ACT Math Formula Sheet: Essential Formulas for Test Success

The mathematics section of the ACT assesses your skills across a range of mathematical content areas. While the test doesn't provide an official formula sheet, having these essential formulas memorized can significantly improve your speed and accuracy. This comprehensive guide covers all the key formulas you need to know for ACT Math success.

Algebra Formulas

Linear Equations

Understanding linear equations is fundamental for the ACT:

Slope-intercept form: y = mx + b Point-slope form: y - y = m(x - x) Standard form: Ax + By = C Slope formula: m = (y - y)/(x - x) Midpoint formula: ((x + x)/2, (y + y)/2) Distance formula: d = [(x - x) + (y - y)]

Quadratic Equations

Quadratic equations frequently appear on the ACT:

Standard form: ax + bx + c = 0 Quadratic formula: x = [-b (b - 4ac)]/2a Axis of symmetry: x = -b/2a Vertex form: y = a(x - h) + k Factoring difference of squares: a - b = (a+b)(a-b)

Exponent Rules

Mastery of exponent rules is essential:

x^m x^n = x^(m+n) x^m x^n = x^(m-n) (x^m)^n = x^(mn) x^(-n) = 1/x^n x^0 = 1

Geometry Formulas

Area and Perimeter

Basic geometric shape formulas:

Rectangle area: l w Rectangle perimeter: 2(l + w) Square area: s Square perimeter: 4s Triangle area: (1/2)bh Trapezoid area: (1/2)(b + b)h Circle area: r Circle circumference: 2r or d

Volume Formulas

Three-dimensional shapes often appear:

Rectangular prism volume: l w h Cylinder volume: rh Cube volume: s Sphere volume: (4/3)r Cone volume: (1/3)rh Pyramid volume: (1/3)Bh (where B is the area of the base)

Triangle Properties

Triangle formulas and theorems:

Triangle sum theorem: The three angles of a triangle always sum to 180 Pythagorean theorem: a + b = c (for right triangles) Isosceles triangle: Two equal sides and two equal angles Equilateral triangle: Three equal sides and three equal 60 angles Similar triangles: Corresponding angles are equal, sides are in proportion

Trigonometry Formulas

Basic Trigonometric Functions

Trigonometry on right triangles:

sin() = opposite/hypotenuse cos() = adjacent/hypotenuse tan() = opposite/adjacent sin() + cos() = 1

Special Right Triangles

Memorize these special triangles:

30-60-90 triangle: sides are in ratio 1:3:2 45-45-90 triangle: sides are in ratio 1:1:2

Statistics and Probability Formulas

Measures of Central Tendency

Understanding averages and data distributions:

Mean (average): sum of values number of values Median: middle value when data is arranged in order Mode: most frequently occurring value Range: difference between highest and lowest values

Probability and Combinations

Advanced counting and probability:

Probability: favorable outcomes total possible outcomes Combination: nCr = n!/[r!(n-r)!] Permutation: nPr = n!/(n-r)! Fundamental Counting Principle: If one event can occur in m ways and a second in n ways, the pair can occur in mn ways

Matrices and Vectors

Basic Matrix Operations

Matrix addition and multiplication rules:

Matrix addition: Add corresponding elements Matrix subtraction: Subtract corresponding elements Scalar multiplication: Multiply each element by the scalar Matrix multiplication: Row of first matrix column of second matrix Determinant of 22 matrix [[a,b],[c,d]]: ad - bc

Vector Operations

Vector addition: (a,b) + (a,b) = (a+a, b+b) Vector magnitude: |v| = (a + b) Dot product: (a,b) (a,b) = aa + bb

Other Important Concepts

Functions

Understanding functions is crucial for success:

Function notation: f(x) Domain: set of all possible input values Range: set of all possible output values Inverse function: f(x) (where f(f(x)) = x) Composite function: (f g)(x) = f(g(x))

Logarithms

log(b) = c means a^c = b log(ab) = log(a) + log(b) log(a/b) = log(a) - log(b) log(a^n) = nlog(a) log(a) = 1 log(1) = 0

Sets

Union: A B (elements in either A or B) Intersection: A B (elements in both A and B) Complement: A' (elements not in A)

By mastering these formulas and understanding how to apply them efficiently, you'll be well-prepared for the ACT Math section. Remember that practice is essentialthe more problems you solve using these formulas, the more intuitive your approach will become on test day. Good luck!