GRE study resources
GRE Math Formula Sheet: Free PDF and Worked Examples
Keep the essential GRE Quant relationships in one place. This free five-page reference includes 55 formula entries, the conditions that make them valid, and a worked example for each topic.
Download the branded GRE formula sheet
Your printable Quant reference
Five pages · 55 formula entries · Five worked examples · Free, no sign-in
Download the formula sheet (PDF)Prefer to read online? The same formulas and examples appear below. In the online tables, ^ means a power, sqrt means square root, and pi means π.
This reference is for the GRE General Test, not the Mathematics Subject Test. It is designed for studying before the exam. Use it alongside the ETS Math Review, which explains the underlying concepts in more detail.
Arithmetic & word problems
Translate the words before calculating. Use consistent units, and identify the original value in every percent problem.
| Relationship | Formula | When it applies / watch out |
|---|---|---|
| Percent of a quantity | part = (p / 100) x whole | p is the percent number. A rate of 12% is 0.12 in a calculation. |
| Percent change | 100 x (new - old) / old | The denominator is the original value, assumed positive. A negative result means a decrease. |
| Successive changes | final = initial x (1 + r) x (1 + s) | Use signed decimal rates. A 20% rise followed by a 20% fall gives 1.2 x 0.8 = 0.96. |
| Ratio shares | A = T x a / (a + b); B = T x b / (a + b) | For a total T split in the ratio a:b, with positive a and b. |
| Distance and rate | distance = rate x time | Convert minutes to hours when the rate is in miles per hour. |
| Average speed | average speed = total distance / total time | Do not average two speeds unless the time spent at each speed is equal. |
| Combined work | 1 / T = 1 / a + 1 / b | Two workers finish alone in a and b time units. Assume constant independent rates; subtract an opposing rate. |
| Simple interest | A = P(1 + rt) | P is principal, r is the annual decimal interest rate, and t is years. Interest earned is Prt. |
| Compound interest | A = P(1 + r / n)^(nt) | n compounding periods per year; no deposits or withdrawals; r is the annual decimal rate. |
| Remainder | N = dq + r; 0 <= r < d | For integer N and positive integer divisor d; q is an integer quotient. |
| GCD and LCM | gcd(a, b) x lcm(a, b) = ab | For positive integers a and b. Prime factors help you find both quantities. |
Worked example
A car travels 60 miles at 30 mph, then 60 miles at 60 mph. What is its average speed?
Show solution and common mistake
Total distance = 120 miles. Total time = 60/30 + 60/60 = 3 hours. Average speed = 120/3 = 40 mph.
Watch out: 45 mph is the mean of the speeds, but more time was spent traveling at 30 mph.
Algebra, exponents & sequences
Keep track of signs and restrictions. Check every candidate answer in the original equation when you square or clear denominators.
| Relationship | Formula | When it applies / watch out |
|---|---|---|
| Exponent rules | a^m x a^n = a^(m+n); (a^m)^n = a^(mn) | Shown for integer exponents; any expression with a negative exponent requires a nonzero base. |
| Division and negatives | a^m / a^n = a^(m-n); a^(-n) = 1 / a^n | a must be nonzero. Also, a^0 = 1 for a nonzero a. |
| Roots | sqrt(a^2) = |a|; sqrt(ab) = sqrt(a)sqrt(b) | The product rule shown requires a and b to be nonnegative real numbers. |
| Difference of squares | a^2 - b^2 = (a - b)(a + b) | The sign between the squared terms must be subtraction. |
| Perfect squares | (a + b)^2 = a^2 + 2ab + b^2 | For (a - b)^2, the middle term is -2ab. Never omit the middle term. |
| Quadratic formula | x = (-b +/- sqrt(b^2 - 4ac)) / (2a) | For ax^2 + bx + c = 0, a != 0. The discriminant b^2 - 4ac determines the number of real roots. |
| Slope and line | m = (y2 - y1) / (x2 - x1); y = mx + b | x2 != x1. A vertical line has undefined slope; a horizontal line has slope 0. |
| Parallel / perpendicular | parallel: m1 = m2; perpendicular: m1 x m2 = -1 | These slope formulas apply to nonvertical lines. A vertical line is perpendicular to a horizontal line. |
| Absolute value | |x - a| = d means x = a + d or x = a - d | For d >= 0. Absolute value measures distance, so it cannot be negative. |
| Arithmetic sequence | a_n = a_1 + (n - 1)d; S_n = n(a_1 + a_n) / 2 | d is the common difference and n is a positive integer. S_n sums the first n terms. |
| Geometric sequence | a_n = a_1 x r^(n-1) | r is the common ratio and n is a positive integer. |
Worked example
If x < 0 and x^2 = 49, what is sqrt(x^2) - x?
