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Math Cheat Sheet

This page is a collection of all the formulas and concepts you need to know for the SAT Math section.

Provided Formulas

The following formulas are provided to you in the BlueBook app while you're taking the test. Since they're provided to you, you don't need to memorize the formulas, but you should memorize which formulas are provided, and familiarize yourself with how to use them.

Algebra

Linear Equation Forms

Slope-Intercept Form
y = mx + b
e.g. y = 2x + 3
m = slope
b = y-intercept
Standard Form
ax + by = c
e.g. 2x + 3y = 4
a = rate/amount per x
b = rate/amount per y
c = total amount of x and y
Point-Slope Form
y - y₁ = m(x - x₁)
e.g. y - 3 = 2(x - 1)
m = slope
(x₁, y₁) = point on line

Concepts

Slope
Steepness/rate of change of line
m = $$\frac{rise}{run}$$ = $$\frac{\Delta y}{\Delta x}$$ = $$\frac{y_2 - y_1}{x_2 - x_1}$$
Y-Intercept
Point where the line crosses the y-axis
Plug in x = 0
X-Intercept
Point where the line crosses the x-axis
Plug in y = 0
Horizontal Line
y = constant
e.g. y = 3
Slope = 0
Vertical Line
x = constant
e.g. x = 3
Slope = undefined
Parallel Lines
Same slope, different intercepts
Never intersect, always same distance apart
Perpendicular Lines
Negative reciprocal slopes
$$m_2 = -\frac{1}{m_1}$$
Intersect at right angle (90°)
Equation/system with no solutions
Parallel lines
Both sides/equations have same slope (x coefficient)
Different y-intercepts
e.g. y = 2x + 3 and y = 2x + 4
or 2x+3 = 2x+4
Solving results in false statement
e.g. 3 = 5
Equation/system with infinite solutions
Both sides/equations represent same line
Same slope, same y-intercept
e.g. y = 2x + 3 and 2y = 4x + 6
or 2x+6 = 2(x+3)
Solving results statement that is always true
e.g. 3 = 3
Equation/system with one solution
Sides/equations have different slopes
e.g. y = 2x + 3 and y = 3x + 2
or 2x+3 = 3x+2
Solving results in x = constant
e.g. x = 3

Formulas

Distance Formula
Distance between two points
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Comes from Pythagorean theorem
Midpoint Formula
Midpoint between two points
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
x = halfway between x₁ and x₂
(avg x coordinate)
y = halfway between y₁ and y₂
(avg y coordinate)

