ACT Math Formulas
What You Need to Know
ACT Math is mostly about recognizing the type of problem fast and grabbing the right formula. You’re rarely deriving anything—you’re plugging in cleanly, tracking units, and avoiding traps.
Key idea: most questions reduce to one of these buckets:
- Algebra (linear/quadratic/exponents/radicals)
- Coordinate geometry (slope, distance, lines)
- Plane geometry (area/perimeter/angles)
- Solid geometry (volume/surface area)
- Trigonometry (right triangles, special triangles, basic trig)
- Stats & probability (mean/median, counting, probability)
Critical reminder: ACT does not provide a formula sheet. You must know the common ones cold.
Step-by-Step Breakdown
A fast “formula triage” method
- Identify what the question is asking for (length, area, volume, angle, probability, equation, x-value, etc.).
- List what you’re given (draw/label a diagram, write known values next to variables).
- Choose the category:
- Length on a coordinate plane → distance/midpoint/slope
- Circle/polygon → circumference/area/angle rules
- 3D figure → volume/surface area
- Exponents/radicals → exponent/radical rules
- “How many ways” → counting/permutations/combinations
- Write the formula before you plug in (prevents mixing up radius/diameter, etc.).
- Solve and sanity-check:
- Units: area should be \text{units}^2, volume \text{units}^3
- Magnitude: does it roughly make sense?
Mini worked walkthrough (coordinate geometry)
Problem type: Find distance between (-2,3) and (4,-1).
- Recognize: coordinate length → distance formula.
- Use d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.
- Compute: d=\sqrt{(4-(-2))^2+(-1-3)^2}=\sqrt{6^2+(-4)^2}=\sqrt{52}=2\sqrt{13}.
Key Formulas, Rules & Facts
Algebra essentials
| Formula / rule | When to use | Notes |
|---|---|---|
| a(b+c)=ab+ac | Distribute | Common in simplifying |
| a^m\cdot a^n=a^{m+n} | Multiply same base | Base must match |
| \frac{a^m}{a^n}=a^{m-n} | Divide same base | a\neq 0 |
| (a^m)^n=a^{mn} | Power of a power | Watch parentheses |
| (ab)^n=a^n b^n | Power of a product | Also (\frac{a}{b})^n=\frac{a^n}{b^n} |
| a^0=1 | Zero exponent | a\neq 0 |
| a^{-n}=\frac{1}{a^n} | Negative exponent | Moves to denominator |
| \sqrt{ab}=\sqrt{a}\sqrt{b} | Radical simplify | For a,b\ge 0 |
| \sqrt{a^2}=|a| | Radical of square | The absolute value trap |
| \frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a} | Rationalize denom | Useful with answer choices |
| ax+b=c \Rightarrow x=\frac{c-b}{a} | Linear equation | Keep signs straight |
| m=\frac{y_2-y_1}{x_2-x_1} | Slope | Undefined if denominator 0 |
| y=mx+b | Slope-intercept form | b is y-intercept |
| y-y_1=m(x-x_1) | Point-slope form | Quick from point + slope |
| ax+by=c | Standard form | Easy intercepts if needed |
Quadratics & polynomials
| Formula / rule | When to use | Notes |
|---|---|---|
| x^2+bx+c=(x+p)(x+q) with p+q=b,\ pq=c | Factoring monic quadratics | Works when leading coefficient is 1 |
| ax^2+bx+c=0 \Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} | Quadratic formula | Memorize perfectly |
| \Delta=b^2-4ac | Discriminant | \Delta>0 two real, \Delta=0 one, \Delta |
| \text{Axis of symmetry: } x=\frac{-b}{2a} | Vertex/graph questions | Plug into quadratic to get vertex y |
| a^2-b^2=(a-b)(a+b) | Difference of squares | Shows up constantly |
| (a+b)^2=a^2+2ab+b^2 | Perfect square | Recognize patterns |
| (a-b)^2=a^2-2ab+b^2 | Perfect square | Sign in middle matters |
Ratios, proportions, percent
| Formula / rule | When to use | Notes |
|---|---|---|
| \frac{a}{b}=\frac{c}{d} \Rightarrow ad=bc | Proportions | Cross-multiply |
| \text{percent} = \frac{\text{part}}{\text{whole}} | Percent problems | Convert percent to decimal |
| \text{new} = \text{old}(1\pm r) | Percent increase/decrease | r as decimal (e.g., 0.15) |
