ to radians; find area of a sector with r=6r=6.
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