Advanced Math

Parabolas/Quadratic Functions

Vertex
The point where the parabola turns around
Either the maximum or minimum point
At the vertex, the slope is 0
X-Intercepts
Points where the parabola crosses the x-axis
Aka the solutions when the quadratic equation equals 0
Can be found by factoring or using the quadratic formula
Standard Form
f(x) = ax² + bx + c
e.g. y = 2x² + 3x + 4
a = coefficient of x²
b = coefficient of x
c = y-intercept
Vertex From Standard Form
If vertex is at (h,k), then:
$$h = \frac{-b}{2a}$$
$$k = f(h)$$
e.g. if $$y = 2x² + 3x + 4$$, then
$$h = \frac{-3}{2(2)} = \frac{-3}{4}$$
$$k = 2\left(\frac{-3}{4}\right)^2 + 3\left(\frac{-3}{4}\right) + 4$$
Standard Form Sum of Solutions
Sum of solutions = $$-\frac{b}{a}$$
e.g. if $$y = 2x^2 + 3x + 4$$, then
solutions add up to $$-\frac{3}{2}$$
Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Gives the solutions to a Standard Form quadratic equation
Aka the x-intercepts
Discriminant
What's inside the radical in the Quadratic Formula
$$D = b^2 - 4ac$$
Tells you how many solutions there are
D > 0: 2 solutions
D = 0: 1 solution
D < 0: no (real) solutions
Factored Form
f(x) = a(x - r)(x - s)
e.g. y = 2(x - 1)(x - 3)
Set each factor equal to 0 for solutions
In above example solutions are:
x = 1 and x = 3
Factored Form from Standard Form
Standard form: ax² + bx + c
Find the product of a and c (ac)
Find two factors of ac that add up to b
We'll call these factors r and s
If a = 1, then factored form is (x - r)(x - s)
If a > 1, then factor by grouping
Factor By Grouping
Find r and s like in previous section
Rewrite as ax² + rx + sx + c
Factor greatest common factor out of first two terms: Ax(Bx + C)
Factor greatest common factor out of last two terms: D(Bx + C)
Rewrite as (Ax + D)(Bx + C)
Example:
$$2x^2 + 5x + 12$$
$$ac = 2*12 = 24$$
$$(8)(-3) = -24$$ and $$(8)+ (-3) = 5$$
$$2x^2 + 5x + 12 = 2x^2 + 8x - 3x + 12$$
$$= 2x(x + 4) - 3(x + 4)$$
$$= (2x - 3)(x + 4)$$
Difference of Squares
When a perfect square is subtracted from another perfect square, it is easily factored
a² - b² = (a + b)(a - b)
e.g. x² - 9 = (x + 3)(x - 3)
Perfect Square Trinomial
A perfect square trinomial is a quadratic equation that can be written as a square of a binomial
If you notice the pattern:
a² + 2ab + b²
It can be factored as (a + b)²
e.g. 9x² + 24x + 16
= (3x)² + 2(3x)(4) + 4²
= (3x + 4)²
Or the pattern:
a² - 2ab + b²
Can be factored as (a - b)²
e.g. 4x² - 12x + 9
= (2x - 3)²
Vertex Form
f(x) = a(x - h)² + k
e.g. y = 2(x + 1)² + 3
(h, k) = vertex
a > 0: opens up
a < 0: opens down
Note the signs before h and k
Vertex is (-1, 3) in example above
Parabola Symmetry
Parabola is symmetric about a vertical line through the vertex
The line is x = h, where h = vertex x-coordinate
X-intercepts are equidistant from vertex (and the line of symmetry)
e.g. if intercepts are at -5 and 3, then vertex is halfway between (x = -1)
Similarly, if any two points have the same y value then the vertex is halfway between them
Or if you know the vertex and one x-intercept, you can find the other (will be same distance from vertex)
Completing the Square
Useful when you want to have a perfect square binomial in the equation
(Like when converting to vertex form or in circle equations)
If we have ax² + bx + c = d, where c and d can represent any constants or additional terms
We can rewrite it as a(x² + b/a x) + c = d
We can change x² + b/a x to (x + b/2a)², but this would add an extra (b/2a)² to the equation
So we subtract (b/2a)² to keep the equation balanced
Ending up with a(x + b/2a)² - (b/2a)² + c = d

Exponents and Radicals

Multiplication Rule
xᵃ * xᵇ = x⁽ᵃ⁺ᵇ⁾
e.g. x² * x³ = x2+3 = x5
Note that this can also be applied in reverse
e.g. 2x+3 = 2x * 23 = 2x * 8
Negative Exponent Rule
x⁻ᵃ = 1/xᵃ
e.g. x⁻² = 1/x²
Note that this can also be applied in reverse
e.g. 3/x⁻² = 3x²
Division Rule
xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾
e.g. x⁵ / x² = x³
This can also be figured out by combining the negative exponent rule and the multiplication rule
xᵃ / xᵇ = xᵃ * x⁻ᵇ = x⁽ᵃ⁻ᵇ⁾
Note that this can also be applied in reverse
e.g. x⁽²ᵃ⁻³ᵇ⁾ = x²ᵃ / x³ᵇ
Power Rule
(xᵃ)ᵇ = xᵃᵇ
e.g. (x²)³ = x⁶
Note that this can also be applied in reverse
e.g. x⁽²ᵃ⁾ = (x²)ᵃ = (xᵃ)²
Zero Exponent Rule
a⁰ = 1, where a is any non-zero number
e.g. 2⁰ = 1
Fractional Exponents
Whenever you have a fractional exponent, you can rewrite it as a radical
xab=xab
e.g. x23=x23
If you forget which number goes where, think of a simple example, like x13=x3

