ACT Math Formula Sheet: 40 Formulas You Must Know
- Edu Shaale
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Algebra · Geometry · Coordinate Geometry · Trigonometry · Statistics · Probability · Worked Examples · Priority Rankings
Published: May 2026 | Updated: May 2026 | ~18 min read
0 Formulas provided by ACT on test day — everything must be memorised | 45 Math questions in 50 min (Enhanced ACT 2025+) — ~67 sec per question | 19.0 National average ACT Math score — knowing formulas is your fastest edge | 40 Must-know formulas in this guide — organised by category and priority |
~35% Algebra + Coordinate Geometry — the single largest formula category | ~25% Geometry questions — area, volume, Pythagorean theorem, circles | ~10% Trigonometry — SOHCAHTOA, Law of Sines/Cosines, identities | 29% Students who meet ACT College Readiness Benchmark (score ≥ 22) |
Table of Contents
Introduction: The ACT Gives You Nothing — Here Is Everything You Need
Most students preparing for the ACT assume they can figure out the math they need on test day — plug in numbers, work backwards, draw a diagram. That assumption is accurate for a handful of questions. It is catastrophically expensive for the rest.
The ACT Math section provides no formula sheet. Unlike the SAT, which gives students a reference box with basic geometry formulas, the ACT operates on the principle that you either know the formula or you do not. On a section where you have roughly 67 seconds per question, stopping to reconstruct a forgotten formula from first principles is not a recoverable mistake — you have already lost the time for the next question before you start it.
The national average ACT Math score is 19.0. Only 29% of students meet the College Readiness Benchmark of 22. This is not primarily a question of intelligence or mathematical ability — it is a question of preparation. Students who know their formulas cold solve routine questions in 20–30 seconds and invest the remaining time in harder problems. Students who do not know their formulas spend that time on recovery. The math section score gap between those two groups is typically 4–6 points — the difference between a 24 and a 28, or a 28 and a 32.
This guide covers all 40 formulas you need to know for the ACT Math section, organised by category, prioritised by frequency, and illustrated with the types of questions they appear in. It is not a textbook chapter on mathematics — it is a strategic reference for a standardised test with a specific format, specific question types, and specific formula demands.
Read it once to understand the framework. Use the formula boxes for daily review. Return to the mistakes section before every practice test. After four weeks of deliberate study using this guide, no ACT Math formula should feel unfamiliar on test day.
1. Why ACT Math Formulas Matter More Than You Think
The argument against memorising formulas is that you can derive them on the fly. That argument misunderstands the ACT.
On a section with 67 seconds per question, derivation is not a strategy — it is a score penalty. Students who know the quadratic formula respond to a quadratic question in 25 seconds. Students who try to re-derive or remember it lose 45 seconds minimum, and frequently make an error in the derivation. The ACT does not reward mathematical insight under time pressure — it rewards fast, accurate formula application.
⚡ THE CRITICAL DIFFERENCE: ACT vs SAT on Formulas |
SAT: Provides a reference box with basic geometry formulas at the start of each Math module. Desmos graphing calculator built-in. ACT: Provides zero formulas. No reference sheet. Calculator permitted (external) but no built-in tool. Implication: ACT formula memorisation is strictly non-negotiable. SAT formula memorisation is beneficial but partially recoverable. If you are studying for the ACT, formula recall speed is a direct scoring variable. |
Why Formula Knowledge Drives ACT Math Scores | The Impact |
No formula sheet provided | Every formula — geometry, trig, statistics — must come from memory |
67 seconds per question average | Derivation under time pressure costs 45+ seconds — unrecoverable |
~35% Algebra questions | Slope, quadratic, distance formulas appear on nearly every test |
~25% Geometry questions | Area, volume, circle, triangle formulas tested directly |
~10% Trigonometry questions | SOHCAHTOA + Law of Sines/Cosines — non-intuitive without memorisation |
Calculator permitted (whole section) | Calculator covers arithmetic — not formula recall or application |
Multiple choice (4 options) | Right formula = right answer; wrong formula = attractive wrong answer |
The students who score 30+ on ACT Math are not necessarily stronger mathematicians than those who score 24. They are faster and more accurate on formula-dependent questions — which frees cognitive capacity for the 10–15 genuinely hard problems where mathematical reasoning matters.
