ACT Math Mastery Guide (2026)

Section Overview

The ACT Math section is the second section of the ACT. You have 60 minutes to answer 60 questions — exactly 1 minute per question on average. A calculator is permitted throughout, but no formula sheet is provided (unlike the SAT). This means you must memorize every formula you will need.

Questions are generally arranged in order of increasing difficulty, with questions 1–20 being easier and questions 51–60 being the hardest. The last 10 questions often test advanced algebra, pre-calculus, and trigonometry concepts.

No Formula Sheet — What You Must Memorize

The ACT provides no reference sheet. These are the formulas you must have memorized before test day:

Algebra & Functions

• Slope formula: m = (y₂ – y₁) / (x₂ – x₁)
• Slope-intercept form: y = mx + b
• Point-slope form: y – y₁ = m(x – x₁)
• Standard form: Ax + By = C
• Quadratic formula: x = (–b ± √(b² – 4ac)) / 2a
• Vertex form: y = a(x – h)² + k; vertex is (h, k)
• Distance formula: d = √((x₂–x₁)² + (y₂–y₁)²)
• Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2)
• Percent change: (new – old) / old × 100

Geometry

• Area of rectangle: A = lw; triangle: A = ½bh; circle: A = πr²
• Circumference: C = 2πr; Perimeter of square: P = 4s
• Pythagorean theorem: a² + b² = c²
• Special triangles: 30-60-90 (1:√3:2) and 45-45-90 (1:1:√2)
• Volume of rectangular prism: V = lwh; cylinder: V = πr²h; cone: V = ⅓πr²h; sphere: V = (4/3)πr³
• Arc length: (central angle / 360°) × 2πr; Sector area: (central angle / 360°) × πr²
• Sum of interior angles of a polygon with n sides: (n – 2) × 180°
• Equation of a circle: (x – h)² + (y – k)² = r²

Trigonometry

• SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
• Unit circle values: sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2
• Pythagorean identity: sin²θ + cos²θ = 1
• Law of Sines: a/sin A = b/sin B = c/sin C
• Law of Cosines: c² = a² + b² – 2ab cos C

Topics by Frequency

Geometry is the largest topic area (~25–30%) — more than on the SAT. If you are weak in geometry, this is your highest-leverage study area. Pre-Algebra questions tend to be easy and fast, so do not lose points there.

Pre-Algebra

Pre-Algebra covers the fundamentals that appear in easier questions (1–25) as well as embedded in harder problems. Topics include:

• Order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
• Fractions, decimals, percents: Converting between them, percent of, percent change, successive percent changes
• Ratios and proportions: Setting up and solving proportions, part-to-part vs. part-to-whole ratios
• Statistics: Mean = sum ÷ count; median = middle value; mode = most frequent; range = max – min
• Probability: P(event) = favorable / total; compound probability; expected value
• Number properties: Even/odd rules, prime numbers, factors and multiples, LCM, GCF
• Sequences: Arithmetic sequences (common difference) and geometric sequences (common ratio)

Algebra

Algebra questions build in difficulty from simple linear equations to quadratics, systems, and function notation.

Linear equations and inequalities

Isolate the variable. For inequalities: flip the sign when multiplying or dividing by a negative. For systems of two linear equations, use substitution or elimination. Know the three cases: one solution (different slopes), no solution (same slope, different intercepts), infinite solutions (same line).

Quadratic equations

Factor when possible. Use the quadratic formula when the expression does not factor cleanly. The discriminant (b² – 4ac): positive = two solutions, zero = one solution, negative = no real solutions. Know vertex form for finding the maximum/minimum of a parabola.

Function notation

f(x) means: substitute the value inside the parentheses everywhere x appears. f(2) = the value when x = 2. Composite functions: f(g(x)) = substitute g(x) into f. The ACT tests these in mid-to-hard questions.

Absolute value

|x| = a means x = a or x = –a. |x| < a means –a < x < a. |x| > a means x > a or x < –a.

Geometry

Geometry is the largest ACT Math topic and covers both plane figures and 3D shapes.

Triangles

Interior angles sum to 180°. Exterior angle = sum of the two non-adjacent interior angles. Similar triangles have proportional sides. Know the Pythagorean theorem and both special triangles (30-60-90 and 45-45-90) without looking them up.

