10 Formulas to Know for SAT Math Section
If you’ve seen the PSAT or SAT recently, you almost certainly have noticed that both SAT math sections begin with a reference key on the page with the instructions. This reference key contains a number of common geometry diagrams and formulas. Unfortunately, the reference information is far less helpful than you might think. Following the 2016 rewrite of the SAT, the College Board removed most of the geometry questions from the test, choosing instead to focus on algebra and data analysis skills. The thinking of the test writers was that these concepts are far more commonly found in everyday life than formal geometry questions.
The result of that change is that students are provided with every geometry formula they might need at the beginning of each section but no geometry questions on which to use those formulas. Instead, the formulas that are actually useful are nowhere to be found. To set you on the path to success on the SAT, here are the ten formulas you need to know for the math sections.
1. Slope formula
Slope is the how you describe the direction and steepness of a line in the coordinate plane. While questions directly referring to slope have become less common since 2016, you will most likely need to refer to the slope formula when you encounter questions on the equation of a line (y = mx + b).
The m is the symbol for slope while the x and y values come from two specific points in the coordinate plane, (x1, y1) and (x2, y2).
Another way to think about slope is rise over run. You are looking for how much the line rises or falls per whole unit of distance as you look from left to right.
2. The definition of a function
A function is a relationship between a starting value and a finishing value. I other words, when I have an equation, I can plug in a value to a variable in that equation to find an outcome to the equation. On the SAT math section, you will encounter numerous different kinds of function problems, since functions can be written as simple algebra equations, come from word problems, or be translated from figures in a coordinate plane.
The basic format of a function is f(x) = y, which reads, “The function of x is equal to y,” where x is the starting value you plug into the equation and y is the result. Any time you see the word function in an SAT math question, immediately write f(x) = y above or beside the question. Then, fill in the information one step at a time.
For more complicated or multi-step functions, always begin with the function furthest inside the notation.
3. The basic exponent rules
An exponents is a notation that describes how many times a number is being multiplied by itself. You will encounter exponents in a variety of questions on the SAT math section. In order to avoid mistakes on exponents, make sure you are familiar with the basic exponent rules.
An exponent is structured such that there a base number, the one that is being multiplied by itself and an exponent, which represents how many times the base is being multiplied by itself. It looks like this x5 or this 55. In this case first x and then 5 are being multiplied by themselves five times.
It is possible to combine and simplify exponents if you have the same base in each item. For example, if you want to multiply 53 and 56, you can simplify them before you calculate an answer. Since they have the same base of 5, you can simply add the exponents like this, 53 + 56 = 59, since 5*5*5 * 5*5*5*5*5*5 would be 5*5*5*5*5*5*5*5*5.
The same is true is you want to divide two values with the same base but different exponents. In this case you would subtract the exponents, like so: y7 ÷ y3 = y4.
Lastly, you can multiply an exponent by itself multiple times, which would look like this: (b3)4 = b12. This works because b*b*b multiple by itself four times would result in twelve iterations of b being multiplied by itself.
4. Mid-point formula
Some SAT questions might ask you to find the coordinates of a midpoint of a line in the coordinate plane. The way to do that is to use the midpoint formula. Like the slope formula, you start with two distinct points, (x1, y1) and (x2, y2), and then plug them into the midpoint formula. The result will be the coordinates of the exact middle of the line between the two points.
5. Distance formula
Similar to the midpoint formula, the distance formula describes something about a line between two points, (x1, y1) and (x2, y2), in a coordinate plane. In this case, it gives you the length of that line.
One of the more challenging topics you will encounter on the SAT math sections is trigonometry. Luckily, the test writers only refer to the basic concepts in the few trig questions on the exam. Instead, the difficulty of the questions tends to depends more on you identifying that the question is in fact referring to trig and remembering the basic rules. For that, utilize the acronym SOHCAHTOA.
Given a right triangle, you are asked to find the sine, cosine, or tangent of a particular angle. For example, if you are asked to find the sine, cosine, or tangent of angle A, follow the patten below.
sin = opposite cos = adjacent tan = opposite
hypotenuse hypotenuse adjacent
In this case, the sine of angle A would be a/c.