2. How the Enhanced ACT Math Section Is Structured (2025–2026)
The Enhanced ACT, introduced for online testing in April 2025 and extended to all test formats from September 2025, changed the Math section's length and composition. Students preparing in 2025–2026 should know both the structure and the formula implications of each category.
Content Category | % of Section | Question Count (~) | What It Tests | Formulas Required? |
Preparing for Higher Math — Number & Quantity | ~57–60% total (combined) | ~2–4 | Complex numbers, rational exponents, vectors, matrices | Moderate — exponent rules, matrix operations |
Preparing for Higher Math — Algebra | Part of PHM | ~7–9 | Linear/quadratic equations, inequalities, systems, functions | HIGH — quadratic formula, slope, distance |
Preparing for Higher Math — Functions | Part of PHM | ~7–9 | Exponential, logarithmic, polynomial, trig functions | HIGH — function forms, trig definitions |
Preparing for Higher Math — Geometry | Part of PHM | ~5–7 | Congruence, similarity, area, volume, circles, trig | HIGH — area/volume/circle/trig formulas |
Preparing for Higher Math — Statistics & Probability | Part of PHM | ~4–6 | Probability, data analysis, inference | Moderate — mean, combinations, probability |
Integrating Essential Skills | ~40–43% | ~18–19 | Rates, percentages, proportions, area, volume in context | HIGH — all geometry + percentage formulas |
Modeling (cross-cutting) | ≥20% (overlapping) | Integrated | Applied real-world math in any category | Same formulas, applied in context |
Enhanced ACT (2025+): What Changed for Math |
Questions: 60 → 45 (paper and online from September 2025) Time: 60 min → 50 min Time per question: 60 sec → ~67 sec (slight improvement) Scoring: Math now counts 1/3 of composite (up from 1/4) when Science is optional Formula policy: Unchanged — no formula sheet provided Calculator policy: Unchanged — approved calculators permitted for the full section |
3. Formula Priority Framework: Must-Know vs Nice-to-Know
Not all formulas carry equal weight on the ACT. The following priority framework reflects question frequency analysis across official ACT practice materials. Use this to sequence your study time.
Priority Level | Definition | Formulas in This Guide | Who Needs Them |
MUST-KNOW | Appears on nearly every ACT administration. Missing it costs multiple questions. | Formulas #1–#22 (Algebra + Geometry core) | Every student — regardless of target score |
HIGH-VALUE | Appears on most administrations. Especially important for 26+ scorers. | Formulas #23–#30 (Trigonometry) | Students targeting 26–32 |
SCORE-BOOSTER | Appears regularly but less frequently. Critical for 30+ scorers. | Formulas #31–#37 (Statistics + Probability) | Students targeting 30–36 |
ELITE | Appears occasionally. Mastery separates 32 from 36. | Formulas #38–#40 (Sequences, Logs, Advanced) | Students targeting 33–36 |
✅ STUDY SEQUENCING RULE |
Do not move to a lower-priority category until the higher-priority formulas are automatic. A student who cannot recall the distance formula in 3 seconds should not be studying logarithm properties. Formula recall must be effortless — any cognitive effort spent remembering a formula is effort taken away from solving the problem. |
CATEGORY 1 — Algebra & Functions | Formulas #1–#10 |
4. Algebra & Functions: The 10 Core Formulas
Algebra accounts for roughly 35% of the ACT Math section when coordinate geometry is included. These ten formulas are the foundation — students who do not know them automatically cannot score above 24 without extraordinary luck on problem selection.
