About 33% of the ACT math section requires you to know and use at least one math formula. This means that remembering your formulas and understanding how to utilize them is of paramount importance.
We’ve put together all the ACT formulas you’ll need to know (prioritized in the order from greatest to least that you’ll see them on the test) as well as how to best use them for test-day.
What Formulas Will You Need on the ACT?
You will NOT be given any formulas on the ACT, so all of your formula knowledge will have to come from your own head.
You will need to know and use a wide range of formulas, including algebraic, geometric, and trigonometric formulas, all of which are laid out in our guide to the 31 formulas you MUST know for the ACT. You’ll notice that we have prioritized them according to “need to know” and “good to know.” This is due to the fact that many ACT questions can be solved long-hand or via more common formulas, rather than forcing you to memorize the more obscure formulas.
For instance, you can solve your sequence questions by either using the formula or by calculating your values long-hand. Though solving the question long-hand will take longer, it is still entirely possible to solve each and every ACT sequence question without the use of a formula. Thus we have classified sequence formulas as “good to know,” not “necessary to know.”
If you feel rusty on any formula or math topic on the list, check out one of our individual math topic guides to see how the formula works (and even why it works), as well as how to recognize when to use it. We’ll also show you the alternatives to using formulas for many questions, including distance questions, sequence questions, and many more.
There are many different "right" paths to solve questions in the ACT math section.
How to Use Your Formulas Effectively
So now that you know what your formulas are, how do you best go about using them? Let’s take a look.
#1: Prioritize memorizing your most crucial formulas
You will have to memorize every formula you’ll use on the ACT, but it’s best to go about this in a systematic and logical way. Spend most of your time and energy memorizing and practicing the most important (common) formulas and less time on the ones that show up rarely, if at all.
Some formulas come up over and over again, while others show up sparingly at best. If you are pressed for time, nervous about memorizing so many formulas, or simply trying to map out your plan of attack, memorize your formulas in the order that they appear most often on the test.
Of your “necessary” formulas, they appear on the test from greatest prevalence to least in roughly this order:
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Law: the sum of the interior degrees of a triangle is 180
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Area of a triangle
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Law: the sum of the degrees of a straight line is 180
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Area of a rectangle (or other quadrilateral)
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Pythagorean Theorem
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Finding slope of a given line (rise/run)
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Finding slope of line connecting two points
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Finding percentages
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Law: the number of degrees of arc in a circle is 360
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Area of a circle
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Circumference of a circle
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Finding averages
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Sine, cosine, tangent (SOH, CAH, TOA)
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Area of a circle’s arc
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Circumference of a circle’s arc
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Finding probabilities
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Finding combinations
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Finding the midpoint of a line
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Volume of rectangular solid
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Volume of cylinder
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Area of a trapezoid
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Equation of a circle
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Rearranging logarithms
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Cosecant, Secant, Cotangent
Of the “good to know” or “shortcut” formulas, you will need them roughly in this order:
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Special right triangle properties, 45-45-90
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Arithmetic sequences
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Geometric sequences
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Distance formula
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$sin^2Θ + cos^2Θ = 1$
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${sinΘ}/{cosΘ} = tanΘ$
#2: Choose NOW which (if any) of your “good to know” formulas you want to memorize
Remembering a formula incorrectly is worse than not remembering the formula at all, so make sure you know your limits when it comes to memorization. For some people, memorizing and using formulas is the easiest way to go. For others, the fewer formulas the better (even if it means taking another step or two to solve a math problem).
There is no right answer in terms of how many formulas you memorize, only what is most comfortable for you personally.
And when it comes to memorizing your formulas, different people do better with different memory techniques. If you're a visual learner, make yourself a set of formula flash cards. If you're a kinesthetic (movement) learner, practice drawing and/or writing them out on a separate piece of paper. And if you're an auditory learner, get a parent or a friend to help you drill your formulas aloud. Once you feel you've got your formulas down, practice using them on actual problems to help you both remember them and learn how to use a particular formula for a particular problem. (We'll give you the opportunity to practice using your formulas on real ACT math questions in the next section.)
