Bonus Material: Test your knowledge with 20 of the Hardest SAT Math Questions ever

If you’re applying to college, you’ll most likely need to score well on the SAT.

Since half of your total score is going to come from the Math section, you’ll want to know exactly what kinds of questions you’re going to see on test day.

In this post, we’ll provide an in-depth explanation of what concepts the SAT Math section tests, and how it tests them. 

This information is crucial to make sure you’re studying the right things for the SAT! Often students waste their time focusing on questions that won’t appear on the SAT at all, or make up a small fraction of the math problems. Follow our advice to study effectively.

Then, test your knowledge with our Hardest SAT Math Questions Quiz.

Download our quiz with 20 of the hardest SAT Math problems

Jump to section:

What Does the SAT Math Section Look Like?
Main Categories of Math Questions
Algebra of Lines
Non-Linear Algebra
Data Analysis and Problem Solving
Foundational Math Skills (and imaginary numbers)
Next Steps

What Does the SAT Math Section Look Like?

The SAT Math Section is split into two portions. The first section contains 20 questions and does not allow you to use a calculator. The second section contains 38 questions for which you can (and usually should) use a calculator. 

You get one total score for both sections combined, based on the raw number of questions you get correct out of the possible 58.

This post will cover the specific content you need to know on the SAT Math. For a more general guide to structure and strategy, check out our post here.

Did you know that the SAT will change significantly in the next two years? The math on the new digital SAT will test similar concepts but with a different structure. Read more about the new digital SAT here, and check if the changes will affect your test dates.

Main Categories of Math Questions

The bad news? The SAT presents math concepts in ways that you may not be familiar with. The great news? Because the SAT has to be consistent from test to test, it is highly predictable and you can absolutely prepare yourself for everything you’ll see. 

Below is a chart that shows how the College Board classifies the four main areas of questions and how often they come up. Because their terminology is a little unspecific (“Heart?” “Passport?”), we’ve translated those into what they actually mean. 

math concepts on the SAT

In a nutshell, the main “families” of questions are: Algebra of Lines; Non-linear Algebra; Data Analysis; Geometry. 

Those categories are helpful to give you an idea of what content the SAT expects you to know. Below, we’ll get more specific: we’ll go through what specific operations the SAT Math section asks you to do.

Aiming for a perfect score on the SAT Math? You’ll have to be able to answer the toughest questions flawlessly. Try out 20 of the toughest SAT Math questions ever with our Hardest SAT Math Questions Quiz.

Algebra of Lines

At the heart of this section is one little equation: y = mx + b. When we talk about “lines” or “linear equations” on the SAT, we’re really talking about y = mx + b. If you can master how the SAT tests the uses of this particular equation, you’ll have the tools to answer 19 of the questions. That’s one third of the Math test!

Perhaps this is old news to you, but if you need a refresh: y is the dependent variable (or output); x is the independent variable (or input); m is the slope (or rate of change); b is the y-intercept (or what you get for y when you plug in 0 for x). 

student working on math problem

But it’s not as simple as just memorizing the equation. There are specific things the SAT will expect you to be able to do, and understanding the elements of this equation is only the starting point. Below is a specific set of tasks the SAT will set for you, roughly in order of difficulty:

  1. Matching a graph of a line to its equation (and vice-versa)
  2. Finding the slope of a line. 
  3. Finding the midpoint of a line. 
  4. Modeling a real-world situation with a linear equation. 
  5. Solving for one variable in a linear equation when given the other elements. 
  6. Finding the equations of parallel and perpendicular lines. 
  7. Modeling a real-world situation with an inequality. 
  8. Graphing an inequality. 
  9. Solving systems of linear equations and inequalities. 
  10. Graphing systems of linear equations and inequalities. 
  11. Graphing and solving equations with absolute values. 

Here’s an example of a classic type of linear function problem that typically shows up on every SAT test:

SAT Math question from test #6

No student should go in to take the SAT without a complete mastery of the linear equations and y = mx + b. Fortunately, there are lots of resources to help you get that mastery. Students can do independent practice on Khan Academy or using the College Board’s official practice tests. For expert guidance on SAT Math concepts, students can work with one of our experienced Ivy-League tutors.

