The GRE does not hand you a formula sheet. That single fact decides how you should study Quant. Unlike some other standardised tests, the GRE expects the formulas to already be in your head, which means a chunk of your score is decided before you ever see a question.
The good news is that the list is short. GRE Quant covers arithmetic, algebra, geometry and data analysis at roughly a strong high-school level, and the formulas that actually earn points fit on one page. This guide is that page, organised by topic, with the traps that come attached to each formula and an honest note on which ones you can skip. When you want to use them under time, take them into GRE Quant practice or a full free mock test.
Memorising formulas is necessary but not sufficient. Most Quant points are lost to misread questions and setup errors, not missing formulas. Pair this sheet with GRE Quant traps and how to beat them.
Does the GRE give you a formula sheet?
No. There is no formula sheet, no reference page, and nothing you may bring into the test centre. What you do get is an on-screen calculator, and the difference between those two things matters more than people expect.
- **You get:** a basic on-screen calculator with the four operations and a square root key. Our GRE calculator guide shows what it can and cannot do.
- **You get:** the question itself, including any figure, and the standard GRE conventions about figures being drawn to scale unless stated otherwise.
- **You do not get:** any formula, definition, conversion, or geometry reference. Not even the area of a circle.
- **You do not get:** scratch paper of your own. Test centres provide scratch paper, and the at-home version provides an erasable whiteboard or equivalent.
So the calculator saves you arithmetic, not thinking. It will divide 4,182 by 17 for you, but it will not tell you that you needed the area of a sector rather than the arc length. That is why this list has to be memorised rather than looked up. For the full picture of what the section contains, see the GRE exam pattern.
How to use this formula sheet
Do not read this page once and feel prepared. A formula you can recognise is not a formula you can retrieve under time pressure, and the GRE only rewards retrieval. Three rules make this list stick.
- **Separate memorise from derive.** Some of these must be instant recall, for example the area of a circle. Others you can rebuild in ten seconds if you understand them, for example the sector area, which is just a fraction of the circle. Mark which is which for yourself, and only drill the recall list.
- **Attach each formula to a mistake.** A formula learned alongside the error it prevents survives much longer than one learned in isolation. Every section below names the trap that goes with it.
- **Test yourself blank-page.** Once a week, write out as much of this sheet as you can from memory on an empty page, then check it. The gaps you find are your real study list.
Arithmetic and number properties formulas
This is the highest-frequency and lowest-glamour area of GRE Quant. Number properties questions look like puzzles rather than calculations, and they reward knowing a handful of structural facts.
| Concept | Formula | The trap |
|---|---|---|
| Average (arithmetic mean) | sum divided by how many terms | Averaging averages. The mean of two group means is only the overall mean if the groups are the same size. |
| Sum from an average | sum = average times count | Forgetting this is reversible. Most average questions are really sum questions in disguise. |
| Sum of first n positive integers | n(n + 1) / 2 | Off-by-one errors on ranges that do not start at 1. Count the terms first. |
| Number of terms in a consecutive list | last minus first, plus 1 | Dropping the plus 1. From 17 to 41 inclusive there are 25 terms, not 24. |
| Sum of an evenly spaced list | average of first and last, times the number of terms | Only valid when spacing is even. Check before using it. |
| Number of factors of n | add 1 to each prime exponent, then multiply | Counting only the obvious factor pairs and missing the rest. |
| Relationship of LCM and GCF | LCM times GCF = product of the two numbers | Applying it to more than two numbers, where it does not hold. |
| Even and odd behaviour | odd times odd is odd; any even factor makes a product even; odd plus odd is even | Assuming the same pattern holds for division. It does not. |
Prime factorisation is the single most useful skill here. Factors, multiples, divisibility, remainders and square roots all become easier once the number is broken into primes.
Percentages, ratios and rates
Percent questions are where careless GRE points go to die. The formulas are trivial; the wording is not. Read for what the percent is taken of, because that is what the question is testing.
| Concept | Formula | The trap |
|---|---|---|
| Percent of a number | part = percent times whole | Mixing up part and whole when the question says of versus more than. |
| Percent change | (new minus old) divided by old, times 100 | Dividing by the new value. The base is always the original. |
| Percent increase as a multiplier | multiply by (1 + rate) | Adding percents across steps instead of multiplying multipliers. |
| Successive percent changes | multiply the multipliers, for example (1.20)(0.80) | Assuming a 20 percent rise then a 20 percent fall returns you to the start. It gives 0.96 of the original. |
| Percent versus percentage points | a move from 5 to 7 percent is 2 percentage points but a 40 percent increase | Treating the two as the same. GRE answer choices punish this. |
| Distance, rate, time | distance = rate times time | Unit mismatches. Convert minutes to hours before multiplying. |
| Average speed | total distance divided by total time | Averaging the two speeds. Equal distances at 30 and 60 give 40, not 45. |
| Combined work rate | add the individual rates, then invert for total time | Adding the times instead of the rates. |
For ratio problems, the reliable method is to write the ratio with a multiplier. If the ratio is 3 to 5, call the parts 3x and 5x. That converts a ratio question into an algebra question, which is easier to control.