#3: Practice ACT math questions at home without looking up your formulas
The only way you’re going to be able to remember your formulas for crunch time (and know which formulas to actually use for the individual problem) is to practice on real ACT math questions without the safety net.
Once you feel that you’ve got your formulas nailed down tight, practice solving ACT math problems without looking up your formulas. We have a list of all the free ACT math practice you can find online and you will be able to use any and all of these problems to not only test your formula knowledge, but also your math topic knowledge.
Once you’ve gone through your practice problems without the benefit of being able to look up your formulas, you will be able to pinpoint your formula strengths and weaknesses. Did you repeatedly forget the Pythagorean Theorem? How about your trigonometry formulas? It’s better to understand where to shift your focus now than being blindsided on the day of the test. Divide your time between rote memorization and practice problems without the use of formulas, and you’ll be able to solidify the knowledge in your head most effectively.
#4: When you reach the math section, immediately write down your formulas
If you’re anxious about forgetting your formulas halfway through the test, or if you simply like the idea of a fallback option, it’s always a good idea to write down your formulas at the very beginning of your math test.
Once you have them written down, you can concentrate on solving your problems without fear of remembering your formulas wrong or forgetting them entirely. So once you open up your math section, write down all your memorized math formulas and take a deep breath. Now you can move on and use them as a reference for the rest of your math section.
#5: Don’t freak out if you forget a formula
Most of all, don’t panic if you forget a formula (or three)! Most every ACT problem can be solved in a multitude of ways, including by means of plugging in answers or plugging in your own numbers.
And if worst comes to worse, and you cannot solve a problem without a formula, you will still likely be able to eliminate at least one or two answer options. Remember--you are not penalized for guessing on the ACT, so always take your chances. And if you can narrow down your options, even better!
Let's look at an example of how to narrow down your answer choices if you forget a formula.
For a question like this, you do not have to understand how trapezoids work or remember any triangle formulas in order to eliminate at least three answer choices. We are being asked to find the distance between our two parallel sides, so draw a straight, perpendicular line between them.
This makes a right triangle. Again, without knowing your formulas, you can just take a stab in the dark and estimate how long the side is. Remember--all figures on the ACT are to scale unless noted otherwise, and this line looks about the same length (maybe more, maybe less) as 5 foot leg we are given. Without knowing any more information, we can eliminate answer choices D and E.
Now maybe you stop here and pick between answer choices A, B, and C. This would give you a 33% chance of guessing the right answer, which isn't too bad at all. But if we go further, you may remember that the side opposite the right angle is the longest side of the triangle, which means that the distance between our parallel lines must be less than 5. We can therefore eliminate answer choice C as well.
So we had a 1 in 3 chance of getting the right answer without any formula knowledge at all and now we have a 1 in 2 chance just by knowing a little bit about how right triangles work. And even now, we can make an educated (rather than a random) guess between our two remaining answer choices. Again, all figures on the ACT are to scale unless noted otherwise and, at a glance, the distance between our parallel bases looks to be longer than the third leg of the triangle. Maybe it is, maybe it isn't, but we have a 50% chance and answer choice B looks to be the best bet between the two.
Without using formulas, we can reach a conclusion that the answer is probably B.
[Note: just so you know, B is totally the right answer. Go you!]
Now it's time to put your formula knowledge to work!
ACT Math Practice Using Formulas
Now that you’ve seen how to best use your ACT formulas, let’s look at a few real ACT math questions that are formula-necessary.
1.
2.
3.
4.
5.
Answers: D, F, J, J, B
Answer Explanations:
1. Because the dog can run on the leash 9 feet in any direction, this means that 9 feet is the radius of the circle in which the dog can run. Now we are asked to find the area of this circle.
If we remember our circle formulas, we know that we find the area by using the formula:
$a =πr^2$
$a =π9^2$
$a = 81π$
$π$ is approximately 3.14 (which we are given), so:
$81(3.14)$
$254.34$
The closest answer to this value is answer D, 254.
