The ISEE math sections do not give you a formula sheet on test day, and calculators are banned, so every formula you need has to live in your head. This complete ISEE math formulas guide pulls together the geometry, algebra, statistics, and arithmetic formulas tested at the Lower, Middle, and Upper Levels, with quick-reference tables, worked examples, and an interactive recall tool to help you commit them to memory.
The single most important fact about ISEE math formulas: E3n (formerly ERB, the Educational Records Bureau) does not provide a formula sheet on test day, at any level. Unlike the SAT, which prints common geometry formulas at the start of each math section, the ISEE assumes you have already memorized every formula you need. That makes this article a true ISEE math formula sheet — a one-stop reference you can study from until the formulas are automatic.
The constraint is double-barreled. Calculators are prohibited on both math sections (Quantitative Reasoning and Mathematics Achievement) unless you have an approved testing accommodation. You will solve every problem with mental math, scratch work in the booklet, and recalled formulas. If you can't pull up the right formula in a few seconds, you can't even start the problem — let alone finish it inside the time limit.
On the Middle and Upper Level ISEE, the Mathematics Achievement section gives you 47 questions in 40 minutes (about 51 seconds per question), and the Quantitative Reasoning section gives you 37 questions in 35 minutes (about 57 seconds per question). Spending five extra seconds trying to remember the volume formula for a cylinder costs you roughly 10% of your total time on that question. Multiply that across the section and the gap between students who have memorized the formula sheet and students who haven't shows up in the score.
The ISEE has two math sections that demand formulas in different ways. Understanding the difference will tell you which formulas to drill hardest depending on whether you're an ISEE Lower Level math formulas student or an ISEE Middle Level math formulas student.
Quantitative Reasoning is designed to test your mathematical reasoning rather than your ability to calculate. Many problems can be solved with little or no calculation if you spot the relationship correctly. On the Lower Level, every QR question is a word problem; on the Middle and Upper Levels, you also see quantitative comparison questions, where you compare two quantities and decide whether one is larger, they're equal, or there isn't enough information.
Mathematics Achievement asks you to apply specific content you have learned in school. Every question is a word problem, but unlike Quantitative Reasoning, you will usually need to do the full calculation. This is where most of your formula application happens — area, perimeter, volume, percent change, mean, median, and algebraic manipulation all show up here.
The number of questions and the time you get differs between Lower Level and Middle/Upper Level. The table below summarizes the structure of both math sections at each level.
| Level | Quantitative Reasoning | Mathematics Achievement | Calculator? |
|---|---|---|---|
| Lower (entering grades 5-6) | 38 questions / 35 min | 30 questions / 30 min | Not allowed |
| Middle (entering grades 7-8) | 37 questions / 35 min | 47 questions / 40 min | Not allowed |
| Upper (entering grades 9-12) | 37 questions / 35 min | 47 questions / 40 min | Not allowed |
Pick your ISEE level to see how many math questions you'll face and how many seconds per question that gives you.
Arithmetic and percent questions account for a large share of every ISEE math section. They also underpin algebra and geometry questions, so locking these formulas in early pays compound interest later in your prep.
Solve in this order: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Multiplication and division share a tier — work them from left to right, not "all multiplication first." Same with addition and subtraction.
Two formulas you will see repeatedly:
Worked Example
Setup: A jacket originally costs $80 and is now on sale for $60. What is the percent decrease in price?
These rules show up in arithmetic, algebra, and scientific-notation questions:
Algebra is heavily tested on Middle and Upper Level ISEE. The ISEE algebra formulas below cluster around three areas: linear equations on the coordinate plane, distance and midpoint between points, and factoring quadratics. These appear especially often in the ISEE Upper Level math formulas you're expected to know.
The slope between two points (x1, y1) and (x2, y2) is:
m = (y2 − y1) / (x2 − x1)
Once you have the slope, you can write the line in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis).
Both formulas operate on two points (x1, y1) and (x2, y2):
Two factoring patterns are worth memorizing cold for the ISEE Upper Level:
Worked Example
Setup: Find the slope of the line through points A(2, 3) and B(8, 15).
Perimeter, area, and angle measures are among the most frequently tested geometry concepts on the ISEE Mathematics Achievement section. The ISEE geometry formulas below appear at every level — the difference is just how many you need to recognize and how creatively the test combines them.
The basic 2D area formulas are short, but the tricky part is matching a given problem to the right one. Note that triangle area uses the perpendicular height from the base, not just any side. The interior angles of any triangle sum to 180°, and the Pythagorean theorem (a² + b² = c²) applies only to right triangles.
| Shape | Area | Perimeter / Circumference |
|---|---|---|
| Rectangle | l × w | 2l + 2w |
| Square | s² | 4s |
| Triangle | ½ × b × h | side1 + side2 + side3 |
| Parallelogram | b × h | 2(a + b) |
| Trapezoid | ½h(b1 + b2) | sum of all four sides |
| Circle | πr² | 2πr (circumference) |
Both circle formulas use the radius r:
A common trap is using the diameter (d = 2r) instead of the radius. Always confirm which dimension the problem gives you before substituting.