Interest formulas: simple and compound
Interest appears often enough to be worth exact memorisation, and the two formulas are easy to confuse under time pressure.
| Type | Formula | Notes |
|---|---|---|
| Simple interest | I = P times r times t | Interest earned only on the original principal. Total value is P(1 + rt). |
| Compound interest | A = P times (1 + r/n) raised to the power nt | P is principal, r the annual rate as a decimal, n the compounding periods per year, t the years in total. |
| Annual compounding | A = P times (1 + r) raised to the power t | The n = 1 case, and the one that appears most. |
| Compound versus simple | compound always exceeds simple after the first period | Useful for quantitative comparison questions where you never need to compute the value. |
In a quantitative comparison, you usually do not need the number. Knowing that compound beats simple after period one, or that a larger n gives a larger result, resolves the comparison without arithmetic. That is often the intended shortcut.
Algebra and exponent rules
The GRE loves algebraic identities because they turn an ugly expression into a clean one in a single step. Recognising the three factoring patterns below is worth more time than any other algebra drill.
| Identity or rule | Formula | Why the GRE uses it |
|---|---|---|
| Difference of squares | a squared minus b squared = (a + b)(a minus b) | The most tested identity on the GRE. It turns 51 squared minus 49 squared into 100 times 2, which is 200, with no calculator. |
| Square of a sum | (a + b) squared = a squared + 2ab + b squared | The middle term 2ab is what gets dropped. That dropped term is usually the wrong answer offered. |
| Square of a difference | (a minus b) squared = a squared minus 2ab + b squared | Same trap, with a sign flip to catch you twice. |
| Product of powers | same base: add the exponents | Multiplying the exponents instead of adding. |
| Power of a power | raise a power to a power: multiply the exponents | Adding instead of multiplying. The mirror image of the rule above. |
| Quotient of powers | same base: subtract the exponents | Sign errors when the bottom exponent is larger. |
| Zero exponent | any nonzero base to the power 0 equals 1 | Assuming it equals 0. |
| Negative exponent | a to the power negative n = 1 over a to the power n | Treating the negative sign as making the value negative. |
| Fractional exponent | a to the power 1 over n is the nth root of a | Useful for rewriting roots as powers so the other rules apply. |
| Quadratic formula | x = negative b, plus or minus the square root of (b squared minus 4ac), all over 2a | Reaching for it too early. Most GRE quadratics factor cleanly. |
| Sum and product of roots | sum = negative b over a; product = c over a | A genuine shortcut when the question asks for the sum of solutions rather than the solutions. |
One warning about inequalities, which is a formula rule rather than a formula: multiplying or dividing an inequality by a negative number flips the direction. This is one of the most common single-step errors in GRE algebra, and it is why you cannot safely multiply an inequality by a variable whose sign you do not know.
Geometry: triangles
Triangles are the densest formula area on the test. The special right triangles in particular are pure memorisation that pays back immediately, because the GRE builds figures out of them constantly.
| Concept | Formula | The trap |
|---|---|---|
| Area of a triangle | one half times base times height | Using a slanted side as the height. The height must be perpendicular to the base. |
| Sum of interior angles | 180 degrees | Forgetting it applies to every triangle, which often supplies the missing angle for free. |
| Pythagorean theorem | a squared + b squared = c squared | Applying it to a triangle that is not right angled, or putting a leg where the hypotenuse belongs. |
| 45-45-90 triangle | sides in ratio 1 to 1 to the square root of 2 | Putting the root 2 on a leg instead of the hypotenuse. |
| 30-60-90 triangle | sides in ratio 1 to the square root of 3 to 2 | Swapping which side faces 30 and which faces 60. The shortest side always faces the smallest angle. |
| Common Pythagorean triples | 3-4-5 and 5-12-13, plus their multiples such as 6-8-10 | Not spotting them and computing the long way. |
| Triangle inequality | each side is less than the sum and more than the difference of the other two | Forgetting the lower bound, which is what most such questions test. |
| Equilateral triangle area | side squared times the square root of 3, divided by 4 | Worth memorising rather than deriving under time. |
| Similar triangles | corresponding sides are proportional; areas scale as the square of the ratio | Scaling area by the side ratio instead of its square. |
On the GRE, figures are drawn to scale unless the question states otherwise, but coordinate figures and figures marked not drawn to scale are the exceptions. Never estimate from a figure that carries that label.