Slope Formula m = (y₂ − y₁) / (x₂ − x₁) | Appears on virtually every ACT. Tested directly and embedded in coordinate geometry questions. Know it in under 3 seconds. |
Slope-Intercept Form y = mx + b (m = slope, b = y-intercept) | Most linear equation questions use or imply this form. Converting to this form from standard form is a frequent sub-task. |
Point-Slope Form y − y₁ = m(x − x₁) | Use when given a point and a slope. Faster than slope-intercept when the y-intercept is not given directly. |
Standard Form of a Line Ax + By = C (A, B, C are integers; A ≥ 0) | Know how to convert to slope-intercept. ACT sometimes gives this form and asks for slope: m = −A/B. |
Quadratic Formula x = [−b ± √(b² − 4ac)] / 2a | Use when a quadratic cannot be factored. The discriminant b² − 4ac tells you the number of real solutions: >0 (two), =0 (one), <0 (none). |
Distance Formula d = √[(x₂ − x₁)² + (y₂ − y₁)²] | The Pythagorean theorem applied to coordinates. Tested in geometry-meets-algebra questions. Remember it as 'diagonal of a right triangle on a grid.' |
Midpoint Formula M = ((x₁ + x₂)/2 , (y₁ + y₂)/2) | Average the x-coordinates; average the y-coordinates. Tested both directly and as a sub-step in geometry problems. |
Vertex Form of a Quadratic y = a(x − h)² + k (vertex at (h, k)) | Appears in function questions asking for the maximum/minimum point. Know that h has the OPPOSITE sign of what appears in the equation. |
Exponential Growth/Decay y = a(1 + r)ᵗ or y = a(1 − r)ᵗ or y = a · bᵗ | a = initial value, r = rate, t = time. Distinguish growth (b > 1) from decay (0 < b < 1). Appears frequently in word problems. |
⚠️ ALGEBRA TRAP: The Sign Error on Vertex Form |
In y = a(x − h)² + k, the vertex is (h, k). But because the formula has MINUS h, students consistently misread the sign. Example: y = 3(x − 4)² + 7 → vertex is (4, 7) ✓ Example: y = 3(x + 4)² + 7 → vertex is (−4, 7) — because x + 4 = x − (−4) ACT wrong answer choices exploit this sign error consistently. Always rewrite to minus form before reading off the vertex. |
CATEGORY 2 — Geometry | Formulas #11–#22 |
5. Geometry: 12 Formulas the ACT Tests Directly\
Geometry is the single largest formula category on the ACT by question count. Questions cover areas, perimeters, volumes, angles, triangles, and circles. Every formula below has appeared in official ACT materials. There is no redundancy in this list.
Area of a Triangle A = ½ × base × height | The height must be PERPENDICULAR to the base — not a slant side. This trap appears in non-right triangles where the slant side looks like a height. |
Area of a Rectangle / Square Rectangle: A = l × w | Square: A = s² | Also used to derive area of parallelogram (#14) and trapezoid (#15). Know the perimeter: P = 2(l + w). |
Area of a Circle A = πr² | Almost always tested with the circumference (#14b) in the same question. Know both together. |
Circumference of a Circle C = 2πr (also written as C = πd) | Arc length for a sector: arc = (central angle/360°) × 2πr. Area of sector: sector = (central angle/360°) × πr². |
Area of Parallelogram / Trapezoid Parallelogram: A = base × height | Trapezoid: A = ½(b₁ + b₂) × h | Trapezoid: b₁ and b₂ are the TWO parallel sides; h is the perpendicular height between them. |
Pythagorean Theorem a² + b² = c² (c is the hypotenuse) | Most frequently tested formula on the entire ACT. Also know Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. |
Special Right Triangles 30-60-90: sides in ratio x : x√3 : 2x | 45-45-90: sides in ratio x : x : x√2 | ACT uses these to create problems that look harder than they are. Recognise the triangle type, apply the ratio, done. |
Volume Formulas Rectangular prism: V = lwh | Cylinder: V = πr²h | Sphere: V = (4/3)πr³ | Cone: V = (1/3)πr²h | Pyramid: V = (1/3)lwh | Surface area of a cylinder: SA = 2πr² + 2πrh. Sphere surface area: SA = 4πr². |
Circle Equation (Standard Form) (x − h)² + (y − k)² = r² | Center: (h, k), Radius: r | The ACT sometimes gives the expanded form and asks for the center or radius. Complete the square to convert if needed. |
Interior Angles of Polygons Sum of interior angles = (n − 2) × 180° | Each angle in regular polygon = [(n − 2) × 180°] / n | n = number of sides. Triangle: 180°. Quadrilateral: 360°. Pentagon: 540°. Hexagon: 720°. |
Parallel Lines & Transversals Alternate interior angles: equal | Corresponding angles: equal | Co-interior (same-side): supplementary (sum = 180°) | ACT diagram problems. Identify whether angles are equal or supplementary before calculating — wrong choice here guarantees wrong answer. |
Similarity Ratios If two similar figures have sides in ratio a:b, then: Areas are in ratio a²:b² | Volumes are in ratio a³:b³ | Tested in ACT geometry questions involving scaled figures. Students frequently apply the wrong power (use ratio directly instead of squaring/cubing). |
✅ GEOMETRY POWER MOVE: Pythagorean Triple Recognition |
When an ACT geometry problem gives you two sides of a right triangle, check whether they match a known Pythagorean triple before using a² + b² = c². 3-4-5 and its multiples: 6-8-10, 9-12-15, 12-16-20 5-12-13 and its multiples: 10-24-26 8-15-17 7-24-25 Recognising 6-8-10 as a 3-4-5 triple saves you 30+ seconds per question. On a 45-question test, that adds up to 3–4 minutes of reclaimed time. |
CATEGORY 3 — Trigonometry | Formulas #23–#30 |
6. Trigonometry: 8 Formulas That Separate 26 from 32
Trigonometry appears in approximately 5–7 questions on every ACT administration — roughly 11–16% of the Math section. This is the category most students under-prepare for, which means formula mastery here produces above-average score gains relative to study time invested.