Show solution and common mistake
x = -7, but sqrt(x^2) = |x| = 7. Therefore sqrt(x^2) - x = 7 - (-7) = 14.
Watch out: The principal square root is nonnegative. Also reverse an inequality sign when multiplying or dividing both sides by a negative number.
Geometry & coordinates
Do not infer lengths or angles from how a diagram looks. Use stated facts and mark radii, heights, and perpendicular lines explicitly.
| Relationship | Formula | When it applies / watch out |
|---|---|---|
| Rectangle / parallelogram | rectangle: A = lw, P = 2(l + w); parallelogram: A = bh | h is the perpendicular height, not necessarily a slanted side. |
| Triangle / trapezoid | triangle: A = bh / 2; trapezoid: A = (b1 + b2)h / 2 | b1 and b2 are the parallel bases of the trapezoid; h is perpendicular to them. |
| Triangle facts | angle sum = 180 degrees; |a - b| < c < a + b | The side inequality holds for a nondegenerate triangle with positive side lengths a, b, c. |
| Right triangle | a^2 + b^2 = c^2 | c is the hypotenuse. Common triples: 3-4-5, 5-12-13, and their positive multiples. |
| Special right triangles | 45-45-90: x, x, x sqrt(2); 30-60-90: x, x sqrt(3), 2x | Lengths appear opposite the angles in increasing order. The final length is the hypotenuse. |
| Circle | A = pi r^2; C = 2 pi r = pi d | d = 2r. Use the radius, not the diameter, in the area formula. |
| Arc / sector | arc = (theta / 360) x 2 pi r; sector = (theta / 360) x pi r^2 | theta is the central angle in degrees; arc is a length and sector is an area. |
| Rectangular solid | V = lwh; surface area = 2(lw + lh + wh) | All three dimensions must use the same length unit. |
| Cylinder | V = pi r^2 h; total surface area = 2 pi rh + 2 pi r^2 | Surface area shown includes both circular ends of a right circular cylinder. |
| Coordinate distance / midpoint | D = sqrt((x2 - x1)^2 + (y2 - y1)^2) | Midpoint = ((x1 + x2)/2, (y1 + y2)/2). Distance is always nonnegative. |
| Similarity / polygon angles | length factor k: area factor k^2; volume factor k^3 | For similar figures/solids and k > 0. An n-sided simple polygon has interior angle sum (n - 2) x 180 degrees. |
Worked example
A circle has radius 6. What is the area of a sector with a 60-degree central angle?
Show solution and common mistake
Sector area = (60/360) x pi x 6^2 = 6 pi square units.
Watch out: The corresponding arc length is 2 pi units. Arc length and sector area are different quantities.
Statistics & data interpretation
Check the sample size, axis scale, units, and whether a chart reports counts or percentages before calculating.
| Relationship | Formula | When it applies / watch out |
|---|---|---|
| Arithmetic mean | mean = sum of values / n; sum = n x mean | n is the number of observations, including repeated values. |
| Weighted mean | weighted mean = sum(weight x value) / sum(weights) | Use nonnegative weights with a positive total. Group means must be weighted by group sizes. |
| Median | odd n: middle value; even n: mean of middle two | Sort the observations first. A median need not be one of the observed values when n is even. |
| Mode / range | mode = most frequent value; range = maximum - minimum | A data set can have more than one mode. Range uses only the two extreme observations. |
| Interquartile range | IQR = Q3 - Q1 | Q1 and Q3 are the first and third quartiles. Use the question's quartile convention when one is specified. |
| Population standard deviation | SD = sqrt(sum((x - mean)^2) / n) | This is the population formula. Sample SD instead divides by n - 1; do not switch definitions silently. |
| Adding a constant | new mean = old mean + c; new SD = old SD | Adding c to every observation shifts the data but does not change the spread or range. |
| Multiplying by a constant | new mean = k x old mean; new SD = |k| x old SD | The range is also multiplied by |k|. Spread cannot become negative. |
| Normal distribution | about 68%, 95%, 99.7% within 1, 2, 3 SD of mean | Use only for a normal distribution. These are approximate proportions, not rules for arbitrary data. |
| Standardized distance | z = (x - mean) / SD | SD must be positive. z tells how many standard deviations a value lies above or below the mean. |
| Percentage points | point change = new percent - old percent | From 20% to 30% is +10 percentage points, but a 50% relative increase. |
Worked example
A group of 10 students has mean score 70. A group of 30 students has mean score 90. What is the combined mean?
