Short answer: To memorize a math formula, first understand where it comes from by deriving it or seeing why it works once. Then write it from memory on a blank page, quiz yourself over several days (spaced repetition), and use it in mixed practice problems. Formulas you understand and retrieve regularly stay with you; formulas you only reread fade within days.
Most students try to memorize formulas as strings of symbols. At Alexander Tutoring we derive our formulas instead, showing where each one comes from, because a formula you can rebuild from an idea is one you never truly lose.
Key points
- Understand it first: Knowing where a formula comes from lets you rebuild it if you forget.
- Write, don’t copy: Writing a formula from memory strengthens recall; copying it does not.
- Review over days: Several short reviews beat cramming formulas the night before.
- Mix problem types: Mixed practice teaches you when to use each formula.
- Check the units: In physics, units tell you whether a remembered formula can be right.
- Mnemonics last: Save memory tricks for formulas with no logic to rebuild, like SOH CAH TOA.
Why does understanding a formula help you memorize it?
A memorized string like “negative b plus or minus the square root…” is a single fragile memory. If one piece slips, the whole thing is gone. A derivation gives you a chain of reasons, and each link can cue the next.
Understanding also tells you when a formula applies and when it does not. Many errors on tests come from using a correct formula in the wrong situation, and our list of common algebra mistakes includes several of these.
There is a motivational payoff too. Students who see where a formula came from tend to care about it more, and they put in the practice time that makes it stick.
“Motivation begets hard work.”
Sean Alexander, founder of Alexander Tutoring
Worked example 1: the area of a circle, A = πr²
Start with what you already know: the circumference of a circle is C = 2πr, which follows directly from the definition of π as circumference divided by diameter.
- Slice the circle, like a pizza, into many thin wedges that all meet at the center.
- Each thin wedge is very nearly a triangle. Its height is the radius r, and its base is a small piece of the circle’s edge.
- The area of each triangle is ½ × base × height = ½ × base × r.
- Add up all the triangles. The heights are all r, and the bases together make up the whole edge of the circle, which is the circumference 2πr.
- Total area = ½ × r × (sum of all bases) = ½ × r × 2πr.
A = ½ × r × 2πr = πr²
The more slices you use, the closer the wedges get to true triangles.
“Packing in infinitely many triangles is actually a first step toward calculus.”
Sean Alexander, founder of Alexander Tutoring
Tutor tip: Once you have seen this, you no longer need to remember “is it πr² or 2πr?” Area is the one with r², because area is measured in square units.
Watch Sean Alexander, founder of Alexander Tutoring, walk through where the area of a circle formula comes from, step by step.
Worked example 2: the quadratic formula from completing the square
The quadratic formula is just completing the square done once, in general, for ax² + bx + c = 0 (with a ≠ 0).
- Divide every term by a: x² + (b/a)x + c/a = 0
- Move the constant to the right: x² + (b/a)x = −c/a
- Add the square of half the x-coefficient, (b/2a)² = b²/4a², to both sides: x² + (b/a)x + b²/4a² = b²/4a² − c/a
- The left side is now a perfect square, and the right side combines over a common denominator: (x + b/2a)² = (b² − 4ac)/4a²
- Take the square root of both sides: x + b/2a = ±√(b² − 4ac) ÷ 2a
- Subtract b/2a: x = (−b ± √(b² − 4ac)) ÷ 2a
x = (−b ± √(b² − 4ac)) ÷ 2a
Notice what the derivation explains for free: the ± comes from taking a square root, the 2a comes from halving the b/a term, and b² − 4ac (the discriminant) must be non-negative for real solutions.
Work through it yourself two or three times on a blank page and the formula will feel inevitable rather than arbitrary.
Where do common math formulas come from?
Most formulas in middle and high school math are short steps from an idea you already know. Learn the idea and the formula comes with it.
| Formula | Where it comes from | What to remember instead |
|---|---|---|
| Area of a triangle: A = ½bh | A triangle is half of a parallelogram (or rectangle) with the same base and height | “Half a rectangle” |
| Area of a trapezoid: A = ½(b₁ + b₂)h | Average of the two bases times the height; two copies form a parallelogram | “Average base × height” |
| Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²) | The Pythagorean theorem applied to the horizontal and vertical gaps | “Pythagoras on a grid” |
| Slope: m = (y₂ − y₁) ÷ (x₂ − x₁) | Rise over run between two points | “How much y changes per unit of x” |
| Arithmetic series: S = n(a₁ + aₙ) ÷ 2 | Pair the first and last terms, second and second-to-last, and so on; each pair has the same sum | “Number of terms × average term” |
| Kinematics: v = v₀ + at | Definition of constant acceleration: change in velocity per unit time | “Starting speed plus how much it changed” |
Geometry is full of these connections, which is also why memorizing without understanding causes so many of the common geometry mistakes we see.