Non-Linear Algebra

The questions in this category test your ability to work with the equations and graphs of quadratics, exponents, and radicals (or roots). Nearly a third of questions belong in this category, but this category covers a far wider range of topics than the previous one.

Still, the core of this section is the quadratic equation. All quadratic equations can (and usually should) be written in the form ax^2 + bx + c = 0. You need to know this equation inside and out: what’s a? What’s b? What’s c? Here’s specifically what the SAT will ask you to do with Quadratics:

  1. Matching a graph of a parabola to its equation (and vice versa).
  2. Factoring/solving quadratics.
    • Factoring normally.
    • Factoring differences of perfect squares.
    • Factoring with the Quadratic Formula.
    • Factoring by grouping.
    • Factoring by completing the square (specifically for the equation of a circle)*. 
  3. Finding the vertex (also known as the minimum/maximum value) from a quadratic equation. 
  4. Solving systems of equations that include at least one quadratic equation. 
  5. Polynomial division of a quadratic.*
  6. Using the discriminant to solve a quadratic equation.*

* Something worth noting: there are a number of questions on the SAT Math that you can reliably expect to see exactly once per test. These tend to be difficult if you’re not expecting them. These are marked with an asterisk (*).

Here’s an example of a more abstract question involving a quadratic equation and the graph of a parabola:

SAT Math question from test #6

And here’s a more challenging problem involving a discriminant:

SAT Math question from test #6
SAT Math question from test #6

But that’s just the quadratic stuff! You’ll also be expected to work with radical and exponential equations. Specifically, you need to understand:

  1. Exponential equations
    • Modeling exponential growth/decay from a word problem. 
    • Graphing exponential equations. 
    • Solving exponential equations. 
    • Exponent rules. 
  2. Radical equations
    • Graphing radical equations. 
    • Solving radical equations (and avoiding extraneous solutions). 
    • Simplifying radicals.
    • Rationalizing a denominator that contains a root (or imaginary number).*
  3. Manipulating polynomials.
    • Finding common denominators for rational expressions. 
    • FOIL-ing. 
    • Simplifying. 

Here’s an example of a real exponent problem from a past SAT:

SAT Math question from test #6

We’ve worked with thousands of students to prepare for the SAT, and we’ve found that students are often a bit rusty with exponential operations. Fortunately, a little bit of targeted practice can iron out any confusions!

There are a lot of concepts tested here, but by reviewing past tests and other resources, you can learn exactly what to expect. As with the previous section, you want to make sure you have a strong grasp on each of these concepts before going in to take the test. If any of these concepts feel a bit shaky, our test prep experts can help. 

Feeling confident about these advanced algebra topics? Try out 20 of the toughest SAT Math questions ever with our Hardest SAT Math Questions Quiz.

Data Analysis and Problem Solving

This subset of math questions tests your ability to perform specific operations that aren’t necessarily linked to linear or non-linear equations. A key element of this section is understanding probabilities and some basic statistical measures

You’ll be tested on your ability to: 

  1. Perform unit conversions.
  2. Create and solve equations with ratios and proportions. 
  3. Perform percentage calculations.
    • Finding percent change given two values. 
    • Finding the original value when given the final value and the percent change. 
    • Finding final value when given the original value and the percent change. 
  4. Understand and find basic measures of center:
    • Finding Mode.
    • Finding Range.
    • Finding Median.
      • They will often ask whether the mean or median of a particular data set will be more affected by an outlier. 
    • Finding Mean/Average (this is the most heavily tested).
      • Solving Mixture problems.
      • Solving equations that include a mean/average.
    • Understanding standard deviation.
      • This one sneaks up on people! You won’t be expected to calculate standard deviation, but you will be expected to know what it is, and which of two number sets has a higher/lower standard deviation. 
  5. Calculate probabilities
    • Finding the probability of a particular outcome. 
  6. Understand graphs and tables
    • Finding a line of best fit. 
    • Scatter-plots
    • Bar graphs
    • Histograms 
    • Stem-and-leaf plots
    • Box-and-whisker plots
  7. Understand basics of experimental design
    • Understand sampling bias. 
    • Understand what makes a study reliable. 

There are a lot of concepts here, but a couple worth paying extra attention to. Many, many students struggle at first with percentages and ratios. These can be highly unintuitive, and you’ll benefit from doing a comprehensive review of these concepts. 