Show solution and common mistake
Total score = 10 x 70 + 30 x 90 = 3,400. Combined mean = 3,400 / 40 = 85.
Watch out: 80 gives both groups equal weight even though one group has three times as many students.
Counting, sets & probability
Decide whether order matters, whether repeats are allowed, and whether events are independent. Those choices determine the formula.
| Relationship | Formula | When it applies / watch out |
|---|---|---|
| Multiplication principle | total outcomes = n1 x n2 x ... x nk | Multiply the number of choices at each stage when each prior choice leaves those numbers of options. |
| Factorial | n! = n(n - 1)...1; 0! = 1 | n is a nonnegative integer. n distinct objects can be arranged in n! orders. |
| Permutations | nPr = n! / (n - r)! | Order matters. Choose r distinct objects from n without replacement, with 0 <= r <= n. |
| Combinations | nCr = n! / (r!(n - r)!) | Order does not matter. Choose r distinct objects from n without replacement. |
| Repeated objects | arrangements = n! / (a! b! ...) | a, b, ... are counts of indistinguishable types and sum to n, such as repeated letters. |
| Equally likely outcomes | P(A) = favorable outcomes / total outcomes | Count this way only when the individual outcomes are equally likely. |
| Complement | P(not A) = 1 - P(A) | For 'at least one,' it is often easier to calculate 1 minus the probability of none. |
| Either event | P(A or B) = P(A) + P(B) - P(A and B) | 'Or' includes both. If events are mutually exclusive, their overlap is zero. |
| Both events | P(A and B) = P(A) x P(B given A) | Only for independent events may you replace P(B given A) with P(B). |
| Two overlapping sets | |A union B| = |A| + |B| - |A intersection B| | Neither = total - |A union B|. Count the overlap once, not twice. |
| Expected value | E(X) = sum(value x probability) | Use all possible values and their probabilities, which must sum to 1. |
Worked example
A bag holds 3 red and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
Show solution and common mistake
P(both red) = (3/5) x (2/4) = 3/10. After one red ball is drawn, 2 red balls remain among 4 total.
Watch out: The draws are not independent. Multiplying (3/5) x (3/5) would describe replacement instead.
How to turn a formula into a usable skill
Start with a topic you have recently missed in practice. Read the formula and its condition, cover the sheet, and explain what each symbol means. Then solve a fresh problem without looking. If you remember the expression but cannot recognize when to use it, return to a worked example and identify the words that signal the relationship.
- Write down the unknown. A question asking for a percent change needs a different setup from one asking for the new amount.
- Check the conditions. Equal-probability outcomes, independent events and right triangles are assumptions, not defaults.
- Keep units consistent. Convert minutes to hours before mixing time with a rate measured per hour. Label areas with square units and volumes with cubic units.
- Estimate before calculating. A discount should reduce the price; a probability must stay between zero and one. An estimate catches many transcription mistakes.
- Practice without the sheet. The reference should gradually become unnecessary for familiar problems.
Continue with GRE math practice, Quantitative Reasoning questions, or targeted guides to percentages, work rates and statistics.
Frequently asked questions
Does the GRE give you a math formula sheet?
Do not expect a general formula reference during the GRE. Learn the core relationships you need and practice applying them. This downloadable sheet is a study aid, not a document to bring into the testing room.
Is this formula sheet for the GRE Mathematics Subject Test?
No. It covers core math for the Quantitative Reasoning measure of the GRE General Test. The Mathematics Subject Test has a different scope.
Is the PDF free to download?
Yes. The Test Ninjas GRE math formula sheet is a free PDF download with no account required. It contains five topic sections, formula conditions and worked examples.
Is memorizing formulas enough for GRE Quant?
No. You also need to translate the question, identify assumptions, compare cases and check whether the result answers what was asked. Practice with fresh problems after reviewing each formula.
Sources and further reading
Official test information checked November 9, 2026. Practice material and study recommendations are by Test Ninjas.