Geometry: circles, quadrilaterals and polygons
| Shape | Formula | Notes |
|---|---|---|
| Circle area | pi times radius squared | Confusing radius with diameter is the single most common circle error. |
| Circle circumference | 2 times pi times radius, or pi times diameter | Both forms appear; know each. |
| Arc length | the central angle over 360, times the circumference | An arc is a piece of the perimeter, measured in length not degrees. |
| Sector area | the central angle over 360, times the area | Same fraction, applied to area instead of circumference. |
| Inscribed angle | half the central angle that subtends the same arc | An angle inscribed in a semicircle is always 90 degrees, which is a frequent hidden right triangle. |
| Rectangle | area = length times width; perimeter = 2(length + width) | Reading a perimeter question as an area question. |
| Parallelogram area | base times height | Using the slanted side rather than the perpendicular height. |
| Trapezoid area | the average of the two parallel sides, times the height | Adding the parallel sides without halving. |
| Sum of interior angles of a polygon | (n minus 2) times 180 degrees | Works for any polygon with n sides, regular or not. |
| One interior angle of a regular polygon | (n minus 2) times 180, divided by n | Only valid when the polygon is regular, meaning all sides and angles equal. |
| Sum of exterior angles | 360 degrees for any convex polygon | A shortcut that surprises people because it does not depend on n. |
Regular polygons deserve one extra note. A regular polygon can always be cut into n identical triangles from its centre, which is how you handle area questions about pentagons and hexagons without a dedicated formula. For a regular hexagon in particular, those triangles are equilateral, so its area is six times the equilateral triangle area above.
Geometry: three-dimensional solids
| Solid | Volume | Surface area |
|---|---|---|
| Rectangular solid | length times width times height | 2 times (lw + lh + wh) |
| Cube | side cubed | 6 times side squared |
| Right circular cylinder | pi times radius squared times height | 2 pi r squared plus 2 pi r h |
- **Diagonal of a rectangular solid:** the square root of (length squared + width squared + height squared). This is the Pythagorean theorem applied twice, and it is the most commonly missed 3D formula.
- **Scaling:** doubling every dimension multiplies surface area by 4 and volume by 8. Expect at least one question that rewards knowing this rather than computing.
- **Cones, spheres and pyramids:** the GRE does not require these volume formulas. If one appears, the question supplies it.
Coordinate geometry
| Concept | Formula | The trap |
|---|---|---|
| Slope | change in y divided by change in x | Inverting it. Rise over run, not run over rise. |
| Slope-intercept form | y = mx + b, where m is slope and b the y-intercept | Reading b as the x-intercept. |
| Distance between two points | the square root of (change in x squared plus change in y squared) | Just the Pythagorean theorem. No separate formula needed. |
| Midpoint | average the x values, average the y values | Subtracting instead of averaging. |
| Parallel lines | equal slopes | Assuming equal intercepts too. |
| Perpendicular lines | slopes multiply to negative 1, so each is the negative reciprocal of the other | Using the reciprocal without the negative sign. |
| Equation of a circle | (x minus h) squared plus (y minus k) squared = r squared, centred at (h, k) | Sign errors on the centre. A plus in the bracket means a negative coordinate. |
| x and y intercepts | set y = 0 for the x-intercept, set x = 0 for the y-intercept | Swapping the two. |
Statistics and data analysis
Data analysis is a quarter of GRE Quant and one of the friendliest areas, because most questions test definitions rather than computation. Standard deviation in particular is almost always conceptual.
| Measure | Definition or formula | What the GRE actually asks |
|---|---|---|
| Mean | sum divided by count | Usually the sum, reached through the mean. |
| Median | the middle value of an ordered list, or the average of the two middle values | Forgetting to sort first. This is the most common data error on the test. |
| Mode | the most frequently occurring value | That a list can have more than one mode, or none. |
| Range | greatest value minus least value | Simple enough that questions hide it inside a longer setup. |
| Standard deviation | the square root of the average of the squared differences from the mean | You almost never compute it. You compare spread: the more clustered set has the smaller standard deviation. |
| Effect of adding a constant | mean shifts by the constant; standard deviation does not change | A classic quantitative comparison. Shifting a set does not change its spread. |
| Effect of multiplying by a constant | both mean and standard deviation scale by that factor | Assuming standard deviation is unaffected, as it is for addition. |
| Interquartile range | third quartile minus first quartile | Confusing it with the full range. |
| Normal distribution | about 68 percent within one standard deviation of the mean, about 95 percent within two | The 68 and 95 figures are the ones worth memorising. |
If a GRE question seems to demand a standard deviation calculation, re-read it. The intended route is nearly always a comparison of spread, or the effect of a transformation, neither of which needs the formula.