The ACT does not test advanced trigonometric identities or calculus-level applications. It tests a specific, learnable set of trig concepts that are fully covered by the 8 formulas below.
SOHCAHTOA (Right Triangle Trig) sin θ = Opposite/Hypotenuse | cos θ = Adjacent/Hypotenuse | tan θ = Opposite/Adjacent | The foundation of all ACT trig. Every trig question either uses SOHCAHTOA directly or builds on it. This must be automatic. |
Reciprocal Trig Functions csc θ = 1/sin θ | sec θ = 1/cos θ | cot θ = 1/tan θ | tan θ = sin θ / cos θ | Tested in simplification questions. The ACT may give an expression with cot and sec and ask for an equivalent — know what to substitute. |
Pythagorean Trig Identity sin²θ + cos²θ = 1 (also: 1 + tan²θ = sec²θ | 1 + cot²θ = csc²θ) | The most important trig identity. Used to simplify expressions and verify equivalences. The primary form (sin²θ + cos²θ = 1) appears on most ACT administrations. |
Law of Sines a/sin A = b/sin B = c/sin C | Use for NON-right triangles when you know AAS (two angles and one side) or ASA. Invaluable for ACT problems involving oblique triangles. |
Law of Cosines c² = a² + b² − 2ab · cos C | Use for non-right triangles when you know SAS (two sides and included angle) or SSS (three sides). Note: when C = 90°, this reduces to the Pythagorean theorem. |
Radian–Degree Conversion π radians = 180° | To convert: degrees × (π/180) = radians | radians × (180/π) = degrees | ACT harder questions use radians. Know the conversion. Also know key angles: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π. |
Sinusoidal Function Form y = A sin(Bx − C) + D or y = A cos(Bx − C) + D | Amplitude = |A| | Period = 2π/B | Phase shift = C/B | Vertical shift = D. Tested in graph interpretation questions. |
Cofunction Identities & Quadrant Signs sin θ = cos(90° − θ) | tan θ = sin θ / cos θ | CAST rule: All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4) positive | CAST tells you the sign of trig functions by quadrant. Tested when an angle is given in non-standard position. |
ACT Trig Question Type | Which Formula Applies |
'Find the length of side x in right triangle ABC' | SOHCAHTOA (#23) — identify which ratio connects x to the known angle and side |
'If cos θ = 3/5, find sin θ' | Pythagorean identity (#25) — sin²θ = 1 − cos²θ → sin θ = 4/5 |
'In triangle ABC, angle A = 35°, angle B = 80°, side a = 12. Find side b.' | Law of Sines (#26) — b/sin 80° = 12/sin 35° |
'Two sides and the included angle are given. Find the third side.' | Law of Cosines (#27) |
'The graph of y = 3sin(2x) + 1 has amplitude ___ and period ___' | Sinusoidal form (#29) — amplitude = 3, period = 2π/2 = π |
'Convert 150° to radians' | Conversion formula (#28) — 150 × π/180 = 5π/6 |
⚠️ TRIG TRAP: SOHCAHTOA Only Works for RIGHT Triangles |
A common and costly error: students apply SOHCAHTOA to a non-right triangle. If the problem involves a triangle without a right angle, you must use Law of Sines or Law of Cosines instead. The trigger: If the triangle diagram has a right-angle symbol (small square), use SOHCAHTOA. If it does not, and two sides + one angle OR three sides are given, use Law of Cosines. If two angles + one side are given, use Law of Sines. |
CATEGORY 4 — Statistics, Probability & Counting | Formulas #31–#37 |
7. Statistics, Probability & Counting: 7 Formulas for 26+ Scorers
Statistics and probability questions account for approximately 8–12% of the ACT Math section. They are disproportionately important for students in the 24–30 score range because they appear in both the Integrating Essential Skills and Preparing for Higher Math categories.
