The Enhanced ACT Math test gives you 50 minutes for 45 questions.

ACT says Math covers material students typically learn through the beginning of grade 12.

Current reporting emphasis:

• Preparing for Higher Math: about 80%
  • Number & Quantity: 10–12%
  • Algebra: 17–20%
  • Functions: 17–20%
  • Geometry: 17–20%
  • Statistics & Probability: 12–15%
• Integrating Essential Skills: about 20%
• Modeling: appears throughout and overlaps other categories

ACT also states that knowledge of basic formulas is assumed, while recall of unusually complex formulas is not the main goal.

This guide focuses on formulas worth knowing and, more importantly, when to use them. For a complete overview of section timing and format, see our Enhanced ACT guide, and for strategy on other sections see the rest of our ACT guides.

Algebra

Slope

[ m=\frac{y_2-y_1}{x_2-x_1} ]

Slope-intercept form:

[ y=mx+b ]

Point-slope form:

[ y-y_1=m(x-x_1) ]

Distance

[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]

Midpoint

[ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) ]

Quadratics

Standard form:

[ ax^2+bx+c=0 ]

Quadratic formula:

[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ]

Vertex x-coordinate:

[ x=-\frac{b}{2a} ]

Discriminant:

[ b^2-4ac ]

• positive → two real solutions
• zero → one repeated real solution
• negative → no real solutions

Exponents

[ a^m a^n=a^{m+n} ]

[ \frac{a^m}{a^n}=a^{m-n} ]

[ (a^m)^n=a^{mn} ]

[ a^{-n}=\frac1{a^n} ]

[ a^{1/n}=\sqrt[n]{a} ]

Percent change

[ \text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times100% ]

Growth multiplier:

[ 1+r ]

Decrease multiplier:

[ 1-r ]

Ratios and proportions

If:

[ \frac{a}{b}=\frac{c}{d} ]

then:

[ ad=bc ]

But do not cross-multiply automatically if a simpler unit-rate method exists.

Rates

[ \text{rate}=\frac{\text{quantity}}{\text{time}} ]

Distance:

[ d=rt ]

Average speed across unequal time intervals is not generally the simple average of the speeds.

Use total distance / total time.

Arithmetic mean

[ \bar{x}=\frac{\text{sum}}{\text{number of values}} ]

Weighted average:

[ \frac{\sum (\text{value})(\text{weight})}{\sum\text{weights}} ]

Probability

[ P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}} ]

For independent events:

[ P(A\text{ and }B)=P(A)P(B) ]

For mutually exclusive events:

[ P(A\text{ or }B)=P(A)+P(B) ]

Use these only when the relevant independence/exclusivity conditions are satisfied.

Geometry

Rectangle

Area:

[ A=lw ]

Perimeter:

[ P=2l+2w ]

Triangle

[ A=\frac12bh ]

Pythagorean theorem

[ a^2+b^2=c^2 ]

Useful triples:

• 3-4-5
• 5-12-13
• 8-15-17

and multiples.

Circle

Circumference:

[ C=2\pi r ]

Area:

[ A=\pi r^2 ]

Arc length in degrees:

[ \frac{\theta}{360}(2\pi r) ]

Sector area:

[ \frac{\theta}{360}\pi r^2 ]

Rectangular prism

[ V=lwh ]

Cylinder

[ V=\pi r^2h ]

Cone

[ V=\frac13\pi r^2h ]

Sphere

[ V=\frac43\pi r^3 ]

Surface area:

[ 4\pi r^2 ]

Angles

Triangle interior angles:

[ 180^\circ ]

Polygon interior sum:

[ (n-2)180^\circ ]

Regular polygon interior angle:

[ \frac{(n-2)180^\circ}{n} ]

Similar figures

Corresponding lengths use scale factor (k).

Areas scale by:

[ k^2 ]

Volumes scale by:

[ k^3 ]

This is a common conceptual trap.

Trigonometry

Right triangle:

[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} ]

[ \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} ]

[ \tan\theta=\frac{\text{opposite}}{\text{adjacent}} ]

Special triangles:

45-45-90

Sides:

[ x,\ x,\ x\sqrt2 ]

30-60-90

Sides:

[ x,\ x\sqrt3,\ 2x ]

Functions

If:

[ f(x)=2x^2-3 ]

then:

[ f(4)=2(16)-3=29 ]

Do not confuse (f(4)) with multiplication.

Function transformations

If (y=f(x)):

• (f(x)+k): up (k)
• (f(x)-k): down (k)
• (f(x-h)): right (h)
• (f(x+h)): left (h)

Horizontal shifts are easy to reverse mentally.

Modeling

ACT emphasizes modeling throughout the Math test.

A model question may ask:

• what slope means;
• what intercept means;
• which equation fits a context;
• what a parameter represents;
• whether an extrapolation is reasonable.

The formula is only half the work.

Calculator strategy

A calculator is allowed on ACT Math under ACT's calculator policy.

Use it for:

• arithmetic;
• roots/powers;
• graphing when useful;
• checking solutions.

But mental/hand work is often faster for:

• simple fractions;
• proportions;
• easy factoring;
• basic geometry.

Do not use the calculator as a substitute for understanding what operation to perform.

Four answer choices matter

The enhanced ACT Math section now uses four answer choices rather than the legacy five-choice format.

That makes elimination slightly more valuable.

If you can eliminate two choices, you have reduced the problem significantly. If you're mapping this review against an upcoming exam, check our guide to ACT test dates.

Original mini-practice

1

A line passes through ((2,5)) and ((8,17)). What is its slope?

2

A jacket's price increases from $80 to $92. What is the percent increase?

3

A right triangle has legs 9 and 12. What is its hypotenuse?

4

If (f(x)=x^2+2x), what is (f(3))?

5

A cube's side length is doubled. By what factor does its volume increase?

Answers

1. 2
   ((17-5)/(8-2)=12/6=2).

2. 15%
   (12/80=0.15).

3. 15
   9-12-15 is a 3-4-5 multiple.

4. 15
   (9+6=15).

5. 8
   Volume scales by (2^3).

The most important formula rule

Do not memorize a formula without knowing:

• what each variable means;
• what units should be;
• when the formula applies;
• what the question is asking for.

ACT Math rewards selection and interpretation as much as recall. If you're wondering whether structured review like this actually moves scores, our look at whether ACT prep works covers the evidence.

Official references

• ACT Math section: https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/act-exam-sections-and-structure.html
• ACT Math description: https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html

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