Updated March 12, 2025
ACT Math Formulas: Everything you need to memorize for the ACT
Bonus Material: 68 ACT Math Formulas Cheatsheet
Scoring well on the ACT is an important part of impressing colleges and securing an admissions offer.
The Math section of the ACT is notoriously tough, and students need to have memorized these 68 essential math formulas in order to earn a high score on the test.
While other test prep sites might simply dump all of these formulas into a list, we’ve used our decades of expertise in ACT prep to indicate which formulas students really need to know for test day.
Read on for an introduction of how to use each key formula. Then download all of the ACT math formulas as a handy guide.
Math formulas in this post:
Algebra: Linear Functions
Exponential Operations
Algebra: Polynomial Functions
Geometry: Angles
Geometry: Triangles
Geometry: Circles
Geometry: Perimeter, Area, and Volume
Trigonometry
Statistics and Probability
Advanced Algebra
Memorizing ACT Math Formulas
Unlike the SAT and other standardized tests that students may have taken, the ACT does not provide students with any resources during the test.
For this reason, students need to memorize a number of math formulas in order to do well on the ACT.
The ACT provides their own formula sheet, but not all of the formulas they list are equally important. We’ve used our decades of expertise with the ACT to organize the formulas so that students can focus on what really matters for the test.
Sometimes the ACT provides more obscure formulas (like the trigonometric identity formula or the volume of a cone) in the text of the individual question. In our downloadable guide, we include only the most important formulas to know for the ACT.
There’s no point in wasting time on “formulas” that don’t actually appear on the ACT!
Use active memorization techniques to make sure that your knowledge of these formulas is rock-solid. Practice copying these formulas out onto a blank sheet of paper from memory. Try explaining the formulas to a friend, or even a pet or inanimate objects (a technique used by engineers at Google).
Of course, it’s important that students feel comfortable not just repeating the formula itself, but implementing the formula on real ACT problems. The best way to hone this skill is to practice with real ACT math problems and real ACT tests. Our Ivy-League ACT tutors can help students practice with the best materials available and learn from their mistakes to boost their test scores.
Algebra: Linear Functions
One of the most common concepts on the ACT is linear equations and functions. The equation for a line is y = mx + b. In this equation, m represents the slope of the line, while b represents the y-intercept. As with all equations, x is the input and y is the output. You may also see this equation written as f(x) = mx + b. This means essentially the same thing!
Equation of a line: y = mx + b
m = slope, b = y-intercept
Slope: m = (y1 – y2) / (x1 – x2 ) = “rise” / “run”
Midpoint: ((x1 – x2 / 2), (y1 – y2 / 2))
We can find the slope of a line by dividing the “rise” by the “run,” or how much the line travels up and down divided by how much the line travels side-to-side. The formula for this is m = y1 – y2 / x1 – x2, where (x1, y1) and (x2, y2) are two different points on the line. It doesn’t actually matter which point you assign to be (x1, y1) vs (x2, y2).
Students should definitely take a few moments to memorize the quadrants of the coordinate plane. This is typically an easy, quick problem if you know which one is which!
Students should also know how to find the midpoint on a line. It’s helpful to think about the midpoint as an average — the middle — between two points. Just like any other time that we calculate an average of two values, we add the two x-values together and divide by two, and then do the same for the y-values. That means that the formula for this concept is (x1 + x2 / 2, y1 + y2 / 2).
Midpoint of a line: ((x1 – x2 / 2), (y1 – y2 / 2))
Some textbooks will make students memorize a separate equation for the distance formula, which finds the distance between two points on the coordinate plane. If students already know this formula, that’s fine. However, we don’t recommend spending time memorizing it, because it’s actually just an application of the Pythagorean Theorem, which we’ll cover below.
To recap, here are the formulas students need to know for linear equations and functions:
- Equation of a line: y = mx + b
- Slope: m = (y1 – y2) / (x1 – x2)
- Quadrants of the coordinate plane
- Midpoint is ((x1 + x2 / 2), (y1 + y2 / 2))
- Bonus: distance formula, but you don’t need to memorize it because it’s just an application of the Pythagorean Theorem
Exponential Operations
We made this a separate section because we know from decades of coaching students on the ACT that many students have a fuzzy knowledge of exponential operations.
These are key concepts for math, and usually with a bit of targeted practice students can gain several ACT points just by solidifying their knowledge of how to work with exponents.
So for example, students should be able to easily simplify the following expressions with exponents:
Algebra: Polynomial Functions
Of course, many problems will use exponents in equations or functions. These are polynomial functions.
One of the most common types of polynomial functions on the ACT is the quadratic function, which makes a parabola shape when graphed on the coordinate plane.
There are two main equations for a quadratic function, and both can be incredibly useful, depending on the context.
