The GRE does not give you a formula sheet. No reference page, no scratch paper with formulas pre-printed. Whatever you need on test day, you need to have memorized before you sit down.
This page organizes every formula you are responsible for, by category. Bookmark it. Come back to it after every practice session. The goal is not to read through this once. The goal is to reach a point where every formula here fires instantly from memory.
Geometry
Triangles
Area of a triangle: $$A = \frac{1}{2} \times base \times height$$
The height is always perpendicular to the base, not the slant side. On the GRE, triangles are not necessarily drawn to scale, so do not eyeball the height.
Perimeter of a triangle: $$P = a + b + c$$
Pythagorean theorem (right triangles only): $$a^2 + b^2 = c^2$$
Where c is the hypotenuse (the side opposite the right angle). Common Pythagorean triples you should recognize on sight: 3-4-5, 5-12-13, 8-15-17. Multiples also apply: 6-8-10, 9-12-15.
30-60-90 triangle side ratios: $$1 : \sqrt{3} : 2$$
The side opposite 30° is the shortest. The hypotenuse is always twice the shortest side. The side opposite 60° is the shortest side times √3.
45-45-90 triangle side ratios: $$1 : 1 : \sqrt{2}$$
Both legs are equal. The hypotenuse is either leg times √2. This comes up constantly in problems involving squares (cut a square diagonally and you get two 45-45-90 triangles).
Sum of interior angles of any triangle: 180°
Sum of interior angles of any polygon: $$(n - 2) \times 180°$$
Where n is the number of sides. A quadrilateral has (4-2) × 180 = 360°. A pentagon has 540°.
Circles
Area of a circle: $$A = \pi r^2$$
Circumference of a circle: $$C = 2\pi r = \pi d$$
Arc length (a portion of the circumference): $$\text{Arc length} = \frac{\theta}{360} \times 2\pi r$$
Where θ is the central angle in degrees.
Area of a sector (a pie slice of the circle): $$\text{Sector area} = \frac{\theta}{360} \times \pi r^2$$
Rectangles and Quadrilaterals
Area of a rectangle: $$A = l \times w$$
Perimeter of a rectangle: $$P = 2l + 2w$$
Area of a square: $$A = s^2$$
Diagonal of a square: $$d = s\sqrt{2}$$
Area of a parallelogram: $$A = base \times height$$
The height is perpendicular to the base, not the slant side.
Area of a trapezoid: $$A = \frac{1}{2}(b_1 + b_2) \times h$$
Where b1 and b2 are the parallel sides.
3D Shapes
Volume of a rectangular solid (box): $$V = l \times w \times h$$
Volume of a cylinder: $$V = \pi r^2 h$$
Surface area of a rectangular solid: $$SA = 2(lw + lh + wh)$$
Algebra
Slope of a line: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$
Slope-intercept form: $$y = mx + b$$
Where m is the slope and b is the y-intercept.
Point-slope form: $$y - y_1 = m(x - x_1)$$
Quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Used when ax² + bx + c = 0. Know this cold. The GRE will sometimes give you a quadratic that factors cleanly, and sometimes it will not. Factoring first is faster when it works; fall back to the formula when it does not.
Difference of squares: $$a^2 - b^2 = (a+b)(a-b)$$
Perfect square trinomials: $$(a+b)^2 = a^2 + 2ab + b^2$$ $$(a-b)^2 = a^2 - 2ab + b^2$$
Exponent rules:
- $a^m \times a^n = a^{m+n}$
- $\frac{a^m}{a^n} = a^{m-n}$
- $(a^m)^n = a^{mn}$
- $a^0 = 1$ (for any nonzero a)
- $a^{-n} = \frac{1}{a^n}$
- $a^{1/n} = \sqrt[n]{a}$
Linear systems: When you have two equations and two unknowns, use substitution or elimination. ETS favors clean integer answers, so if the arithmetic gets messy, check your setup.
Arithmetic and Number Properties
Percent change: $$\text{Percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100$$
A positive result is an increase. A negative result is a decrease.
Percent of a number: $$\text{Part} = \text{Percent} \times \text{Whole}$$
Or equivalently: Part / Whole = Percent.
Simple interest: $$I = P \times r \times t$$
Where P is principal, r is the annual rate (as a decimal), and t is time in years.
Compound interest: $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
Where n is the number of compounding periods per year. The GRE rarely tests compound interest with complex values. Recognize the formula; the numbers will usually be manageable.
Key divisibility rules:
- Divisible by 2: last digit is even
- Divisible by 3: sum of digits is divisible by 3
- Divisible by 4: last two digits form a number divisible by 4
- Divisible by 5: last digit is 0 or 5
- Divisible by 9: sum of digits is divisible by 9
Properties of zero and one:
- Zero is even, not positive, not negative
- One is not prime
- The smallest prime is 2
Statistics and Probability
Mean (arithmetic average): $$\bar{x} = \frac{\text{sum of all values}}{n}$$
Median: The middle value when all values are ordered. For an even number of values, the median is the average of the two middle values.
Mode: The value that appears most frequently.
Range: $$\text{Range} = \text{maximum} - \text{minimum}$$
Standard deviation: The GRE does not ask you to calculate standard deviation from scratch. You need to understand what it measures (spread around the mean) and how to compare two sets. A set with values clustered near the mean has a lower standard deviation than one with values spread far from the mean.
Basic probability: $$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$
Probability of A and B (independent events): $$P(A \text{ and } B) = P(A) \times P(B)$$
Probability of A or B: $$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$
Combinations (order does not matter): $$nCr = \frac{n!}{r!(n-r)!}$$
Permutations (order matters): $$nPr = \frac{n!}{(n-r)!}$$
The distinction: choosing 3 people for a committee from 10 is a combination. Arranging 3 books on a shelf from 10 is a permutation.
Rate, Work, and Distance
Distance formula: $$D = r \times t$$
Where D is distance, r is rate (speed), and t is time. Know the rearrangements: $r = D/t$ and $t = D/r$.
Average speed (when the same distance is covered at two different speeds): $$\text{Average speed} = \frac{2 r_1 r_2}{r_1 + r_2}$$
Note: average speed is not the simple average of the two speeds unless equal time (not equal distance) is spent at each speed.
Combined work rate (two workers completing one job together): $$\text{Time together} = \frac{xy}{x + y}$$
Where x is the time for worker A alone and y is the time for worker B alone. For three workers: find the combined rate (1/x + 1/y + 1/z = 1/T) and solve for T.
How to Use This Page
Reading formulas is not the same as knowing them. After you go through this list, close it and write out every formula from memory. Anything you cannot write from memory is a gap.
Then use concept lessons to see each formula applied in context with worked examples. Once you have the underlying concept, run practice problems by category until the formulas are automatic.
The formulas that show up most often on the GRE, in rough order: Pythagorean theorem, percent change, D=rt, combined work, probability, area of a circle, triangle area, and the special right triangles. If you have limited time, build fluency in those first.