Counting and probability
| Concept | Formula | The trap |
|---|---|---|
| Fundamental counting principle | multiply the number of independent choices at each stage | The workhorse. Most counting questions need only this. |
| Permutations | n factorial divided by (n minus r) factorial | Using it when order does not matter. |
| Combinations | n factorial divided by r factorial times (n minus r) factorial | The reverse error. Ask whether rearranging the same selection makes a new outcome. |
| Basic probability | favourable outcomes divided by total outcomes | Miscounting the total, not the favourable. |
| Probability of not A | 1 minus the probability of A | The best shortcut on the test. At least one questions are almost always easier backwards. |
| A and B, independent | multiply the probabilities | Applying it when the events are dependent, such as drawing without replacement. |
| A or B | add the probabilities, then subtract the probability of both | Forgetting the subtraction when the events can happen together. |
The order versus no order question decides almost every counting problem. A committee of three from ten is a combination, because the same three people are the same committee. Three finishing positions from ten runners is a permutation, because first and second are different outcomes.
The formulas you do not need to memorise
Almost every formula sheet you will find online is padded. Knowing what is off the syllabus saves real study time, and the GRE Quant syllabus is genuinely narrower than people fear.
- **Trigonometry.** No sine, cosine or tangent. Right triangle questions are solved with the Pythagorean theorem and the two special triangles.
- **Calculus.** No derivatives, integrals or limits of any kind.
- **Imaginary and complex numbers.** Not tested. Every number on the GRE is real.
- **Formal proofs and geometric construction.** Not tested.
- **Cone, sphere and pyramid volumes.** Not required. If such a solid appears, the question gives you the formula.
- **Advanced sequences.** No general formula for geometric series is required. Sequence questions are usually solved by writing out terms.
- **The distance and midpoint formulas as separate facts.** Both fall out of the Pythagorean theorem and averaging, so understanding beats memorising here.
If you want the authoritative scope rather than our summary, ETS publishes a free Math Review PDF and a Math Conventions PDF covering exactly what is assumed. Those two documents, plus the free official practice tests described in our GRE PowerPrep guide, are the best free calibration available.
The one-page core: the formulas that carry most of the test
If you memorise nothing else, memorise these. In our experience of the question types, this short list touches the large majority of formula-dependent GRE Quant questions.
- Average = sum over count, and therefore sum = average times count.
- Percent change = (new minus old) over old.
- Successive percent changes multiply as multipliers, they do not add.
- Distance = rate times time, and average speed = total distance over total time.
- Difference of squares: a squared minus b squared = (a + b)(a minus b).
- (a + b) squared = a squared + 2ab + b squared, and the same with a minus sign.
- Exponents: add when multiplying same bases, multiply when raising a power to a power.
- Pythagorean theorem, plus the 3-4-5 and 5-12-13 triples.
- 45-45-90 sides are 1 to 1 to root 2; 30-60-90 sides are 1 to root 3 to 2.
- Triangle area = one half base times height, with height perpendicular to base.
- Circle area = pi r squared; circumference = 2 pi r.
- Arc length and sector area are both the central angle over 360, times circumference or area.
- Interior angles of an n-sided polygon sum to (n minus 2) times 180.
- Rectangular solid volume = lwh; its diagonal is root of (l squared + w squared + h squared).
- Slope = change in y over change in x; perpendicular slopes are negative reciprocals.
- Median requires sorting the list first.
- Adding a constant shifts the mean and leaves standard deviation unchanged.
- Probability of at least one = 1 minus the probability of none.
- Combinations when order does not matter, permutations when it does.
- Simple interest is P r t; compound interest is P times (1 + r over n) to the power nt.
Twenty lines is a realistic memorisation target for a week. Write them from memory each morning and you will have them cold well before test day. Our GRE study plan shows where this fits in a 4, 8 or 12 week schedule.
How to actually memorise these
Rereading a formula sheet feels productive and does very little. What works is retrieval, spacing and immediate application, in that order.
- **Blank page recall, daily.** Five minutes writing the core list from memory. The act of failing to recall and then checking is what builds the memory, so do not peek early.
- **Space the review.** Review the same formula after a day, then three days, then a week. Spacing beats volume for anything you need on demand.
- **Apply within an hour.** After drilling geometry formulas, do ten geometry questions immediately. A formula used once in a real question is worth twenty passive readings.
- **Keep an error log.** Every time a missed question traces to a formula, write the formula and the specific misuse. That log becomes a personalised version of this page, and it is far more useful than the generic sheet.
- **Test under time.** Recall speed matters, not just recall. Take a timed mock test and note any formula you had to reconstruct mid-question, because those are the ones not yet automatic.
One realistic caveat. Formula recall raises your floor, not your ceiling. It removes a category of avoidable loss, which for most people is worth a few points, but a high Quant score is built on reading questions correctly and choosing efficient approaches. Once these twenty lines are automatic, put your time into Quant practice rather than more memorisation.