The SAT math test is unlike any math test you've ever taken before. It's designed to take concepts you're used to and apply them in new (and often weird) ways. It's complicated, but attention to detail and knowledge of the basic formulas and concepts covered by the test can improve your score.
So what formulas do you need to memorize for the math portion of the SAT before test day?In this comprehensive guide, I will cover all the important formulas you MUST know before taking the exam. I'll also explain them in case you need to refresh your memory on how a formula works. Understanding all of the formulas on this list will save you valuable time on the test and likely answer a few extra questions correctly.
Formulas on the SAT, explained
This is exactly what you will see at the beginning of both math sections (the calculator and non-calculator sections). This can easily be overlooked, so familiarize yourself with the formulas now to avoid wasting time on test day.
You get 12 formulas in the test itself and three laws of geometry. It can be helpful and save time and effort to remember the given formulas, butin the end it's unnecessaryas given in all the mathematical sections of the SAT.
They only give you geometry formulas, so make it a priority to memorize your algebra and trigonometry formulas before test day (we'll cover that in the next section).You should be concentrating most of your study effort on algebra anyway, since geometry has been emphasized in the new SAT and now accounts for only 10% (or less) of the questions in each test.
However, you need to know what the given geometric formulas mean. The explanations for these formulas are as follows:
area of a circle
$$A=πr^2$$
- π is a constant that can be written as 3.14 (or 3.14159) for SAT purposes.
- Ris the radius of the circle (any line drawn from the center to the edge of the circle)
circumference of a circle
$C=2πr$ (or $C=πd$)
- Dis the diameter of the circle. It is a line that bisects the circle and touches two ends of the circle on opposite sides. It's double the radius.
area of a rectangle
$$A = low$$
- UEis the length of the rectangle
- Cis the width of the rectangle
area of a triangle
$$A = 1/2bh$$
- Bis the length of the base of the triangle (the edge of one side)
- His the height of the triangle
- In a right triangle, the height is one side of the 90 degree angle. For non-right triangles, the altitude falls on the inside of the triangle (unless otherwise noted) as shown above.
The Pythagorean theorem
$$a^2 + b^2 = c^2$$
- In a right triangle, the two shorter sides (AmiB) are square. Their sum is equal to the square of the hypotenuse (c, the longest side of the triangle).
Special properties of the right triangle: Isosceles triangle
- An isosceles triangle has two equal sides and two equal angles opposite those sides.
- An isosceles right triangle always has one 90 degree angle and two 45 degree angles.
- The lengths of the sides are determined by the formula: $x$, $x$, $x√2$, where the hypotenuse (90 degrees opposite side) has the length of one of the shorter sides *$√2$.
- For example, an isosceles right triangle might have side lengths of $12$, $12$, and $12√2$.
Special properties of the right triangle: 30, 60, 90 degree triangle
- A 30, 60, 90 triangle describes the degrees of the three angles of the triangle.
- The side lengths are determined by the formula: $x$, $x√3$ and $2x$
- The side opposite 30 degrees is the smallest, with measure $x$.
- The opposite side of 60 degrees is the average length with the measure $x√3$.
- The side 90 degrees opposite is the hypotenuse (longer side) with length $2x$.
- For example, a 30-60-90 triangle can have side lengths of $5$, $5√3$, and $10$.
Volume of a rectangular body
$$V = lwh$$
- UEis the length of a side.
- His the height of the figure.
- Cis the width of a page.
volume of a cylinder
$$V=πr^2h$$
- $r$ is the radius of the circular side of the cylinder.
- $h$ is the height of the cylinder.
volume of a sphere
$$V=(4/3)πr^3$$
- $r$ is the radius of the sphere.
volume of a cone
$$V=(1/3)πr^2h$$
- $r$ is the radius of the circular side of the cone.
- $h$ is the height of the pointed part of the cone (measured from the center of the circular part of the cone).
volume of a pyramid
$$V=(1/3)lwh$$
- $l$ is the length of one of the edges of the rectangular part of the pyramid.
- $h$ is the height of the figure at its top (measured from the center of the rectangular part of the pyramid).
- $w$ is the width of one of the edges of the rectangular part of the pyramid.
Law: The number of degrees in a circle is 360
Law: The number of radians in a circle is $2π$
Law: The number of degrees in a triangle is 180
Prepare that brain because here are the formulas to remember.
Formulas not given in the test
Most of the formulas on this list simply require you to dig in and memorize them (sorry). While some of them can be useful, they don't need to be memorized because their results can be calculated in other ways. (These are still useful to know, so treat them seriously.)
