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Most ACT Math questions do not ask you to recite a formula — they test whether you recognize the right situation to use it. The list below covers the formulas that appear most often on the enhanced ACT, grouped by topic. For each one we show the formula, when it applies, when it does not, and a short worked example with a verified answer. If you are building a broader study plan, pair this with our enhanced ACT guide and math preparation walkthrough.
On the digital ACT, the Math section includes a built-in Desmos graphing calculator. You may also bring an approved calculator, but certain models with computer algebra systems (CAS) are prohibited. Check ACT’s current calculator policy before test day.
Algebra essentials
These rules handle lines, quadratics, exponents, and systems. The common mistake is not the formula itself — it is using it before the problem is in the right form.
| Formula | Use it when... | Watch out... | Worked example | Answer |
|---|---|---|---|---|
| Slope: m = (y₂ − y₁) / (x₂ − x₁) | You have two points and need the steepness or parallel/perpendicular check. | x₁ cannot equal x₂; a vertical line has undefined slope. | Find the slope of the line through (1, 2) and (5, 10). | m = (10 − 2)/(5 − 1) = 8/4 = 2 |
| Slope-intercept: y = mx + b | You know the slope m and y-intercept b, or you need to read them from an equation. | Vertical lines (x = a) cannot be written this way. | Write the equation with slope 3 and y-intercept −2, then find y when x = 4. | y = 3x − 2; when x = 4, y = 10 |
| Point-slope: y − y₁ = m(x − x₁) | You know one point and the slope. | Vertical lines need x = a, not this form. | Line through (2, 5) with slope 3; find y when x = 4. | y − 5 = 3(x − 2); at x = 4, y = 11 |
| Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2) | You need the center of a line segment. | It divides in a 1:1 ratio only; other ratios need section formulas. | Midpoint of the segment from (2, 4) to (8, 10). | ((2 + 8)/2, (4 + 10)/2) = (5, 7) |
| Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²) | You need the straight-line distance between two points. | It is just the Pythagorean theorem; it gives straight-line distance, not travel distance along a path. | Distance from (1, 2) to (4, 6). | d = √(3² + 4²) = √25 = 5 |
| Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a | You need exact roots of ax² + bx + c = 0 and factoring is not obvious. | The equation must be set equal to 0 first; if it is not, rearrange before substituting. | Solve x² − 5x + 6 = 0. | x = (5 ± √(25 − 24))/2 = (5 ± 1)/2, so x = 3 or x = 2 |
| Exponent product rule: xᵃ · xᵇ = xᵃ⁺ᵇ | You are multiplying powers with the same base. | Bases must match; 2³ · 3⁴ does not combine this way. | Simplify 2³ · 2⁴. | 2³ · 2⁴ = 2⁷ = 128 |
| Linear system: ax + by = c, dx + ey = f | You have two linear equations and need the (x, y) pair that satisfies both. | Systems of parallel lines have no solution; identical lines have infinitely many. | Solve x + y = 5 and x − y = 1. | Adding gives 2x = 6, so x = 3 and y = 2 |
Functions and graphs
Function questions usually test notation, evaluation, or the meaning of a rate of change. Treat f(x) as a rule that sends an input to exactly one output.
| Formula | Use it when... | Watch out... | Worked example | Answer |
|---|---|---|---|---|
| Function notation: f(x) = expression | A problem gives a rule and asks for the output at a specific input. | Not every equation is a function; a function must give one output per input. | If f(x) = 2x + 1, find f(4). | f(4) = 2(4) + 1 = 9 |
| Average rate of change: (f(b) − f(a)) / (b − a) | You need the slope between two points on a curve or the change per unit over an interval. | This is the secant slope, not the instantaneous rate (which needs calculus). | For f(x) = x², find the average rate of change from x = 1 to x = 3. | (9 − 1)/(3 − 1) = 8/2 = 4 |
Geometry
Geometry questions reward two habits: drawing the figure and labeling what you know. The formulas below cover the shapes ACT uses most often.
| Formula | Use it when... | Watch out... | Worked example | Answer |
|---|---|---|---|---|
| Area of a triangle: A = ½bh | You know (or can find) a base and the perpendicular height. | The height must be perpendicular to the chosen base; it is not always a side of the triangle. | Triangle with base 8 and height 5. | A = ½ · 8 · 5 = 20 |
| Pythagorean theorem: a² + b² = c² | You have a right triangle and know two sides. | c must be the hypotenuse; the theorem does not apply to non-right triangles. | Right triangle with legs 3 and 4; find the hypotenuse. | 3² + 4² = 9 + 16 = 25, so c = √25 = 5 |
| 30-60-90 triangle: x, x√3, 2x | The angles are 30°, 60°, and 90°. | Requires the exact angle set; do not assume it for a generic right triangle. | Short leg = 5; find the hypotenuse and long leg. | Hypotenuse = 10; long leg = 5√3 ≈ 8.66 |
| 45-45-90 triangle: x, x, x√2 | The triangle is a right isosceles triangle. | Only works when the two legs are equal. | Leg = 7; find the hypotenuse. | Hypotenuse = 7√2 ≈ 9.90 |
| Area of a circle: A = πr² | You know the radius. | r is the radius, not the diameter; if you are given the diameter, divide by 2 first. | Circle with radius 3. | A = 9π ≈ 28.27 |
| Circumference of a circle: C = 2πr | You need the distance around a circle. | Same radius-versus-diameter issue as area. | Circle with radius 4. | C = 8π ≈ 25.13 |
| Area of a rectangle: A = lw | You have a rectangle with known length and width. | For a general parallelogram, use base × perpendicular height, not side × side. | Rectangle with length 6 and width 4. | A = 6 · 4 = 24 |
| Perimeter of a rectangle: P = 2l + 2w | You need the total distance around the rectangle. | Perimeter is linear distance, not area. | Rectangle with length 6 and width 4. | P = 2(6) + 2(4) = 20 |
Trigonometry basics
ACT trigonometry stays close to right triangles. If you can label opposite, adjacent, and hypotenuse relative to the angle in question, SOH-CAH-TOA handles most problems.
