51 SAT and ACT Math Formulas You Need to Know
51 SAT and ACT Math Formulas You Need to Know

51 SAT and ACT Math Formulas You Need to Know

Here is a list of (almost) all the formulas you will need for the SAT math section and the ACT math section. Although the SAT math section is changing with the digital SAT, the SAT math formulas listed below can be used. The ACT math formulas will not change very much this year. You can use this formula sheet to review and memorize the formulas. It is recommended that you have a firm grasp of concepts like calculating degree measures, linear equations, and other things you have learned in your high school math classes. Knowing the ACT and SAT math formulas can help raise your SAT math test score as well as help you on the ACT math section. Let’s get down to business!

PEMDAS

PEMDAS

  1. Do anything inside the parentheses first
  2. Do all exponents
  3. Do multiplication and division from left to right
  4. Do addition and subtraction from left to right.

Factors and Multiples

Rules for Factors and Multiples

  1. Definition: A factor is a number that divides another number completely, leaving no remainder. Conversely, a multiple is a number that can be represented as the product of another number and an integer.
  2. Relation: If A is a factor of B, then B is a multiple of A.
  3. Number One:1′ is a factor of every number, and every number is a multiple of 1.
  4. Self Relation: Every number is a factor as well as a multiple of itself.
  5. Multiples: The multiples of a number continue indefinitely. For example, the multiples of 2 are 2, 4, 6, 8, 10, and so on.
  6. Even Numbers: Every even number has 2 as a factor, and hence, every even number is a multiple of 2.
  7. Factor Limit: The factors of a number cannot be larger than the number itself.
  8. Prime Numbers: Prime numbers have only two distinct positive factors: 1 and the number itself.

Probability, Combination, and Permutation

  1. Probability: Probability is the likelihood of an event happening and is measured between 0 (not possible) and 1 (will happen). Add the probabilities of all possible outcomes to get 1. When dealing with independent events multiply the individual probabilities of both events to get the probability of both events occurring.
  2. Combinations: Combinations are where the order of choice does not matter. C(n, r) = n! / [(r!)(n – r)!]
  3. Permutations: Permutations are where order matters. P(n, r) = n! / (n – r)!
  4. Independent and Dependent Events: If the outcome of one event does not affect the outcome of another, the events are independent. If the outcome of one event does affect the outcome of another, the events are dependent.
  5. Mutually Exclusive: If the occurrence of one event precludes occurence of the other event, they are mutually exclusive

Rules for Percentages

The math formulas for percentages are relatively straightforward.

  1. Percentage increase and decrease: When calculating a percentage increase or decrease, the percentage is always relative to the original amount, not the changed amount.
  2. How to go from decimals to percentages: multiply by 100.
  3. How to go from percentages to decimals: divide by 100.

Geometry Formulas

Geometry math formulas are numerous. Here are some of the most common ACT and SAT math formulas you will need

Distance formula

The distance formula is used to determine the distance, denoted as ‘d’, between two points in a coordinate system. Given two points, (x1, y1) and (x2, y2), the distance formula is represented as:

d = √[(x2 – x1)² + (y2 – y1)²]

In this formula, `√` represents the square root sign, and the values inside the square root are squared, as indicated by the superscript ‘2’.

Midpoint Formula

M = [(x1 + x2)/2 , (y1 + y2)/2]

Area of triangle

A = 1/2 * base * height

Pythagorean Theorem (for Right Triangle only)

a² + b² = c²

30-60-90 Right Triangle Formula

If shortest side = a,

Longer leg = a√3

Hypotenuse = 2a

`

45-45-90 Right Triangle Formula

  • Other 2 legs = a
  • Hypotenuse = a√2

SOHCAHTOA

`SOHCAHTOA should be broken into soh cah toa. Now memorize soh cah toa individually.

Sine = Opposite over Hypotenuse (SOH),

Cosine = Adjacent over Hypotenuse (CAH),

Tangent = Opposite over Adjacent (TOA).