We split the list into"I need to know"mi"Good to know,"Depending on whether you are a formula loving examiner or a formulaless examiner type, the better.
gradients and graphs
I need to know
- slope formula
Given two points, $A (x_1, y_1)$,$B (x_2, y_2)$, find the slope of the connecting line:
$$(y_2 - y_1)/(x_2 - x_1)$$
The slope of a straight line is ${\rise (\vertical \change)}/ {\run (\horizontal \change)}$.
- How do you write the equation of a line
- The equation of the line is written as: $$y = mx + b$$
- If you get an equation that is NOT in this form (e.g. $mx-y = b$), please rewrite it in this format!It is very common for the SAT to give an equation in a different form and then ask if the slope and intercept are positive or negative. Unless you rewrite the equation as $y = mx + b$ and misunderstand what the slope or intercept is, you will miss the question.
- Metrois the slope of the line.
- Bis the y-intercept (the point where the line touches the y-axis).
- If the line goes through the origin $(0,0)$, the line is written as $y = mx$.
- The equation of the line is written as: $$y = mx + b$$
Good to know
- midpoint formula
Given two points, $A (x_1, y_1)$, $B (x_2, y_2)$, find the midpoint of the line connecting them:
$$({(x_1 + x_2)}/2, {(y_1 + y_2)}/2)$$
- distance formula
- Given two points, $A (x_1, y_1)$,$B (x_2, y_2)$, find the distance between them:
$$√[(x_2 - x_1)^2 + (y_2 - y_1)^2]$$
You don't need this formula, since you can easily graph your points and create a right triangle from them. The distance is the hypotenuse, which you can find using the Pythagorean theorem.
circles
Good to know
- length of an arc
- Given a radius and a measure in degrees of an arc from center, find the length of the arc.
- Use the formula for the circumference times the angle of the arc divided by the total measure of the circle's angle (360).
- $$L_{\arc} = (2πr)({\Grad \Measure \center \of \arc}/360)$$
- For example, an arc of 60 degrees is 1/6$ of the total circumference, since 60/360$ = 1/6$
- Area of an arc sector
- Given the radius and measure in degrees of an arc from the center, find the area of the arc sector.
- Use the formula for the area times the angle of the arc divided by the total measure of the angle of the circle
- $$A_{\arc \sector} = (πr^2)({\grad \measure \center \of \arc}/360)$$
- Use the formula for the area times the angle of the arc divided by the total measure of the angle of the circle
- Given the radius and measure in degrees of an arc from the center, find the area of the arc sector.
- An alternative to memorizing the "formula"it's easy to stop and think logically about arc circles and arc areas.
- You know the formulas for the area and perimeter of a circle (because they are in the test equation table).
- You know how many degrees a circle is (because it's in the equation box in the text).
- Now join the two:
- If the arc spans 90 degrees of the circle, it must be 1/4$ of the total area/perimeter of the circle since $360/$90 = $4. If the arc makes an angle of 45 degrees, then it's 1/8$ of the circle because $360/45 = $8.
- The concept is exactly the same as the formula, but it can be helpful to think of it this way rather than memorize it as a "formula".
Algebra
I need to know
- quadratic equation
- Given a polynomial of the form $ax^2+bx+c$ solve for x.
$$x={-b±√{b^2-4ac}}/{2a}$$
Just plug in the numbers and solve for x!
Some of the polynomials you encounter in the SAT are easy to factor (e.g. $x^2+3x+2$, $4x^2-1$, $x^2-5x+6$, etc.), but some of them are more difficult to factor and almost impossible to obtain with simple mental trial and error. In these cases, the quadratic equation is your friend.
Make sure you don't forget to create two different equations for each polynomial: one with $x={-b+√{b^2-4ac}}/{2a}$ and another with $x={-b - √ {b^2-4ac}}/{2a}$.
Monitoring:if you know howcomplete the square, so you don't have to remember the quadratic equation. However, if you're not entirely comfortable completing the square, it's relatively easy to memorize the quadratic formula and have it ready. I recommend memorizing it to the tune of "Pop Goes the Weasel" or "Row, Row, Row Your Boat."
media
I need to know
- The mean is equal to the mean
- Find the mean/average of a set of numbers/terms
- find average speed
$$\speed = {\total\distance}/{\total\time}$$
opportunities
I need to know
- Probability is a representation of the likelihood of something happening.
$$\text"Probability of an outcome" = {\text"Number of desired outcomes"}/{\text"Total number of possible outcomes"}$$
Good to know
- A probability of 1 is guaranteed to occur, a probability of 0 will never occur.
percent
I need to know
- Find x percent of a given number n.
$$n(x/100)$$
- Find out what percentage one number n is of another number m.
$$(n100)/m$$
- Find out what number n is x percent of.
Trigonometry
Trigonometry is a new addition to the new math section of the SAT 2016. Although it accounts for less than 5% of math questions, you cannot answer the trigonometry questions without knowing the following formulas.