Mean, Median, Mode, Range Mean = (sum of all values) / (count of values) | Median = middle value (sort first) | Mode = most frequent | Range = max − min | The ACT tests all four in the same question cluster. Also tested: effect on mean/median when a value is added or removed. |
Probability — Basic P(event) = (favourable outcomes) / (total possible outcomes) | P(A or B) = P(A) + P(B) − P(A and B) | The ACT distinguishes 'or' from 'and.' For mutually exclusive events: P(A or B) = P(A) + P(B). For independent events: P(A and B) = P(A) × P(B). |
Combinations & Permutations Combinations (order does NOT matter): C(n,r) = n! / [r!(n−r)!] | Permutations (order matters): P(n,r) = n! / (n−r)! | Read the problem carefully: 'how many ways to CHOOSE' = combination; 'how many ARRANGEMENTS' = permutation. This distinction determines the formula. |
Counting Principle If task A has m ways and task B has n ways, both can be done together in m × n ways | Also called the Fundamental Counting Principle. Extended to any number of independent choices: multiply all options together. |
Percent Change % change = [(new value − original value) / original value] × 100 | Note the direction: if new > original, it is an increase. If new < original, it is a decrease. The denominator is always the ORIGINAL value. |
Rates, Proportions & Unit Conversions rate × time = distance | unit price = total cost / quantity | proportion: a/b = c/d → cross-multiply | The most commonly tested applied-math concept in the Integrating Essential Skills category. Set up dimensional analysis for unit conversion questions. |
Standard Deviation (Conceptual) SD measures spread: larger SD = more spread out data. SD is zero when all values are identical. | The ACT does NOT ask you to calculate SD from scratch. It asks you to reason about it: which data set has larger/smaller SD? Add a constant to all values → mean shifts, SD unchanged. Multiply all values → both mean and SD scale. |
✅ STATISTICS SHORTCUT: The Mean Trap |
ACT statistics questions frequently involve this scenario: a data set has a known mean, one value is added, and the new mean is asked. Method: Sum = mean × count. Add the new value to the sum, divide by the new count. Example: 5 values with mean 12 → sum = 60. Add the value 18 → new sum = 78, new count = 6 → new mean = 78/6 = 13. Students who try to average 12 and 18 get 15 — and select the wrong answer choice. |
CATEGORY 5 — Sequences, Logarithms & Advanced Topics | Formulas #38–#40 |
8. Sequences, Logarithms & Advanced Topics: 3 Elite Formulas
These three formula groups appear on approximately 3–5 questions per ACT. They are essential for students targeting 30+, and students targeting 33–36 should know them without hesitation.
Arithmetic & Geometric Sequences Arithmetic nth term: aₙ = a₁ + (n−1)d | Sum: Sₙ = (n/2)(a₁ + aₙ) | Geometric nth term: aₙ = a₁ · rⁿ⁻¹ | d = common difference (arithmetic). r = common ratio (geometric). The ACT asks for the nth term OR the sum of the first n terms — know both formulas for both types. |
Logarithm Properties log_b(b) = 1 | log_b(1) = 0 | log(mn) = log m + log n | log(m/n) = log m − log n | log(mⁿ) = n log m | log_b(x) = ln(x)/ln(b) | The ACT tests logarithm simplification and equation solving. Know how to convert log₂(8) = 3 (because 2³ = 8). Change-of-base formula for calculator use. |
Matrix Operations Matrix addition: add corresponding elements | Scalar multiplication: multiply every element | Matrix multiplication: (AB)_{ij} = row i of A · column j of B | The ACT rarely tests matrix multiplication beyond 2×2. It more commonly tests whether a product is defined (columns of A must equal rows of B) and scalar multiplication. |
9. Worked Examples: How the ACT Tests Each Category
Example 1 — Algebra (Formula #5: Quadratic Formula)
Question type: 'Which of the following gives all values of x that satisfy 3x² − 5x − 2 = 0?'
Step 1: Identify a = 3, b = −5, c = −2.
Step 2: Apply the quadratic formula: x = [5 ± √(25 + 24)] / 6 = [5 ± √49] / 6 = [5 ± 7] / 6
Step 3: x = 12/6 = 2 or x = −2/6 = −1/3.
Answer: x = 2 and x = −1/3.