Coordinate geometry
| Formula / rule | When to use | Notes |
|---|---|---|
| d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} | Distance | Pythagorean in the plane |
| \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) | Midpoint | Average coordinates |
| m_1m_2=-1 | Perpendicular lines | Slopes are negative reciprocals |
| m_1=m_2 | Parallel lines | Same slope |
Plane geometry (perimeter/area/angles)
Area & perimeter
| Shape | Formula | Notes |
|---|---|---|
| Rectangle | A=lw, P=2l+2w | |
| Square | A=s^2, P=4s | |
| Triangle | A=\frac{1}{2}bh | Height is perpendicular to base |
| Equilateral triangle | A=\frac{\sqrt{3}}{4}s^2 | Useful with 30\text{-}60\text{-}90 |
| Parallelogram | A=bh | Height ⟂ base |
| Trapezoid | A=\frac{1}{2}(b_1+b_2)h | Bases are parallel sides |
| Circle | C=2\pi r=\pi d, A=\pi r^2 | Don’t mix r and d |
| Arc length | s=\frac{\theta}{360}\cdot 2\pi r | \theta in degrees |
| Sector area | A=\frac{\theta}{360}\cdot \pi r^2 | Degree-based on ACT |
Angle facts
| Fact | Formula / rule | Notes |
|---|---|---|
| Straight line | 180^\circ | Linear pair sums to 180^\circ |
| Full circle | 360^\circ | |
| Triangle sum | A+B+C=180^\circ | Any triangle |
| Exterior angle | \text{ext} = \text{remote int}_1+\text{remote int}_2 | Triangle exterior angle theorem |
| Polygon interior sum | (n-2)\cdot 180^\circ | n sides |
| Regular polygon interior angle | \frac{(n-2)\cdot 180^\circ}{n} | Each angle equal |
| Regular polygon exterior angle | \frac{360^\circ}{n} | Fast for n |
| Parallel lines | alternate interior angles equal | Also corresponding equal |
Solid geometry (volume & surface area)
| Solid | Volume | Surface area | Notes |
|---|---|---|---|
| Rectangular prism | V=lwh | SA=2(lw+lh+wh) | |
| Cube | V=s^3 | SA=6s^2 | |
| Cylinder | V=\pi r^2h | SA=2\pi r^2+2\pi rh | Two bases + lateral |
| Cone | V=\frac{1}{3}\pi r^2h | SA=\pi r^2+\pi r\ell | \ell is slant height |
| Sphere | V=\frac{4}{3}\pi r^3 | SA=4\pi r^2 | Classic ACT favorite |
Right triangles & trigonometry
| Concept | Formula | Notes |
|---|---|---|
| Pythagorean theorem | a^2+b^2=c^2 | c is hypotenuse |
| 45-45-90 | legs =x, hyp =x\sqrt{2} | Ratio 1:1:\sqrt{2} |
| 30-60-90 | short =x, long =x\sqrt{3}, hyp =2x | Ratio 1:\sqrt{3}:2 |
| SOHCAHTOA | \sin\theta=\frac{\text{opp}}{\text{hyp}}, \cos\theta=\frac{\text{adj}}{\text{hyp}}, \tan\theta=\frac{\text{opp}}{\text{adj}} | Right triangles (degrees) |
Sequences
| Type | General term | Sum (common ACT) | Notes |
|---|---|---|---|
| Arithmetic | a_n=a_1+(n-1)d | S_n=\frac{n}{2}(a_1+a_n) | Constant difference d |
| Geometric | a_n=a_1 r^{n-1} | S_n=a_1\frac{1-r^n}{1-r} | Constant ratio r, r\neq 1 |
Statistics & probability
| Topic | Formula / rule | Notes |
|---|---|---|
| Mean | \bar{x}=\frac{\text{sum}}{n} | Average |
| Weighted mean | \bar{x}=\frac{\sum w_ix_i}{\sum w_i} | Like grade averages |
| Median | middle value | Sort first; if even, average two middles |
| Probability | P(A)=\frac{\text{favorable}}{\text{total}} | “Equally likely” outcomes |
| Complement | P(A^c)=1-P(A) | Often faster |
| Independent “and” | P(A\cap B)=P(A)P(B) | Only if independent |
| Mutually exclusive “or” | P(A\cup B)=P(A)+P(B) | Only if disjoint |
| General “or” | P(A\cup B)=P(A)+P(B)-P(A\cap B) | Avoid double-count |
Counting (ways)
| Concept | Formula | When to use |
|---|---|---|
| Fundamental counting principle | multiply choices | Sequential choices |
| Permutations | \,{}_nP_r=\frac{n!}{(n-r)!} | Order matters |
| Combinations | \,{}_nC_r=\frac{n!}{r!(n-r)!} | Order doesn’t matter |
| Factorial | n!=n(n-1)(n-2)\cdots 1 | Count arrangements |
Examples & Applications
Example 1: Circle sector + arc length
A circle has radius 6 and central angle 120^\circ.