Exponential Functions

General Format
f(x) = a * b^x, where a and b are constants
e.g. f(x) = 2 * 3^x
The value will increase by a factor of b for every time the exponent increases by 1
In the example above, the value will increase by a factor of 3 for every time x increases by 1
But in the example f(x) = 2 * 3^(2x), the value will increase by a factor of 3 for every time 2x increases by 1 (or when x increases by 1/2)
Or we could think of it as increasing by a factor of 3^2 for every time x increases by 1 (because we could rewrite as 2 * (3^2)^x)
Exponential Growth vs Decay
If the function is $$f(x) = a * b^x$$:
If b > 1, then it's an exponential growth function
If 0 < b < 1, (b is a positive fraction less than 1) then it's an exponential decay function
Exponential growth functions increase more rapidly towards infinity as x increases
Exponential decay functions decrease and approach 0 as x increases
Applications
Exponential functions are used to model growth and decay in many real-world situations
e.g. population growth or decay, radioactive decay, investment growth, technological growth, etc.
If something is increasing or decreasing by a constant factor for every unit of time, it's an exponential function
e.g. if a population doubles every 5 years, that's exponential growth
If a material loses 6% of its mass every 10 years, that's exponential decay
Interest Formulas
The simple interest formula (when interest is compounded annually) is $$A = P(1 + r)^t$$
A is the final amount
P is the principal (initial amount)
r is the annual interest rate (e.g. 0.05 for 5%)
t is the time in years
e.g. if you invest $1,000 at 5% interest for 10 years:
$$A=1,000(1+0.05)^{10}=\$1628.89$$
If interest is compounded more than once per year, then the formula is $$A = P(1 + r/n)^{nt}$$
Where n is the number of times interest is compounded per year
e.g. if you invest $1,000 at 5% interest for 10 years, compounded monthly:
$$A=1,000\left(1+\frac{0.05}{12}\right)^{12*10}=\$1647.01$$
Normally on the SAT, you'll only need to know the simple formula

Transformations

Vertical Shift
If you have a function f(x)
The function f(x) + c shifts the graph up by c units
e.g. if you have f(x) = x², and you want to shift it up by 3 units, you can add 3 to the function to get f(x) = x² + 3
Horizontal Shift
If you have a function f(x)
The function f(x - c) shifts the graph to the right by c units
e.g. if you have f(x) = x², and you want to shift it to the right by 3 units, you can replace all instances of x with (x-3) to get f(x) = (x - 3)²
Other Transformations
-f(x) reflects the graph over the x-axis (flips it upside down)
f(-x) reflects the graph over the y-axis (flips it left to right)
c*f(x) stretches the graph vertically by a factor of c (makes it skinnier if c > 1, or fatter if c < 1)
Graphs of Common Functions

Problem-Solving and Data Analysis

Statistics and Probability

Mean
aka average
Mean = Sum of all values / Number of values
e.g. if you have the numbers 1, 2, 3, 4, 5, the mean is (1+2+3+4+5) / 5 = 3
Median
The middle value in a set of numbers when ordered from least to greatest
e.g. if you have the numbers 1, 2, 3, 4, 5, the median is 3
If there's an even number of values, the median is the average of the two middle values
e.g. if you have the numbers 1, 2, 3, 4, 5, 6, the median is (3+4) / 2 = 3.5
Mode
The most frequently occurring value in a set of numbers
e.g. if you have the numbers 1, 2, 2, 3, 4, the mode is 2
Range
The difference between the largest and smallest values in a set of numbers
e.g. if you have the numbers 1, 2, 3, 4, 5, the range is 5 - 1 = 4
Standard Deviation
A measure of how much the values in a data set vary from the mean
In other words, how spread out the data is
If all the data points are clustered around the mean, the standard deviation is low
If the data points are spread out, the standard deviation is high
You won't need to actually calculate the standard deviation on the SAT, but they might test you on the concept
Margin of Error
The margin of error is the amount of error that is allowed in a survey or experiment
It's usually expressed as a percentage
e.g. if a survey estimates that 35% of people support a certain candidate, and the margin of error is 5%, then the true value is within 5% of the estimated value
This means the true value is between 30% and 40%
Surveys and Sampling
Surveys are used to collect data from a sample of a population
Conclusions from a survey should only be made about the population that the sample represents
e.g. if you survey 20 random athletes at a school and ask how many hours they study per week, you can only draw conclusions about how much athletes at that school study, not the entire student body
Probability
Probability is the likelihood of an event occurring
Probability = Number of favorable outcomes / Total number of outcomes
e.g. a bowl has 3 red marbles and 2 blue marbles. The probability of picking a red marble is 3/5
The probability of two events both happening is the product of their probabilities
e.g. the probability of rolling a 3 and then rolling a 4 is (1/6) * (1/6) = 1/36
The probability of either of two events happening is the sum of their probabilities
e.g. the probability of rolling a 3 or rolling a 4 is (1/6) + (1/6) = 1/3
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has occurred
Take what they say is given, and ignore all possibilities that are inconsistent with that
e.g. say white bucket has 3 red marbles and 2 blue marbles, and a black bucket has 2 red marbles and 3 blue marbles. If you're asked the probability of picking a red marble given that you're picking from the white bucket, you can ignore the black bucket.
The probability of picking a red marble from the white bucket is 3/5