Linear Functions

Slope of a Line

The formula for the slope of a line when given two points, (x1, y1) and (x2, y2), on the line is

m = (y2 – y1) / (x2 – x1)

Point Slope Form

y – y1 = m(x – x1)

Slope Intercept Form

y = mx + b`

  • The essence of the slope formula is rise/run

Lines

Parallel lines

Parallel lines have the same slope and never meet.

m1 = m2

Intersecting lines

Intersecting lines meet and do not have the same slope.

m1 ≠ m2

Perpendicular lines have slopes that are negative reciprocals of each other.

Circle Formulas

Diameter of a Circle

D = 2r`

Circumference of a Circle

C = πD = 2πr

Area of a Circle

A = πr²

Equation of a circle

Equation of a Circle

The equation of a circle x² + y² = r² if the circle is centered at the origin. If the center of the circle is at any point (h,k), the equation of the circle will be (x – h)² + (y – k)² = r²

Squares

Perimeter of a Square

P = 4s

Area of a Square

A = s²

Rectangles

Perimeter of a Rectangle

P=2l+2w

Area of a Rectangle

A = l * w

Other Quadrilaterals

Parallelograms

Area of a Parallelogram

A = b * h

Rhombuses

Area of a Rhombus

A = (p*q)/2 where ‘p’ and ‘q’ are the lengths of the diagonals.

Trapezoids

Area of a Trapezoid

A = 1/2(a+b) * h where ‘a’ and ‘b’ are the lengths of the parallel sides and ‘h’ represents the height.

Volume Formulas

Volume Formulas

Sphere

V = 4/3 * π * r³

Cube

V = s³

Cone

V = 1/3 * π * r² * h

Right Cylinder

V = π * r² * h

Quadratic Formulas

x = [-b ± √(b² – 4ac)] / 2a

Factoring

Factoring with FOIL

To factor a quadratic expression using FOIL (First, Outer, Inner, Last), you first have to ensure the expression is in the format of ax² + bx + c. FOIL is the reverse process of expanding brackets. Here are the steps:

  1. F (First): Look for two numbers that multiply to ‘a’ (the coefficient of x²) and at the same time, add up to ‘b’ (the coefficient of x). These two numbers will be the ‘first’ terms in each of two binomials.
  2. O (Outer) and I (Inner): Check the ‘outer’ and ‘inner’ products of the binomials to make sure they add up to ‘b’. The ‘outer’ product is the product of the first term in the first binomial and the second term in the second binomial. The ‘inner’ product is the product of the second term in the first binomial and the first term in the second binomial.
  3. L (Last): Finally, double-check that the ‘last’ terms in each binomial multiply together to give ‘c’ (the constant term).

Imaginary Numbers

Rules for Imaginary Numbers

  1. Defining ‘i’: The imaginary unit ‘i’ is defined as the square root of the negative one (√-1).
  2. Square of ‘i’: The square of ‘i’ (i^2) is equal to -1.
  3. Product of ‘i’: The product of ‘i’ with itself for four times (i^4) gives a result of 1. This means i, i^2, i^3, i^4 cycle continually as i, -1, -i, 1.

Rules for Exponents

  1. Product of Powers: x^z * y^w = x^(y+x).
  2. Quotient of Powers: x^y / x^z = a^(y-z).
  3. Power of a Power: (x^y)^z = z^(y*z).
  4. Power of a Product: (ab)^x = a^x * b^x.
  5. Power of a Quotient: (a/b)^x = a^x / b^x.
  6. Zero Exponent:. Any number or variable raised to 0 is 1 except 0 itself. z^0 = 1 for any z ≠ 0.
  7. Negative Exponent: x^-n = 1/x^n.

Rules for Logarithms

  1. Product Rule: log_a (xy) = log_a(x) + log_a(y).
  2. Quotient Rule: log_a(x/y) = log_a(x) – log_a(y).
  3. Power Rule: log_a(x^y) = y * log_a(x).

Now you have an overview of the major math formulas and definitions. Thls list of ACT math formulas is equally applicable to the SAT math as well. Regardless of whether you will be taking the ACT math or the SAT math, knowing thise formulas can make a big difference on test day.

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