Standard form: f(x) = ax2 +bx + c
c = y-intercept
The first form of the quadratic function is standard form: ax2 + bx + c = 0. In this equation, a, b and c are known values. Furthermore, a can’t be 0. If a is positive, the graph of the parabola will open upwards. If a is negative, the graph of the parabola will open downwards. Higher values of a will make the parabola narrower, while values between -1 and 1 will stretch the parabola horizontally.
Positive a = the graph opens upwards.
Larger a = the graph is narrower.
Negative a = the graph opens downwards.
Smaller a = the graph is wider.
When our quadratic function is in standard form, we can use the formula h = -b / 2a to find the x-value of the vertex. (This is also the axis of symmetry.) Once we have this, we can plug in h for x in the equation to find the y-value of the vertex.
X-value of vertex (“axis of symmetry”): h = -b / 2a
If this same equation is rearranged (using algebra) into this format, we can learn other things from the equation:
Vertex form: f(x) = a(x-h)2 + k
(h, k) is the vertex, axis of symmetry is x=h
a indicates the direction and width of the graph
In this form, whatever numbers end up in the h and the k positions tell us what the vertex is for the parabola. That’s the lowest or highest part of the curve.
The vertex form of a quadratic function is incredibly powerful, because we can immediately know the location of the vertex without doing any other work!
Rarely, the ACT will also expect students to know the Quadratic Formula. That’s perhaps the most complicated formula for the ACT:
Quadratic formula:
when ax2 + bx + c = 0
a, b, c = constants, where a ≠ 0
x = the unknown
Ideally, students have this memorized. However, in our experience, usually problems can be solved without the Quadratic Formula, by doing some clever factoring (sometimes using the “completing the square” approach instead. So if students are tight on time, don’t prioritize memorizing this complicated formula.
Students should also know how to match the graph of a polynomial function with an equation. The most important thing to know about this is that the graph will have one fewer “turning point” than the value of the biggest exponent. So the graph of an equation with x^2 will have one turning point, the graph of x3 will have two turning points, the graph of x4 will have three turning points, and so on.
Finally, students are expected to know the equation of a circle when graphed on the coordinate plane:
Equation of a circle: (x-h)2+(y-k)2 = r2
(h, k) = center of the circle,
r = radius
This is typically necessary for one problem on each ACT Math section — so it’s lower priority than the essential rules of linear functions and basic geometry, but it will reliably be relevant on every ACT.
To recap, here are the formulas we need for polynomial functions:
- Quadratic function standard form: ax2+bx+c = 0
- Find the x-value of the vertex (also the axis of symmetry) from standard form: h = -b/2a
- Find the zeros / solutions f and g by factoring from standard form: (x-f)2+(y-g)2=0
- Quadratic function vertex form: y=a(x-h)2+k, where (h, k) is the vertex
- Graphing polynomial functions: the graph will have one fewer “turning point” than the value of the biggest exponent
- Circle function: (x-h)2+(y-k)2=r2, where r is the radius and (h, k) is the center
- Bonus: quadratic formula, which is a classic but tends not to be necessary for ACT problems
Geometry: Angles
Students familiar with the SAT may be surprised that the ACT has significantly more geometry than the SAT.
The most common geometry formulas or rules that students need to know involve angles. Students need to know that supplementary angles add up to 180 degrees, and complementary angles add up to 90 degrees.
Vertical angles (also called opposite angles) are always congruent:
Adjacent angles are supplementary and add up to 180 degrees:
When two parallel lines are crossed by a third line, the corresponding angles, exterior angles, interior angles, and alternate angles are all congruent.
Geometry: Triangles
A key rule about triangles is that all of the interior angles of a triangle will add up to 180 degrees.
There are a handful of “rules” that seem a bit self-explanatory if you take a pencil and paper and try drawing some triangles. For example, any two sides of a triangle must sum to be greater than the third side, and the biggest angle is always the longest side of the triangle. No single angle of a triangle can be more than 180 degrees.
Right triangles (triangles where one angle is 90 degrees) are very important in ACT geometry. One of the most commonly-used formulas for working with the side lengths of triangles is the Pythagorean theorem. This formula states that a2 + b2 = c2, where a and b are the lengths of the shorter sides of a right triangle and c is the hypotenuse (the longest side).
Pythagorean theorem = a2 + b2 = c2
While the Pythagorean Theorem can be used to re-derive many individual triangles, it can be very handy to recognize three famous right triangles in particular.
The first of these famous triangles is the 3-4-5 triangle, referring to a right triangle with side lengths of 3 and 4 and a hypotenuse with length 5. On the ACT, students are more likely to see a slightly-disguised version of this famous triangle, like 6-8-10 or 12-16-20.
The other two special right triangles make reference to the angle measurements.