Our final answer is D, 254.
2. If we remember our trig formulas, we know our mnemonic SOH, CAH, TOA. The tangent of an angle is thus the opposite/adjacent.
In this case, we are looking for the tangent of angle B. The adjacent side is the side that touches the angle that is NOT the hypotenuse. In this case, the adjacent side for angle B is 2, which means it is our denominator. This means we can eliminate answer choices H, J, and K.
If we use the Pythagorean Theorem, we can find our missing side measure.
$a^2 + b^2 = c^2$
$a^2 + 2^2 = 5^2$
$a^2 + 4 = 25$
$a^2 = 21$
$a = √21$
Our opposite side will be $√21$ and we have already established that our adjacent side is 2.
Our final answer is F, $√21/2$
3. We can solve this problem in one of two ways--by using the more common triangle formulas, or by using a more obscure one.
If we use the common triangle formulas, we know that the interior angles of a triangle will always add up to 180 degrees. This means we can find the missing angle measure by subtracting all of our known angles from 180.
$180 - 72 - 57 = 51$
Now, we also know that every straight line is 180 degrees as well. We can therefore find the exterior angles by subtracting each of our given angles from 180.
$y = 180 - 72$
$y = 108$
$x = 180 - 57$
$x = 123$
$z = 180 - 51$
$z = 129$
Now we can find the sum of $y$, $x,$ and $z$.
$108 + 123 + 129 = 360$
Our final answer is J, 360.
Alternatively, we can use our more obscure triangle, which is that every exterior angle is the sum of the two opposite interior angles. For instance,
$z = 72 + 57$
$z = 129$
From here, we can solve the problem the same way we did above.
$x = 72 + 51$
$x = 123$
$y = 51 + 57$
$y = 108$
$x + y + z = 129 + 123 + 108$
$= 360$
Either way, our final answer is J, 360.
4. If we remember our slope formulas, we know that the equation of a line is:
$y = mx + b$
$m$ represents the slope of the line, and the bigger the $m$, the larger the slope.
In our first given equation, $a$ stands in place of our $m$ and our slope. In our second equation, $c$ stands in place of $m$ and our slope. So if the slope of our first equation is larger than the slope of our second equation, then $a$ must be larger than $c$.
Our final answer is J, $a > c$.
5. Now our shape in the middle is a square, which means all of the sides are equal. The two sides of the square that make up part of the perimeter are $8 + 8 = 16$.
Now we just need to find the part of the perimeter made up by the two half circles. If we put them together, we can one full circle circumference. We know that the circumference of a circle is:
$c = πd$
The dimeter of our circle is 8, so our full circumference will be $8π$
Now let's put our two values together:
$16 + 8π$
Our final answer is B, $16 + 8 π$
Whoo! You did it!
Image: Sean MacEntee/Flickr
The Take-Aways
Knowing (and knowing how to utilize) your formulas is one of the foundational elements of doing well on the ACT math section, but it is still only one part. Though formulas are used in some capacity or another on approximately 33% of ACT math test, that still leaves 66% of your questions that do NOT require formulas at all.
So do take care to understand (and memorize!) your most important formulas, but don’t think that's all you need to do to succeed on the ACT math section. You still must understand the ins and outs of all of the ACT math topics that you will see on the ACT, so don't neglect the rest of your ACT math study. A balanced study plan, a knowledge of your formulas, and a more-than-passing familiarity with all your ACT math topics will help get your math score to where you want it to be.
What’s Next?
Want to brush up on a particular ACT math topic? Check out our individual math topic guides for all your ACT math needs.
Running out of time on the ACT math section? We'll show you how to beat the clock and maximize your score before time runs out.
Been procrastinating on your ACT math study? Our guide will help you balance out your study time and beat back the urge to procrastinate.
Aiming for a perfect score? Check out our guide to getting a 36 on the ACT math section, written by a perfect-scorer.