For any polygon with n sides, the interior angles sum to:
(n − 2) × 180°
That's 180° for a triangle, 360° for a quadrilateral, 540° for a pentagon, and 720° for a hexagon. Memorize the formula rather than the values — the formula works for any number of sides.
Worked Example
Setup: A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
3D solids show up on Middle and Upper Level ISEE. Volume and surface area formulas can blur together under time pressure, especially the cylinder, cone, and sphere group. Use the table below as your daily reference, then drill until you can recall each formula in under five seconds.
For a rectangular prism with length l, width w, and height h: volume = l × w × h, and surface area = 2(lw + lh + wh). A cube is a special case where l = w = h = s, so volume = s³ and surface area = 6s².
These three rounded solids share πr² in their formulas but combine it with height differently:
| Solid | Volume | Surface Area |
|---|---|---|
| Cube | s³ | 6s² |
| Rectangular prism | l × w × h | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πrh + 2πr² |
| Cone | ⅓πr²h | πrs + πr² |
| Sphere | (4/3)πr³ | 4πr² |
Pick a shape and enter its dimensions to see the formula applied step-by-step. Use this to drill formula recall.
Worked Example
Setup: A cylindrical can has radius 3 cm and height 10 cm. Find its volume in terms of π.
On the Upper Level, data analysis is the most frequently tested concept area in Mathematics Achievement, which makes statistics formulas a high-leverage study target for older students. (On the Lower Level, fractions, decimals, and area dominate; on the Middle Level, polygons and arithmetic word problems are most common.) The probability formulas are short — the trick is recognizing when a problem calls for them.
When a problem mentions multiple legs of a trip, average speed is rarely the simple average of the two speeds. Use the formula:
Average speed = total distance / total time
Two formulas cover most ISEE probability questions:
Worked Example
Setup: Five test scores are 78, 82, 85, 90, and 95. Find the mean and median.
Knowing every formula on this ISEE math cheat sheet only matters if you can recall them under timed pressure. Memorization is a skill, not just a willingness to read the same page over and over. Here's the routine that actually transfers to test day.
Make one flashcard per formula. Put the formula name on one side ("Pythagorean theorem", "cylinder volume", "midpoint formula") and the formula itself on the back. Review them daily for two weeks, then drop to every other day, then twice a week. Spaced repetition forces your brain to retrieve the formula instead of merely recognizing it.
Memorizing isn't enough — you have to know which formula a question is asking for. Practice with mixed problem sets (geometry + algebra + statistics in the same drill) so that recognition becomes automatic. This is exactly how the actual ISEE will mix topics on test day.
The week before your test, sit down with a blank piece of paper and reproduce the entire formula sheet from memory. Do this every other day. If you can write all 30+ formulas correctly in under ten minutes, you're ready. If a formula keeps slipping, that's the one to put back in heavy rotation.
| Formula Topic | Lower Level | Middle Level | Upper Level |
|---|---|---|---|
| Order of operations (PEMDAS) | Yes | Yes | Yes |
| Percent of change | Yes | Yes | Yes |
| Simple interest (I = prt) | Limited | Yes | Yes |
| Area / perimeter of basic shapes | Yes | Yes | Yes |
| Pythagorean theorem | No | Yes | Yes |
| Slope and slope-intercept form | No | Limited | Yes |
| Distance and midpoint | No | No | Yes |
| Surface area / volume of 3D solids | Limited | Yes | Yes |
| Exponent rules (negative, zero) | No | Yes | Yes |
| Mean, median, mode, range | Yes | Yes | Yes |
| Independent-event probability | No | Yes | Yes |
No. The ISEE does not give students a formula sheet on test day at any level. Students must memorize every formula they need for both the Quantitative Reasoning and Mathematics Achievement sections. Because calculators are also not allowed, knowing formulas cold is essential to finishing the math sections within the time limit.
No. Calculators are prohibited on both ISEE math sections (Quantitative Reasoning and Mathematics Achievement). The test is designed to be solved with mental math, scratch work in the test booklet, and memorized formulas. The only exception is when a calculator is granted as an approved testing accommodation in advance.
Lower Level students (entering grades 5-6) should master arithmetic operations, fractions and decimals, percent of change, basic area and perimeter for rectangles and triangles, mean and median, and the order of operations. They are not asked to know slope, the Pythagorean theorem, or 3D surface area at this level.
Upper Level (entering grades 9-12) introduces more advanced algebra and coordinate geometry: distance and midpoint formulas, slope and slope-intercept form, factoring quadratics, scientific notation, factorials, and 3D solid surface area and volume formulas for cones, cylinders, and spheres.
About 25-35 core formulas cover the vast majority of ISEE math questions. The exact list depends on your level. Lower Level students need around 10-15, Middle Level students need 20-25, and Upper Level students need 30-35 including coordinate geometry and 3D geometry formulas.
Use spaced-repetition flashcards, group formulas by topic (geometry, algebra, statistics), and practice with mixed problem sets so you train recognition of which formula applies. Writing each formula by hand 10-15 times the week before the test also helps cement recall under timed conditions.