ACT application: The answer choices will include distractors with sign errors (−2 and 1/3) and calculation errors. Students who know the formula find both roots in under 45 seconds. Students who try to factor by trial-and-error frequently take 90+ seconds and make errors.
Question type: 'In right triangle ABC, angle A = 60°, angle C = 90°, and BC = 8. What is AB?'
Step 1: Recognise this is a 30-60-90 triangle (angles 30°, 60°, 90°).
Step 2: Recall the ratio: sides opposite 30°, 60°, 90° are in ratio x : x√3 : 2x.
Step 3: BC (opposite 60°) = 8 = x√3. Therefore x = 8/√3 = 8√3/3.
Step 4: AB (the hypotenuse, opposite 90°) = 2x = 16√3/3.
ACT application: Students who do not know the 30-60-90 ratio must use SOHCAHTOA with a calculator: sin 60° = 8/AB → AB = 8/sin 60° = 8/(√3/2) = 16/√3. Same answer, more steps, more time. The special triangle ratio is faster.
Example 3 — Trigonometry (Formula #26: Law of Sines)
Question type: 'In triangle PQR, angle P = 40°, angle Q = 75°, and side p (opposite P) = 15. What is the length of side q (opposite Q), to the nearest integer?'
Step 1: Identify this as a Law of Sines problem — two angles and one side given (AAS).
Step 2: Apply: q/sin 75° = 15/sin 40°
Step 3: q = 15 × sin 75° / sin 40° ≈ 15 × 0.966 / 0.643 ≈ 22.5 ≈ 23.
Answer: q ≈ 23.
ACT application: Students who try to use SOHCAHTOA on this problem fail immediately — there is no right angle. Law of Sines is the only path. Calculator permitted; the formula is the gate.
Example 4 — Statistics (Formula #33: Combinations)
Question type: 'A committee of 3 is to be chosen from 8 candidates. How many different committees are possible?'
Step 1: Order does not matter (a committee is a committee regardless of selection order) → use combinations.
Step 2: C(8,3) = 8! / [3! × 5!] = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.
Answer: 56 committees.
ACT application: Students who use the permutation formula get 336 and select the wrong answer. The word 'committee' (or 'group,' 'team,' 'selection') signals combination. The words 'arrangement,' 'order,' 'rank' signal permutation.
10. Common Formula Mistakes That Cost Students Points
The wrong answers on the ACT are not random. They are engineered to trap students who know the formula but apply it incorrectly. These are the seven most expensive formula mistakes, based on official ACT question analysis.
Mistake | Formula Involved | What Students Do | What They Should Do | Score Impact |
Using base × slant height for triangle area | #11 — Triangle Area | Count the slant side as the height | Drop a perpendicular from vertex to base; that is the height | 1–2 questions per test |
Reading vertex form sign incorrectly | #08 — Vertex Form | y = (x − 3)² → vertex at (−3, 0) | Vertex is at (3, 0) — opposite sign of what appears in the bracket | 1 question per test |
Applying SOHCAHTOA to a non-right triangle | Use sin/cos ratios on an oblique triangle | Identify no right angle → use Law of Sines or Cosines | 1–2 questions per test | |
Choosing permutation for a combination problem | #33 — Combinations | 'How many groups of 3 from 8?' → P(8,3) = 336 | Order doesn't matter → C(8,3) = 56 | 1 question per test |
Using slant height instead of radius in circle sector | #14 — Circle | Arc length = (angle/360) × diameter instead of radius | Arc length = (angle/360) × 2πr | 1 question per test |
Squaring instead of cubing for volume similarity | #22 — Similarity | Similar figures with ratio 2:3 → volume ratio is 4:9 | Volume ratio = 2³:3³ = 8:27 | Occasionally tested |
Forgetting to square both sides when using distance formula | #06 — Distance | d = (x₂−x₁) + (y₂−y₁) instead of square root of summed squares | d = √[(x₂−x₁)² + (y₂−y₁)²] | 1 question per test |
11. How to Memorise 40 Formulas Efficiently: A 4-Week System
Memorising 40 formulas in one session is neither realistic nor useful. The following 4-week system builds recall through spaced repetition and application — which is how test-taking automaticity is built.