While there are usually only one or two trigonometry questions on a given SAT, they tend to fluster students. If you see any of the terms associated with trig, immediately write the word SOHCAHTOA above or adjacent to the question, and fill in the information you have to discover the answer they want.
7. Standard form of a quadratic
One of the more common academic math concepts that appear among the harder questions on the math sections is quadratics. Whether the test writers have framed the question in terms of a figure of a parabola in a coordinate plane, a word problem, or an obvious algebraic equation, quadratics represent a concept which you should know well. Generally, you will be asked to one of two things with a quadratic: factor it out of standard form or FOIL it into standard form.
Standard from of a quadratic looks like this: Ax2 + Bx + C = y, where A, B, & C are constants and x and y represent points on the parabola. The factored form of a quadratic looks like this: (x + y)(x + y) = 0, such that the x’s represent the points at which the parabola crosses the x-axis.
You can move from factored form to standard form by following the rules for FOIL: first, inside, outside, last. In order, multiply the x values of each parenthetical (x + y)(x + y) = 0, then the values on opposite ends of the equation (x + y)(x+ y) = 0, next the adjacent values in the middle of the equation (x + y)(x + y) = 0, and finally the y values of each parenthetical (x + y)(x + y) = 0. This will result in an equation which looks like this: x2 + xy + xy + y2, which can be simplified through collecting like terms back into standard form.
8. Quadratic formula
Factoring and FOILing quadratics is the simplest and most efficient way to solve questions that include them. Unfortunately, factoring and FOILing only work well with whole number results. In your math class, most quadratics do not result in whole number solutions. In those cases, you must rely on the quadratic formula.
Using standard form again (Ax2 + Bx + C = y) where y = 0, you can plug the A, B, and C values into the quadratic formula to find the x-values of the solutions, the places at which the parabola crosses the x-axis. The good news is that the quadratic formula is rarely needed on the math sections, since the SAT tends to focus on challenging puzzles that utilize basic math knowledge.
9. Equation of a Circle
Most of the formulae regarding circles can be found in the reference key at the beginning of every SAT math section: specifically the formulae for area and circumference of a circle. Every so often, however, the SAT test writers will ask a question about solving for a circle in a coordinate plane. To do that, you need the equation of a circle.
The equation reads (x – h)2 + (y – k)2 = r2, where (x, y) represents a point on the circle, (h, k) represents the center of the circle, and r represents the radius of the circle. While this concept can be challenging for most students, the questions that refer to the equation of a circle tend to, once again, focus less on the math and more on the puzzle. So long as you have the equation memorized and know the basics of what to plug in and where to do so, you should be able to solve these questions. If you find these questions difficult, the good news is that they don’t appear on the math sections every iteration of the SAT.
10. The Unit Circle
Perhaps the single most challenging concept you may encounter on an SAT math section, the Unit Circle is a two-dimensional representation of a wave form completing a a full period. In your high school math classes, the Unit Circle allows you to solve complex trigonometric functions, usually at the pre-Calculus level.
On the SAT, the test writers are mostly interested in determining if you remember and/or recognize the Unit Circle, and if you know the basic rules and properties of it.
Often, the test writers will ask you to translate between the coordinate values in radicals, the circumference values in π, and the measure values in degrees. Memorize the most common values and the basic rules (such as the circumference of the Unit Circle is always 2π).
While Unit Circle questions are invariably difficult, they are also rare. These questions don’t show up on every SAT, and in fact may only turn up on the math sections two or three times per calendar year. While that may seem like a relief, it also means that you won’t necessarily get enough practice on this concept just by taking the SAT or practice tests.
The most important thing to remember about the SAT math section is that this is not really a math test. The point of the questions is not to analyze your raw math skill, but instead to utilize math as a tool for crafting complicated puzzles to confound you. It is far more useful to develop good problem solving and critical thinking skills to beat the math section on the SAT, however, memorizing, and then mastering these ten formulas will prevent the SAT test writers from completely tricking you. This is especially true on the more difficult math questions.