I need to know
- Find the sine of an angle given the lengths of the sides of the triangle.
$sin(x)$= dimension of the opposite side of the angle / dimension of the hypotenuse
In the figure above, the sine of the labeled angle would be $a/h$.
- Find the cosine of an angle given the lengths of the sides of the triangle.
$cos(x)$= dimension of the side adjacent to the angle / dimension of the hypotenuse
In the figure above, the cosine of the labeled angle would be $b/h$.
- Find the tangent of an angle given the lengths of the sides of the triangle.
$tan(x)$= measure of the side opposite the angle / measure of the side adjacent to the angle
In the figure above, the tangent of the labeled angle would be $a/b$.
- A useful memory trick is an acronym: SOHCAHTOA.
Smine is the sameÖOppositeHHypotenuse
CThe Osina is the sameAnext to itHHypotenuse
Tagent is the sameÖOppositeAunderlying
SAT Math: Beyond Formulas
Although that's allformulasyou need (the ones you were given and the ones you need to memorize), this list does not cover all aspects of SAT Math. You must also understand how to factor equations, how to manipulate and solve for absolute values, how to manipulate and use exponents, and more. these topicsThey are all covered here.
Another important thing to remember is that while it is important to remember the formulas in this article that were not given to you on the test, knowing this list of formulas does not mean you are qualified for SAT Math to be ready.You also need to practice using these question-answer formulas so you know when it makes sense to use them.
For example, if you're asked to find the probability of a white marble being drawn from a jar containing three white marbles and four black marbles, it's easy to see that you need to use this probability formula:
$$\text"Probability of an outcome" = {\text"Number of desired outcomes"}/{\text"Total number of possible outcomes"}$$
and use it to find the answer:
$\text"Probability of a white marble" = {\text"Number of white marbles"}/{\text"Total number of marbles"}$
$\text"Probability of a white marble" = 3/7$
However, in the math part of the SAT you can also find more complex probability questions like these:
Dreams remembered for a week
none | 1 a 4 | 5 or more | In total | |
Group X | 15 | 28 | 57 | 100 |
Group Y | 21 | 11 | 68 | 100 |
In total | 36 | 39 | 125 | 200 |
The data in the table above was compiled by a sleep researcher who studied the number of dreams people recall when asked to record their dreams for a week. Group X consisted of 100 people who went to bed earlier and Group Y consisted of 100 people who went to bed later. If a person is randomly selected from those who remember at least 1 dream, what is the probability that they belong to group Y?
A) $ 68/100 $
b) $ 79/100 $
c) $79/164 $
D) $ 164/200 $
There is a lot of information to summarize in this question: a data table, a two-sentence explanation of the table, and finally what you need to solve.
Unless you've practiced these types of problems, you won't necessarily realize that you need the probability formula you've memorized, and it may take you a few minutes to play around the table racking your brains to figure out how to solve them. solve it. get the answer—Minutes that you can't use now for other problems in this section or to review your work.
However, having practiced these types of questions, you can quickly and effectively implement the memorized probability formula and solve the problem:
This is a probability question, so you'll probably (ha) have to use this formula:
$$\text"Probability of an outcome" = {\text"Number of desired outcomes"}/{\text"Total number of possible outcomes"}$$
Okay, so the number of results you want is everyone in group Y who remembers at least one dream. These are the bold cells:
none | 1 a 4 | 5 or more | In total | |
Group X | 15 | 28 | 57 | 100 |
Group Y | 21 | 11 | 68 | 100 |
In total | 36 | 39 | 125 | 200 |
And then the total number of possible outcomes is all the people who remembered at least one dream. To do this, I need to subtract from the total number of people (200) the number of people who don't remember at least one dream (36). Now I put everything back into the equation:
$\text"Outcome Probability" = {11+68}/{200-36}$
$\text"Outcome Probability" = {79}/{164}$
The correct answer is C)$ 79/164 $
The conclusion of this example:Once you memorize these SAT math formulas, you should learn when and how to use themdigs inpractical questions.
What's next?
Now that you know the SAT's critical formulas, it might be time to review themComplete list of SAT math skills and knowledge you need before test day. And for those of you who score extra goals, check out our article onHow to get an 800 on the SAT Mathfor a perfect SAT scorer.
Do you currently reach midfield in mathematics?Look no further than our article abouthow to improve your score if it is below the 600 range.
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Courtney Montgomery
About the author
Courtney earned the 99th percentile on the SAT in high school and graduated from Stanford University with a BA in Cultural and Social Anthropology. She is passionate about providing education and tools for the success of students of all backgrounds and walks of life as she believes open education is one of society's great levelers. She has years of experience as a tutor and writes creative works in her spare time.
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