The statistical concepts don’t usually require much math or calculation on your part. But you do need to understand each of these concepts thoroughly to answer the questions presented. If you can’t immediately think of how to quickly compare median and mean for a data set, or if you aren’t sure what a histogram looks like, it’s probably time to study up or reach out to a tutor.

SAT math answer key

Here’s an example of a common type of question asking students to find the mean of a small data set, and then use that to solve a short algebra problem:

SAT Math question from test #6


About 10% of the questions on the SAT will test your knowledge of geometric figures, mostly triangles and circles. Though lots of students stress out over geometry, there’s actually a fairly limited number of concepts you need to master to succeed on these questions. 

Here’s what you’ll have to know:

  1. Triangles
    • Solving right triangles with the Pythagorean theorem
    • Solving for Sin/Cos/Tan in right triangles
    • Solving similar triangles (they always ask about these!)
    • Using special right triangles (30-60-90 and 45-45-90)
    • Solving for area of a triangle
    • Using triangle properties to solve for an angle or side
      • Interior angles sum to 180
      • The biggest angle is always opposite the longest side, etc. 
      • Any two sides of a triangle must sum to be greater than the third
  2. Circles
    • Using the equation of a circle to solve for center or radius*
    • Finding area and circumference
    • Converting between radians and degrees
    • Solving for the angle measure/arc length/area of a slice of a circle*
  3. Miscellaneous Geometry
    • Solving for the area and perimeter of rectangles and squares
    • Solving for the area of other polygons (this can always be done by cutting up the polygon into rectangles and triangles). 
    • Solving for missing angles in transversals
    • Finding the interior angle measure of any regular polygon (hint: there’s an equation!)
    • Solving for the volume of regular shapes (the equations will be given)
    • Comparing the areas/volumes of shapes

Here’s an example of a classic geometry problem implementing similar triangles and the Pythagorean Theorem:

SAT Math question from test #6

Ready for a challenge? Try some more advanced geometry problems on our Hardest SAT Math Questions Quiz.

Foundational Math Skills (and i)

In addition to asking you to do all of the things outlined above, the SAT also checks your knowledge of foundational math skills. These are basically baked into the rest of the problems, and you likely do some or all of these things without thinking about them:

  1. Order of operations
  2. Simplifying expressions
  3. Finding common denominators to combine fractions
  4. Isolating variables
  5. Understanding i and imaginary numbers
    • This gets tested every time: do you know what i represents? Do you know what to do when it’s in the denominator?
math chalkboard

Next Steps

If you’ve read through this list carefully, you now know just about everything the SAT will ask you to do.

Our SAT experts have guided thousands of students through studying for the SAT in the most effective way, focusing on the types of questions that, when mastered, will improve scores the most. We’ve shared some of those insights here.

Now, it’s time to make sure you can handle these math concepts yourself! We recommend Khan Academy for math review. If you feel like an expert, test your skills with our popular quiz featuring the 20 hardest SAT Math questions ever found on real SAT tests.

If some of those questions stump you, or if some of the concepts listed above aren’t crystal clear, there’s nothing better than an experienced one-on-one tutor to help you master the SAT Math. Our Ivy-League tutors will make sure that you know exactly how to prepare for the SAT most efficiently and effectively. Our co-founder, Kevin, also offers a limited number of small-group SAT MasterClasses that help students to reach their full potential on the SAT.

To reserve a spot in a SAT MasterClass or start one-on-one SAT tutoring today, set up a quick free consultation with our team.

In the meantime, happy prepping!

Download our quiz with 20 of the hardest SAT Math problems

Bonus Material: Download our quiz with 20 of the hardest SAT Math problems ever

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Bonus Material: Download our quiz with the Hardest SAT Math Questions from real SAT tests


Emily graduated summa cum laude from Princeton University and holds an MA from the University of Notre Dame. She was a National Merit Scholar and has won numerous academic prizes and fellowships. A veteran of the publishing industry, she has helped professors at Harvard, Yale, and Princeton revise their books and articles. Over the last decade, Emily has successfully mentored hundreds of students in all aspects of the college admissions process, including the SAT, ACT, and college application essay.