The first is the 45-45-90 triangle. This could also be described as an isosceles right triangle, because two of the angles and side lengths are identical. If the sides of the triangle are x, then the hypotenuse of the 45-45-90 triangle is x times √2.
The last special right triangle is the 30-60-90 triangle. If the shortest leg of this triangle is x, then the other leg is x times √3, and the hypotenuse is 2x.
While there is often a longer way of solving problems involving special triangles, if students can quickly spot the special triangle, they’ll have a huge advantage and be able to solve the problem much more quickly.
If triangles (or any other shapes) are similar, that means that their angles are the same, and their side lengths are proportional to each other.
The formula for the area of a triangle is the length * width * ½. This makes sense if you think about it, because two triangles together will make up a rectangle (for which the area is length * width).
Area of a triangle = ½ * length * width
So to recap,
- The most important formula for triangles is the Pythagorean Theorem, a2 + b2 = c2.
- All of the angles of a triangle will add up to 180 degrees.
- Students should also recognize the special right triangles and know their proportional side lengths: the 3-4-5 triangle, the 45-45-90 triangle, and the 30-60-90 triangle.
- The area of a triangle is ½ * length * width.
Geometry: Circles
Questions about circles also appear frequently on the ACT.
There are a small handful of circle formulas that are absolutely essential to know for the ACT. There’s no way to re-derive these formulas a longer way on the text — either you know them, or you don’t.
Fortunately, the most important circle formulas are very short!
The radius is the distance from the center of the circle to the outer edge.
No matter which direction you draw the radius, it will always be the same.
The diameter is the distance all the way across the circle at the widest point — it’s the same as 2 * the radius.
Diameter = 2 * radius
Students need to know that for a circle with radius r, the perimeter of the circle will be equal to 2πr, and the area of the circle will be equal to πr2.
Perimeter = 2πr
Area = πr2
Remember that π (a symbol which is pronounced pi, like “pie”) is a special number that goes on forever, but can be estimated with 3.14. Don’t worry about memorizing digits of π for the ACT, because students will always be able to use their calculators, which always have π stored.
Now, in addition to being able to find the radius, diameter, area, and perimeter of a circle, students should be able to work with arcs and sectors.
Central angle = corresponding arc
Meanwhile, inscribed angles (angles that hit the back wall of the circle) will be half of the measure of the corresponding arc or central angle. No matter where you place the inscribed angle, if the endpoints on the circle are the same, then it will always have the same measure:
Inscribed angles = ½ * corresponding arc
One cool trick is that any angle subtended by the diameter of the circle will be 90 degrees, though this fact is rarely necessary for the ACT:
Technically, there are a few more equations for these concepts that students can memorize:
Length of an arc:
arc length = θ*r
where θ = central angle in radians, r = radius
Area of a sector:
area = (θ/360°)*r2
where θ = central angle in degrees, r = radius
However, for both of these concepts, it’s possible to work them out using logic and the knowledge that the circle is made up of 360 degrees (or 2π radians). For that reason, we don’t recommend prioritizing memorizing these formulas.
Students will also occasionally be expected to convert between degrees and radians. It’s not necessary to have the entire unit circle memorized. The essential thing to know is that 180 degrees = π radians.
To recap, the important circle formulas to know for the ACT are:
- Circle diameter: d = 2πr
- Circle area: A = πr2
- Circle has 360 degrees
- Converting between degrees and radians: 180 degrees = π radians
- An inscribed angle has half the measure of the corresponding central angle or arc
- Bonus: any angle subtended by the diameter of the circle will be 90 degrees
- Bonus: length of an arc
- Bonus: area of a sector
Geometry: Perimeter, Area, and Volume
There are a handful of other formulas for perimeter, area, and volume that may be useful on the ACT. However, as long as students know that the general way to find area is to multiply length by width, and that for volume you just need to multiply the area by the height (to make the shape three-dimensional). Similarly, the area of a three-dimensional shape can almost always be figured out on the test by drawing a little picture.
That means that students can memorize the following formulas individually, but it’s also possible to puzzle them out with logic on the test:
For example, if students encounter a trapezoid on the test, they can simply split it into two triangles and a rectangle in the middle. Sure, it’s possible to apply a special trapezoid formula, but it’s totally unnecessary.
That said, it is good to know the formula for the area and volume of a sphere, which is not reasonable to figure out with logic. (We covered the perimeter and area of a circle above — those are definitely key formulas to know for the ACT!)
Volume of sphere = (4/3)πr3
Surface area of sphere = 4πr2
In theory, students might want to also know the formula for the area and volume of a cone. However, in practice, the ACT usually gives students these more obscure formulas as part of the question.
Area of cone = πr2 + πrs
Volume of cone = (πr2h) / 3
It’s also good to know that the interior angles of a polygon with n sides sum up to ((n-2) x 180°)/n. If students forget this formula, they can remember two familiar shapes: a triangle (with 3 sides) sums up to 180°, and a square (with 4 sides) sums up to 360°. Following the pattern, just add 180° for each new side on the polygon!