Percentages, Ratios, and Proportions

Percentages
Percentages are a way to express a fraction as a number out of 100
e.g. 25% is the same as 25/100 or 0.25
Percent "Of"
"Percent of" means that you're taking a percentage of a number
To find p% of q, multiply q by p/100
e.g. 25% of 80 is (25/100) * 80 = 20
Percent Increase/Decrease
If a number q increases by p%, then the new number is q * (1 + p/100)
If a number q decreases by p%, then the new number is q * (1 - p/100)
e.g. if a shirt costs $20 and it increases by 10%, then the new price is $20 * (1 + 10/100) = $20*1.1 = $22
e.g. if a shirt costs $20 and is on sale for 10% off, then the new price is $20 * (1 - 10/100) = $20*0.9 = $18
Working the other way, if you are told the initial value and the final value, you can find the percent change by using the formula:
Percent Change = (New Value - Old Value) / Old Value
e.g. if a shirt costs $20 and increases to $22, the percent increase is (22 - 20) / 20 = 10%
NOTE: If something increases by 200%, it is not doubling.
The new value is the initial value plus 200% of the initial value, so really it's tripling.
Pre-Percentage Amount
Sometimes you might be told the amount after a percentage has been applied, and you need to find the original amount.
The easiest way to intuitively set up this formula is:
final amount = original amount * (1 + percentage)
e.g. if a shirt costs $22 after a 10% increase, then:
$22 = original amount * (1.1)
So, ultimately the equation for finding the original amount is:
$$original = \frac{final}{1 + percentage}$$
e.g. $$original = \frac{22}{1.1}$$
Ratios
A ratio is a comparison of two quantities by division
e.g. if you have 3 apples and 2 oranges, the ratio of apples to oranges is 3:2 or 3/2
If you know two things have the same ratio, you can set up a proportion to find the unknown quantity
e.g. if two similar triangles have corresponding sides of 3 and 5 on one triangle, and 6 and x on the other, you can set up the proportion:
$$\frac{3}{5} = \frac{6}{x}$$
Cross-multiplying gives:
$$3x = 30$$
Then divide both sides by 3 to get:
$$x = 10$$
Unit Conversion
Multiply the original quantity by the conversion ratio to get the new quantity
e.g. if 1 mile = 5,280 feet, the conversion ratio can be written as $$\frac{5280\ feet}{1\ mile}$$ or $$\frac{1\ mile}{5280\ feet}$$
Since the numerator and denominator represent the same distance, the conversion ratio is equal to 1, so multiplying by it doesn't change the original quantity
e.g. if you want to convert 10 miles to feet, you can multiply 10 miles by the conversion ratio:
$$10\ miles * \frac{5280\ feet}{1\ mile} = 52,800\ feet$$
You want to set the numerator and denominator so that the correct units cancel out, leaving you with the desired units
e.g. in the example above, we want to cancel out miles and end up with feet, so we put miles in the denominator
Chemical Solutions/Mixed Materials
Some problems deal with combining two substances, each consisting of mixed elements
e.g. combining 1L of saline that is 10% salt with 2L of saline that is 20% salt
The ratio might be given as percent by volume, or percent by mass, but the concept is the same
It could also be given as a ratio instead of a percent
The general formula for this type of problem is:
$$V_1 * P_1 + V_2 * P_2 = (V_1 + V_2) * P_f$$
e.g. In the above formula, V1 represents the volume of the first substance, P1 represents the percent of the first substance, Pf represents the percent of the final combined substance, and so forth
To demonstrate with the saline example from above, we would plug in the values as follows:
$$1 * 0.1 + 2 * 0.2 = (1 + 2) * P_f$$
This yields a final solution of 16% salt
Mixed Substances/Solutions
When you combine two substances (e.g. saline solutions with different concentrations), the final concentration of the substance is a weighted average of the concentrations of the two substances
$$C_f = \frac{C_1V_1 + C_2V_2}{V_1 + V_2}$$
Cf = final concentration
C1 = concentration of the first substance
V1 = volume of the first substance
C2 = concentration of the second substance
V2 = volume of the second substance
e.g. if you have 100g of saline that is 10% salt and 200g of saline that is 20% salt, the final concentration is (100*10 + 200*20) / (100 + 200) = 15%
The same concept applies when it's percent by mass instead of percent by volume
Distance, Rate, and Time
Distance = Rate * Time
Easiest way to remember: speed is given in miles per hour: speed = miles/hour
Since speed is the rate, miles is the distance, and hours is the time: r = d/t
This can be manipulated to either d = r*t or t = d/r