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What are the best ways to memorize math formulas?
Understanding comes first. These seven methods then lock the formula into long-term memory, and most of them draw on the techniques rated most useful in the review of learning strategies by Dunlosky et al. (2013).
1. Derive it once yourself
Work through the derivation with your notes, then again without them. You do not need to re-derive every formula on every test, but knowing that you could removes the panic when memory fails.
“Once you go through this process, you’ll never forget πr² again.”
Sean Alexander, founder of Alexander Tutoring
2. Write it from memory on a blank page
Copying a formula ten times is mostly passive. Instead, cover your notes, write the formula and what each symbol means, then check.
This is retrieval practice, and practice testing was one of only two techniques rated high utility in the Dunlosky review.
3. Space your reviews with flashcards
Put the formula on one side and its name or a typical problem on the other. Review the next day, a few days later and about a week later.
Apps like Anki handle the scheduling, showing cards you miss more often and spacing out cards you know.
4. Use formulas in mixed practice problems
Doing a page of identical problems teaches you to apply one formula, not to choose it. In a study of math practice, Rohrer and Taylor (2007) found that students who practiced several problem types shuffled together scored far better on a test one week later than students who practiced each type in its own block.
Tutor tip: Our SAT tutors often see students who know a faster method but default to “the long way,” which is slower and leaves more room for errors. When you review a mixed set, ask which formula or shortcut would have solved each problem most directly, not just whether you got it right.
5. Check the units
Especially in physics, units act as a built-in memory check. If you are unsure whether distance is ½at² or ½at, check units: acceleration (m/s²) times time squared (s²) gives meters, so ½at² is right.
Our physics study tips go further into this habit.
6. Explain it out loud
Say what the formula means in plain words: “the slope is how much y changes for each step in x.” If you cannot explain a symbol, that is the part you will forget. Self-explanation was rated moderate utility in the same review.
7. Use mnemonics only for the truly arbitrary
Some facts have no derivation you can reasonably rebuild under time pressure. For these, a mnemonic is fine:
- SOH CAH TOA: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, tangent = opposite ÷ adjacent. Some students prefer a sentence such as “Some Officers Have Curly Auburn Hair Till Old Age.”
- Order of operations: PEMDAS, remembering that multiplication and division (and addition and subtraction) are done left to right.
For a deeper look at the research behind these methods, see our guide to the 8 best memorization techniques.
How do you prepare formulas for a big test?
- Two weeks out: list every formula the test could use and confirm you understand where each comes from.
- Every few days: rewrite the full list from memory on a blank page and mark anything you missed.
- Throughout: do mixed problem sets drawn from all units, not just the latest one.
- The day before: one last blank-page recall, then sleep instead of cramming.
Tutor tip: If you blank on a formula during a test, go back to the idea it came from. For the area of a circle, sketch one thin wedge: height r, with all the bases adding up to 2πr.
For standardized tests, check the official test materials to see which formulas, if any, appear on a provided reference sheet, and know everything else cold. If you are preparing for the ACT, review which math topics the ACT covers, and for AP courses see our AP math success guide.
Tutor-written guide · PDF
Tips for Performing Well During STEM Exams
The game plan our tutors give students before a big math or science test. Enter your name and email and the guide opens right away.
- The night-before routine that beats cramming
- A first-five-minutes plan: formula dump, read-through, quick wins
- Halfway and 10-minute checkpoints so time never runs out
- Calm-down steps for test anxiety you can use mid-exam
Frequently asked questions
What is the fastest way to memorize math formulas?
Understand where the formula comes from, then write it from memory on a blank page and check it, repeating over a few short sessions. Self-testing works faster and lasts longer than rereading or copying the formula.
Should you memorize formulas or understand them?
Both, but understanding comes first. A formula you understand is easier to memorize, harder to misuse, and can be rebuilt if you forget it under pressure.
How do you remember the quadratic formula?
Derive it by completing the square on ax² + bx + c = 0 two or three times, which shows where the ±, the square root and the 2a come from. Then write it from memory every few days until it is automatic.
Why do I forget math formulas right after studying?
Rereading creates a sense of familiarity that fades within days, and memory naturally drops fastest right after learning. Testing yourself on formulas over several days, instead of reviewing them once, makes them last.
Do learning styles matter for memorizing formulas?
A major review in Psychological Science in the Public Interest found no adequate evidence that matching instruction to a learning style improves results. Methods like retrieval practice, spacing and pairing formulas with diagrams help nearly every student.
Which math formulas should I memorize?
Prioritize the formulas your course and tests use most, and that are not provided on a reference sheet. For most high school students that includes the quadratic formula, slope and linear equation forms, the Pythagorean theorem, basic area and volume formulas, and the trigonometric ratios.
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