A polygon with n sides sum to ((n-2) x 180°)/n.
So to recap, here are the formulas for perimeter, area, and volume that students should know for the ACT:
- Area of a rectangle
- Area of a rectangular prism
- Volume of a rectangular prism
- Area of a triangle
- Area of a triangular prism
- Volume of a triangular prism
- Area of a sphere
- Volume of a sphere
- Interior angles of a polygon with n sides are ((n-2) x 180°)/n
- Bonus: area of a cone
- Bonus: volume of a cone
Trigonometry: SOHCAHTOA
The ACT does include some questions about basic trigonometry — notably more than the SAT does, for students familiar with the SAT.
The most crucial trigonometry formulas for students to know are sine, cosine, and tangent:
Of course, most students know the SOHCAHTOA acronym for this information that makes it much easier to remember the difference between sine and cosine.
Students should also be familiar with cosecant, secant, and cotangent:
There are a handful of more advanced trigonometric concepts that regularly appear on the ACT. Students should know how to graph sine, cosine, or tangent.
They should know how to identify the period and amplitude of a wave function.
Students should know where sin, cos, and tan are positive/negative on the coordinate plane.
More rarely, students might find questions with trigonometric functions like sin2θ + cos2θ = 1 or sin / cos = tan. However, these will usually be given to students in the question, so it’s not necessary to have them memorized.
To recap, the most important trigonometry formulas that will definitely be on the SAT are SOHCAHTOA. More advanced trigonometric concepts are fair game, so it’s a good idea to be familiar with the rest of these formulas, especially for ambitious students who are trying for a high score on the ACT.
Statistics and Probability
In addition to algebra and geometry, the ACT also tests basic skills in statistics and probability.
Students should be comfortable working with percentages. They should know that the percent is equal to the part divided by the whole, multiplied by 100%.
X% of y = (x/100) * y OR (y/100) * x.
The “percent change” is the value of the change divided by the original value.
Percent change = change / original
The most important statistical formula is a simple one: the average (or mean) is equal to the sum of the terms divided by the number of terms.
Mean = (sum of terms) / (# of terms)
The range of a dataset is the biggest value minus the smallest value.
Range = (largest value) – (smallest value)
The probability of an event is the number of desired terms divided by the total number of terms.
Probability = (# of desired terms) / (# of terms)
The probability that an event does happen plus the probability that an event doesn’t happen always adds up to 1.
(probability event happens) + (probably event doesn’t happen) = 1
The probability of two separate events both happening is equal to the probability of the first event multiplied by the probability of the second event. In order to find the probability of more than two separate events all happening, simply continue the pattern and multiply all the individual probabilities together!
More rarely, the ACT may ask questions about permutations and combinations.
Permutations:
Combinations:
Advanced Algebra
There are a handful of additional concepts that can appear on the ACT Math section, especially in the last 20 problems, which are always the hardest and the most advanced.
There are often problems involving exponential growth functions. Students should know how to set up these functions:
Exponential growth:
It’s also possible for questions about logarithms to appear on the ACT. The definition of logarithms is that if loga(b) = c, then ac = b.
If loga(b) = c, then ac = b.
If it’s necessary to change the base of a logarithm, then use the change of base rule:
Rarely, there may be questions about arithmetic sequences or geometric sequences on the ACT. The formulas for these are:
Arithmetic sequence: tn = t1 + d(n – 1)
Geometric sequence: tn = t1 * r(n-1)
Finally, every few tests there’s a question about matrices. Students who are chasing a high score on the ACT should make sure that they know how to add and multiple matrices:
Given two matrices A and B
The sum C=A+B is:
The product C=A*B is:
Not sure how to work with these more advanced topics? Our Ivy-League tutors have helped thousands of students boost their math knowledge for the ACT. One-on-one tutoring is the ideal setting for filling in any holes in a student’s math knowledge.
Next Steps
Now that you’ve seen all of the important formulas and rules that you need to know for the ACT, the next step is to make sure you’ve memorized them thoroughly. Download our handy guide with all of the math formulas necessary for the ACT.
Learning these ACT formulas is a crucial step of the ACT prep process. The next step is to practice implementing these formulas with practice on real ACT math problems.
Not sure how to practice these ACT math concepts? Hit a plateau with your practice? Through targeted practice and carefully-planned lessons, our experienced ACT tutors have helped thousands of students boost their ACT scores.
Regardless of whether you’re filling in some holes in your general math knowledge (especially common these days after interruptions in schooling due to Covid) or chasing that high score in the 30s, our Ivy-League tutors will match your individual needs.
Schedule a short educational consultation to get started with high-quality tutoring.
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