Week | Focus Area | Formulas to Master | Daily Practice Method |
Week 1 | Algebra & Geometry Core | Formulas #1–#15 (slope, quadratic, distance, area, Pythagorean) | Write each formula from memory. Apply each to 3 practice problems. Drill daily for 15 minutes. |
Week 2 | Geometry Advanced + Trig Foundation | Formulas #16–#25 (volumes, circle eq., special triangles, SOHCAHTOA, Pythagorean identity) | Add new formulas to Week 1 deck. Time yourself: each formula recalled in under 5 seconds. |
Week 3 | Trig Advanced + Statistics | Formulas #26–#37 (Law of Sines/Cosines, sinusoidal, probability, combinations) | Apply each trig and stats formula to 5 real ACT-style questions. Focus on trigger recognition. |
Week 4 | Elite Formulas + Full Integration | Formulas #38–#40 + all categories under timed conditions | Take a full ACT Math section. After scoring: identify every wrong answer by formula category. Drill that category. |
✅ THE MOST IMPORTANT HABIT: Trigger Recognition |
Knowing a formula is necessary but not sufficient. You also need to recognise the question type that triggers each formula — in under 5 seconds. Drill this: Flash yourself the question type, not the formula. 'Non-right triangle, two angles + one side given' → instantly think 'Law of Sines.' 'Two coordinate points, find distance' → instantly think 'Distance formula.' The trigger must come before the formula. This is the difference between knowing math and scoring on the ACT. |
12. Score Band Strategy: Which Formulas to Prioritise by Target Score
Not every student needs to master every formula at the same depth. The following table maps formula categories to score band priorities to help you allocate study time strategically.
Current Score | Target Score | Priority Categories | Formulas to Master First | Expected Gain |
14–18 | 20–22 | Algebra core + basic geometry | +3 to +5 points | |
19–22 | 24–26 | All algebra + full geometry | +3 to +5 points | |
23–26 | 28–30 | Algebra + Geometry + Trig foundation | +4 to +6 points | |
27–30 | 32–34 | All categories + advanced trig + stats | #1–#37 complete | +3 to +5 points |
31–34 | 35–36 | Full formula mastery + elimination of careless errors | #1–#40, zero formula errors | +2 to +3 points |
SCORE REALITY CHECK: Where Formula Errors Actually Hit |
Students scoring 19–22: Losing points primarily on Algebra and Geometry formula errors — the highest-frequency categories. Fixing these produces the largest score jumps. Students scoring 23–26: Algebra and Geometry formulas mostly solid. Losing points on Trig questions (refusing to apply Law of Sines/Cosines) and Statistics questions (permutation vs. combination confusion). Students scoring 27–30: Formula knowledge strong. Losing points on speed — spending too long reconstructing formulas they know but cannot recall instantly. Solution: fluency drills, not more learning. |
Take a Free ACT Math Diagnostic Before drilling all 40 formulas, find out exactly which formula categories your ACT Math score is losing points in. EduShaale's free diagnostic test at testprep.edushaale.com gives you a category-by-category breakdown so your study time targets the highest-return formulas first.
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13. Frequently Asked Questions
Does the ACT provide any math formulas during the test?
No. The ACT provides zero formulas. Some individual questions may include a formula within the question stem itself if solving the problem requires an unusual relationship — but this is the exception. The vast majority of ACT Math questions assume you already know the relevant formulas. This is a fundamental difference from the SAT, which provides a basic geometry formula reference box. If you are switching from SAT to ACT preparation, formula memorisation should be your first priority
Are calculators allowed on the ACT Math section?
Yes. A calculator is permitted for the entire ACT Math section. Approved calculators include graphing calculators (TI-84, TI-Nspire in non-CAS mode), scientific calculators, and basic calculators. The calculator helps with arithmetic — it does not help you recall or apply formulas. Students who believe a calculator makes formula memorisation unnecessary consistently underperform, because the bottleneck is formula recall, not arithmetic execution.
How many trigonometry questions are on the ACT?
Approximately 5–7 questions per ACT administration involve trigonometry — roughly 11–16% of the 45-question Math section. These questions disproportionately trip up students who have not deliberately studied trig for the ACT, because the ACT tests specific applications (SOHCAHTOA, Law of Sines/Cosines, trig identities, sinusoidal functions) rather than full trig curriculum. With 8 formulas (covered in this guide) and 2–3 weeks of practice, trig questions become manageable even for students who found trig difficult in school.
What is the most important formula on the ACT Math section?