Geometry and Trigonometry

Provided Formulas

The SAT provides you with many geometry and trigonometry formulas. Since they're provided, it's not necessary to memorize them. However, it's important to familiarize yourself with which ones are provided, so you know where to find them. We already showed these earlier on the page, so we won't repeat them here.

Terminology

Supplementary Angles
Supplementary angles are two angles that add up to 180 degrees
Complementary Angles
Complementary angles are two angles that add up to 90 degrees
Congruent
Means equal and is represented by the symbol ≅
Congruent angles have the same measure
Congruent lines have the same length
Congruent polygons have all sides and all angles equal
Similar
Similar means proportional and is represented by the symbol ∼
Similar polygons have all corresponding angles equal and all corresponding sides proportional
e.g. a right triangle with sides 3, 4, and 5 is similar to a right triangle with sides 6, 8, and 10

General Angle Rules

Vertical Angles
Vertical angles are angles that are opposite each other when two lines intersect
Vertical angles are equal in measure
60°60°
Supplementary and Complementary Angles
Supplementary angles are two angles that add up to 180 degrees
Complementary angles are two angles that add up to 90 degrees
Linear Pair
Linear pairs are two angles that are adjacent and together form a straight line
Linear pairs are supplementary, meaning they add up to 180 degrees
e.g. in the example below, the 60° angle and the 120° angle form a linear pair
60°120°
Parallel Lines With Transversal
Parallel lines are lines that never intersect
A transversal is a line that cuts through two parallel lines, creating 8 angles
Unless the transversal is perpendicular to the parallel lines, you'll have 4 acute angles (<90°) and 4 obtuse angles (>90°)
All 4 acute angles are congruent (equal in measure), and all 4 obtuse angles are congruent
abbaabba
Radians
Radians are another way to measure angles (instead of degrees)
2π radians = 360 degrees
(or π radians = 180 degrees)
To convert from one unit to the other, set up a proportion:
degrees180=radiansπ