- Arc length: s=\frac{120}{360}\cdot 2\pi(6)=\frac{1}{3}\cdot 12\pi=4\pi
- Sector area: A=\frac{120}{360}\cdot \pi(6^2)=\frac{1}{3}\cdot 36\pi=12\pi
Exam variation: they may give diameter d; convert to radius first: r=\frac{d}{2}.
Example 2: Quadratic roots and discriminant
For 2x^2-4x+k=0 to have exactly one real solution:
- Need \Delta=b^2-4ac=0.
- Here a=2,b=-4,c=k: (-4)^2-4(2)(k)=16-8k=0 \Rightarrow k=2.
Exam variation: “no real solutions” means \Delta
Example 3: Similar triangles / scaling (area vs length)
A triangle is scaled by factor 3 (all side lengths triple). What happens to area?
- Area scales by the square of the scale factor: 3^2=9.
So new area =9\times old area.
Why this shows up: ACT loves “scale drawings” and “similar figures.”
Example 4: Counting with restrictions
How many 3-letter codes can you make from A, B, C, D, E with no repeats?
- Order matters, no repeats: \,{}_5P_3=5\cdot 4\cdot 3=60.
Exam variation: If repeats allowed, it’s 5^3=125.
Common Mistakes & Traps
Radius vs. diameter confusion: You plug d into \pi r^2 or use 2\pi d. Wrong because circle formulas are built on r. Fix: write r=\frac{d}{2} immediately.
Forgetting units and powers: You report area in linear units or volume in square units. Wrong because dimensions change. Fix: always label \text{units}^2 for area, \text{units}^3 for volume.
Height not perpendicular: You use a slanted side as the triangle/trapezoid height. Wrong because height must be perpendicular to the base. Fix: draw the right angle marker.
Sign errors in the quadratic formula: You drop the \pm or misplace parentheses: \frac{-b\pm\sqrt{b^2-4ac}}{2a}. Fix: substitute carefully with parentheses around b and 2a.
Mixing up slope formulas: You do \frac{x_2-x_1}{y_2-y_1} or swap points inconsistently. Fix: memorize “rise over run”: \frac{\Delta y}{\Delta x}.
Absolute value from square roots: You simplify \sqrt{x^2} to x. Wrong because \sqrt{x^2}=|x|. Fix: if solving, split cases or think sign.
Probability ‘or’ vs ‘and’: You add when you should multiply, or double-count overlaps. Fix: “and” often multiplies (independent), “or” uses addition with overlap rule: P(A\cup B)=P(A)+P(B)-P(A\cap B).
Permutation vs combination: You use \,{}_nC_r when order matters (like seating). Fix: ask: “Do different orders count as different outcomes?” If yes → permutation.
Memory Aids & Quick Tricks
| Trick / mnemonic | What it helps you remember | When to use |
|---|---|---|
| SOHCAHTOA | \sin,\cos,\tan definitions | Right-triangle trig |
| “Circle: A then C” | A=\pi r^2 and C=2\pi r | Circle problems |
| Special triangles | 45\text{-}45\text{-}90: 1:1:\sqrt{2} and 30\text{-}60\text{-}90: 1:\sqrt{3}:2 | Fast side lengths |
| “Distance = Pythagorean on coordinates” | d=\sqrt{(\Delta x)^2+(\Delta y)^2} | Coordinate geometry |
| “Perimeter adds, Area multiplies” | Perimeter is sum of side lengths; area is 2D measure | Avoid mixing up |
| “Scale factor squares/cubes” | Similar figures: area factor k^2, volume factor k^3 | Similarity/3D scaling |
| “Keep-Change-Flip” | Dividing fractions: \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c} | Fraction division |
| “FOIL” | Multiply binomials: First-Outer-Inner-Last | Algebra expansion |
Quick Review Checklist
- You can write instantly: A=\pi r^2, C=2\pi r, a^2+b^2=c^2.
- You remember special triangles: 1:1:\sqrt{2} and 1:\sqrt{3}:2.
- You know coordinate basics: m=\frac{\Delta y}{\Delta x}, distance, midpoint.
- You can deploy the quadratic formula and discriminant without errors.
- You can compute polygon angle sums: (n-2)180^\circ and regular exterior \frac{360^\circ}{n}.
- You know core volumes: prism lwh, cylinder \pi r^2h, sphere \frac{4}{3}\pi r^3.
- You can distinguish permutations vs combinations: \,{}_nP_r vs \,{}_nC_r.
- You check units and reasonableness before choosing an answer.
You’ve got the tools—now it’s just pattern recognition and clean execution.