Circles

Area = πr², circumference = 2πr. Arc length and sector area use the central angle fraction. Inscribed angle = half the central angle that subtends the same arc. Know the standard form of a circle equation and how to complete the square to convert to standard form.

Coordinate geometry

Lines: slope, intercepts, parallel (same slope), perpendicular (negative reciprocal slopes). Parabolas open up when a > 0, down when a < 0. The vertex is the maximum or minimum point.

3D figures

Volume formulas for rectangular prisms, cylinders, cones, spheres, and pyramids must be memorized. Surface area: add the areas of all faces. These appear in mid-to-hard questions.

Trigonometry

Trig is 5–10% of the test — only 3–6 questions — but they are often among the hardest. Students who know trig gain points that most competitors leave on the table.

SOH-CAH-TOA

In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Most ACT trig questions at the basic level ask you to find a missing side using these ratios or to find an angle given two sides (use the inverse trig function on your calculator).

Unit circle basics

Know the sine and cosine values for 0°, 30°, 45°, 60°, and 90°. Know that sine is the y-coordinate and cosine is the x-coordinate on the unit circle. Know the signs of sin/cos/tan in each quadrant (All Students Take Calculus: All positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4).

Trig identities (last few questions)

• Pythagorean identity: sin²θ + cos²θ = 1
• tan θ = sin θ / cos θ
• Reciprocal identities: csc = 1/sin, sec = 1/cos, cot = 1/tan
• Complementary angles: sin(θ) = cos(90° – θ)
• Double angle formulas (rare but possible): sin(2θ) = 2 sin θ cos θ

Law of Sines and Cosines

These appear in the last 10 questions for non-right triangles. Law of Sines: a/sin A = b/sin B. Law of Cosines: c² = a² + b² – 2ab cos C. If you know two sides and the included angle, use Cosines. If you know two angles and a side, use Sines.

The Last 10 Questions

Questions 51–60 are significantly harder than the first 40. They often test:

• Trig identities and non-right triangle formulas (Law of Sines/Cosines)
• Complex function composition and inverse functions
• Matrices and determinants (rare but tested)
• Logarithms and exponential equations
• Complex numbers (i = √(–1))
• Conic sections other than parabolas (ellipses, hyperbolas)
• Sequences and series (arithmetic and geometric sums)

Time allocation for the last 10

If you are targeting a score of 28–32, it is acceptable to spend 1.5 minutes on questions 51–60 and skip any you cannot start within 30 seconds. Answer all 10 (guess if needed — no penalty) and move on. The points from the first 50 questions are more reliably earned.

If you are targeting 33–36, you need most of the last 10. Drill trig, logarithms, complex numbers, and function composition specifically in your preparation.

Common Traps

Calculator Strategy

A calculator is allowed throughout the ACT Math section. Use it wisely:

• Use for: Arithmetic with large numbers, confirming factored quadratics, evaluating trig functions, checking arithmetic in multi-step problems
• Do NOT use for: Simple arithmetic that takes less than 5 seconds by hand — calculator entry errors waste time
• Graphing calculators: Highly useful for graphing parabolas, finding intersections, and verifying answers to quadratic or trig questions
• Practice with your actual calculator before test day — know where the trig functions, exponent keys, and parentheses are without looking
• Do not rely on the calculator for understanding. If you do not understand the concept, the calculator cannot tell you what equation to type.

Time Strategy

At 1 minute per question on average, you have almost no buffer. Use this approach:

• Questions 1–30: Aim for under 50 seconds per question. These should be straightforward. If one takes over 90 seconds, skip it and return.
• Questions 31–50: Budget 1–1.5 minutes each. Medium-hard — work carefully but keep moving.
• Questions 51–60: Budget 1.5–2 minutes each. If you cannot make progress in 30 seconds, make your best guess and move on.
• Leave no blanks: With 5–10 minutes remaining, fill in an answer for every question you have not answered — there is no penalty for wrong answers.
• Check your work on questions 1–15 if you finish with time to spare — careless errors on easy questions cost the most points relative to effort.

ACT Math Study Plan