Triangles

Sum of Interior Angles
The sum of the interior angles of a triangle is 180 degrees
Types of Triangles
Right Triangle - One angle is 90 degrees
Equilateral Triangle - All sides are equal, and all angles are 60 degrees
Isosceles Triangle - Two sides are equal, and two angles are equal
Scalene Triangle - No sides are equal, and no angles are equal
Right Triangle - One angle is 90 degrees
Acute Triangle - All angles are less than 90 degrees
Obtuse Triangle - One angle is greater than 90 degrees
Right Triangles Concepts
Hypotenuse - The longest side of a right triangle, opposite the right angle
Legs - The two shorter sides of a right triangle, adjacent to the right angle
Most triangle problems on the SAT will deal with right triangles
The Pythagorean Theorem (covered later) applies only to right triangles
Questions involving Sine, Cosine, and Tangent (covered later) will only be asked about right triangles
Pythagorean Theorem
Applies only to right triangles
When you have the lengths of two sides of a right triangle, you can use the Pythagorean Theorem to find the length of the third side
a2+b2=c2
a and b are the lengths of the legs of the triangle (the two shorter sides adjacent to the right angle)
c is the length of the hypotenuse (the longest side, opposite the right angle)
This is provided on the SAT formula sheet, but you should still memorize it
abc
Pythagorean Triples
Pythagorean triples are special right triangles with side lengths that neatly work out to whole numbers
Recognizing them will save you time, so that you don't have to apply the Pythagorean Theorem
The two most common ones have side lengths 3-4-5 and 5-12-13
So, for example, if you see a triangle with a leg of 5 and a hypotenuse of 13, you will know that the other leg is 12
You might also see a scaled up version of 3-4-5, like 6-8-10
SOH-CAH-TOA
SOH-CAH-TOA is a way to remember the three basic trigonometric functions: Sine, Cosine, and Tangent
sin=OppositeHypotenuse
cos=AdjacentHypotenuse
tan=OppositeAdjacent
Special Right Triangles
There are two right triangles with "special" angles that show up often on the SAT
The 30-60-90 triangle:
x√3x2x30°60°
The 45-45-90 triangle (aka isosceles right triangle):
xxx√245°45°
They are provided on the SAT formula sheet, memorizing their side ratios will help you recognize them on the test
Sine-Cosine Relationship
The sine of an angle is equal to the cosine of its complement
sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
Similar Triangles
Have all the same angles
They might be different sizes, but the ratio of their side lengths is the same
There are three ways to prove that two triangles are similar:
AA - Two angles are congruent (equal)
SSS - All three sides are proportional
SAS - Two sides are proportional and the angle between them is congruent
Triangle Altitude
The altitude of a triangle is the perpendicular distance from one vertex to the opposite side
Sometimes the altitude is inside the triangle:
Sometimes the altitude is outside the triangle:
The altitude can be used as the "height" to calculate the area of a triangle
Area of a Triangle
Area=12·base·height
In a right triangle, the two legs are the base and height
In any other triangle, you can use any side as the base, and its altitude as the height
Right Triangle Similar Subtriangles
If you have a right triangle, and you draw an altitude from the right angle to the hypotenuse, you will create two smaller right triangles
These two smaller right triangles are similar to the original triangle and to each other
ABCDabba
Trianges ABC, ACD, CBD are all similar
Circles
Radius and Diameter
The radius of a circle is the distance from the center to a point on the circle
The diameter of a circle is the distance across the circle through the center
The diameter is twice the radius
Circumference and Area of a Circle
The circumference of a circle is the distance around the circle (the perimeter)
The circumference of a circle is given by the formula:
C=2πr
The area of a circle is the amount of space inside.
The area of a circle is given by the formula:
A=πr2
Both formulas are provided on the SAT formula sheet, but doesn't hurt to memorize them!
Circle Arcs and Sectors
An arc is a part of the circumference of a circle
A sector is a part of the area of a circle
You can set up proportions to find unknowns
angle360=arc lengthcircumference=sector areacircle area
Circle Tangents
A tangent is a line that touches the circle at exactly one point
The radius and tangent line form a right angle
Inscribed and Central Angles
An inscribed angle is an angle with its vertex on the circle
A central angle is an angle with its vertex at the center of the circle
The measure of an inscribed angle is half the measure of the central angle
x2x
Polygons
Regular Polygon
A regular polygon is a polygon where all sides and all angles are equal
Examples include equilateral triangles, squares, and octagons shaped like stop signs
Interior Angles
If n represents the number of sides of a polygon, the sum of the interior angles is given by the formula:
Sum=180(n-2)
For regular polygons, each interior angle is the sum divided by the number of sides:
Angle=180(n-2)n
Exterior Angles
An exterior angle is the angle between one side of a polygon and the extension of an adjacent side (see diagram below)
The sum of the all the exterior angles of a polygon is 360 degrees
α
Parallelograms
Opposite sides are parallel and equal in length (congruent)
Opposite angles are congruent
Consecutive angles are supplementary
The diagonals bisect each other (cut each other in half - see diagram below)
aabb
The sum of the interior angles is 360 degrees
Area = base x height
heightbase