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, $\Delta<0$ none real
$\text{Axis of symmetry: } x=\frac{-b}{2a}$	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 $\Delta<0$ .

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.