| Formula | Use it when... | Watch out... | Worked example | Answer |
|---|---|---|---|---|
| SOH-CAH-TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj | You have a right triangle and need a ratio involving an acute angle. | Only applies to right triangles; the labels depend on which angle you choose. | Right triangle with opposite side 3 and hypotenuse 5; find sin θ. | sin θ = 3/5 = 0.6 |
| Pythagorean identity: sin²θ + cos²θ = 1 | You know one trig ratio and need the other for the same angle. | Sign matters in advanced settings; on the ACT, acute angles keep both positive. | If sin θ = 3/5, find cos θ using the identity. | cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 |
Probability and statistics
Statistics questions are usually light on computation and heavy on interpretation. Know the difference between mean and median, and read probability questions carefully to count the right outcomes.
| Formula | Use it when... | Watch out... | Worked example | Answer |
|---|---|---|---|---|
| Mean: sum of values / number of values | You need the arithmetic average. | Mean is pulled up or down by outliers; it is not the same as median or mode. | Find the mean of 4, 7, 8, and 11. | (4 + 7 + 8 + 11)/4 = 30/4 = 7.5 |
| Median: middle value when data are ordered | You need the central value resistant to outliers. | Data must be ordered first; with an even count, average the two middle values. | Find the median of 3, 5, 8, 12, 20. | The middle value is 8 |
| Probability: P(event) = favorable outcomes / total outcomes | All outcomes are equally likely and you can count them. | If outcomes are not equally likely, this ratio is wrong. | Probability of rolling a 3 on a fair six-sided die. | P = 1/6 ≈ 0.167 |
| Fundamental counting principle: m · n | One choice has m options and an independent second choice has n options. | The choices must be independent; overcounting is common when choices overlap. | 3 shirts and 4 pairs of pants; how many outfits? | 3 · 4 = 12 outfits |
| Percent change: ((new − original) / original) · 100% | You need the percentage increase or decrease. | Always divide by the original value, not the larger value. | Price rises from $80 to $100; what is the percent increase? | ((100 − 80)/80) · 100% = 25% |
How to practice formulas (not memorize)
Memorizing the list above is a start, but the ACT tests recognition, not recall. A formula you know becomes a formula you miss when you use it in the wrong situation. Build fluency with this loop:
- Classify the trigger. Before you solve, name what the problem is asking: slope, area, roots, ratio, average, or probability.
- State the limitation. Ask yourself what has to be true for the formula to work — right triangle? zero on one side? perpendicular height? independent choices?
- Solve, then audit. Plug your answer back into the original statement or solve the same problem a second way.
- Log the real error. Record whether each miss was "did not know the formula," "knew it but used the wrong situation," or "arithmetic slip." Most students over-index on the first category.
If you want a structured schedule, our ACT study plan spreads formula work across two weeks with mixed review built in. When you are ready to test yourself under realistic timing, move to the free ACT practice area.
All facts in this article were verified against ACT's official publications on October 1–2, 2026.
References
Frequently asked questions
No. ACT Math does not provide a formula sheet, so you need to know the common formulas for lines, quadratics, right triangles, circles, and basic statistics before test day.
Slope and line forms, the quadratic formula, area and perimeter of triangles and rectangles, the Pythagorean theorem, special right triangles, SOH-CAH-TOA, mean/median, and basic probability appear most frequently.
Memorize them, but prioritize knowing when each one applies. The most common error is using the right formula in the wrong situation — for example, applying the Pythagorean theorem to a non-right triangle or using the distance formula for travel along a curved path.
The digital ACT includes a built-in Desmos graphing calculator for Math, which can help you graph functions, find intersections, and check tables. It does not remove the need to recognize the right formula or setup, and it does not replace algebraic reasoning for non-graphable questions.
No. ACT publishes a calculator policy that prohibits CAS calculators and specific models such as the TI-89, TI-92, TI-Nspire CAS, HP Prime, and Casio ClassPad. Always check the current ACT calculator policy before test day.
Work mixed sets of official-style questions, classify each miss by error type, and re-solve missed problems a second way when possible. Pair formula practice with a study plan and full timed sections rather than isolated formula drills.
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