By question frequency: the Pythagorean theorem (a² + b² = c²) appears in more questions than any other single formula, either tested directly or embedded in multi-step geometry problems. By impact on score: the slope formula and quadratic formula collectively drive the largest share of algebra points, and missing them produces consistent errors across 8–12 questions. If you can only memorise three formulas, memorise these: Pythagorean theorem, slope formula, and quadratic formula.
How is the Enhanced ACT (2025+) different for Math?
The Enhanced ACT reduces the Math section from 60 questions in 60 minutes to 45 questions in 50 minutes. The time per question increases slightly (from 60 to ~67 seconds), but the section content and formula requirements are unchanged. The most significant scoring change is that Math now counts for one-third of the composite score (rather than one-fourth) when the Science section is optional — making Math score improvement more valuable for composite score targets than it was previously.
Do I need to memorise the Law of Sines and Law of Cosines?
Yes, if you are targeting a score of 26 or higher. These two formulas appear on most ACT administrations in the context of non-right triangle problems. Students who do not know them are forced to skip these questions entirely or guess — typically costing 1–2 questions per test. For students targeting 24–25, focusing on SOHCAHTOA and the Pythagorean identity first is more efficient. For students targeting 28+, Law of Sines and Cosines are non-negotiable.
How do I know whether to use combinations or permutations?
The key word is 'order.' If the problem is about selecting or choosing a group, committee, or team where the order of selection does not matter — use combinations: C(n,r) = n!/[r!(n-r)!]. If the problem involves arrangement, ranking, seating order, or a process where the order of selection changes the outcome — use permutations: P(n,r) = n!/(n-r)!. ACT wrong answer choices are precisely engineered to catch students who apply the wrong formula, so this distinction is consistently tested.
What ACT Math score do I need for selective colleges?
Score targets vary significantly by institution. Highly selective universities (Harvard, MIT, Stanford, Yale) typically see admitted students with Math scores of 34–36. Selective universities (top-30 programs) often see 28–33. Strong state universities generally accept students in the 24–28 range. Note that many top colleges, including Harvard, Yale, MIT, and Caltech, have reinstated test requirements for 2025–2026 applicants after the test-optional period. Always verify current requirements at the institution's admissions page.
How long should I study formulas before a real ACT?
The minimum effective memorisation timeline is 4 weeks of consistent daily practice (15–20 minutes per day). Students who begin formula study 1–2 weeks before the test typically achieve basic recall but not automatic recall — they can remember formulas with effort but lose time doing so. Automatic recall requires 4+ weeks of spaced repetition. Students with 8+ weeks before their test date should prioritise formula mastery in weeks 1–4 and apply those formulas to official practice problems in weeks 5–8.
Is the ACT Math section harder than SAT Math?
The difficulty profiles are different, not simply harder or easier. ACT Math covers a broader range of topics (including trigonometry, which SAT does not test extensively), requires formula memorisation with no reference sheet, and moves faster. SAT Math is more focused, provides a formula reference, and has a built-in Desmos graphing calculator for digital test takers. Students who are strong at geometry and trig often prefer the ACT; students who prefer calculator-integrated problem solving and narrower topic coverage often prefer the SAT. Taking a full practice test for each is the most reliable way to determine which suits you.
Are there formulas the ACT will never test?
Yes. The ACT does not test: calculus (derivatives, integrals), advanced statistics beyond combinations/basic probability, complex number arithmetic beyond identification, and most matrix operations beyond 2×2. Despite appearing in many ACT prep books, double-angle formulas and sum-to-product identities almost never appear on official ACT administrations. The formula list in this guide is calibrated to what actually appears — not what theoretically could appear.
How do I use this formula guide alongside official ACT practice tests?
After completing a full official ACT Math section, categorise every wrong answer by the formula group it belongs to (algebra, geometry, trig, stats). Then cross-reference that category against this guide. If you missed 3 algebra questions, you do not have an algebra problem — you have a specific formula problem within algebra. Identify which formula, drill it with 10–15 official questions, then retest. This targeted approach produces faster score gains than re-doing full practice tests without error analysis.
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EduShaale's core ACT Math observation: The students who improve most on ACT Math are not those who practice the most problems — they are those who have automated every formula and can identify the right formula from a question in under 5 seconds. That automation comes from active recall drills applied to official ACT questions — not from reading formula lists passively. Start with the diagnostic. Know exactly where your formula gaps are. Drill those first. Book your free session: edushaale.com/contact-us |
15. References & Resources
Official ACT Resources
ACT Math Formula & Content Resources
EduShaale ACT Resources
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