Every Formula the Digital SAT Math Section Expects You to Know
Every Formula the Digital SAT Math Section Expects You to Know
Many students assume the Digital SAT reference sheet covers everything. It doesn't.
The sheet in Bluebook is mostly geometry: area, volume, and triangles. That's only one of the four Math domains, and it's the smallest one. Most Math questions come from Algebra, Advanced Math, and Problem-Solving & Data Analysis, and the sheet gives you almost nothing for them.
This guide has three parts: 1. What the reference sheet gives you, so you can stop memorizing it 2. The roughly 25 formulas it leaves out, grouped by domain 3. When to use each formula, and when Desmos is faster
Part 1: What the Reference Sheet Gives You
You can open the reference sheet at any time in either Math module. Here's what's on it:
| Category | Formulas provided |
|---|---|
| Circle | Area A = πr², circumference C = 2πr |
| Rectangle | A = lw |
| Triangle | A = ½bh |
| Pythagorean theorem | c² = a² + b² |
| Special right triangles | 30-60-90: x, x√3, 2x; 45-45-90: s, s, s√2 |
| Rectangular prism | V = lwh |
| Cylinder | V = πr²h |
| Sphere | V = (4/3)πr³ |
| Cone | V = (1/3)πr²h |
| Pyramid | V = (1/3)lwh |
| Facts | A circle has 360° (2π radians); a triangle's angles sum to 180° |
What to do with it: You don't need to memorize these. Do learn the special right triangle ratios well enough that you never have to open the sheet for them. Students who open the sheet 6–8 times per module can lose 2–3 minutes in total. That's a full question's worth of time on a test that gives you about 1 minute 35 seconds per question.
Part 2: The ~25 Formulas Not on the Sheet
Algebra (about 35% of Math)
1. Slope: m = (y₂ − y₁) / (x₂ − x₁) When to use: Any time you're given two points, or a table with x and y values. In word problems, slope is the rate: dollars per hour, gallons per minute.
2. Slope-intercept form: y = mx + b When to use: Interpreting what the numbers mean. The b is the starting value (the value when x = 0). The SAT asks "what does 45 represent?" questions constantly.
3. Point-slope form: y − y₁ = m(x − x₁) When to use: You have one point and a slope. It's faster than solving for b.
4. Standard form: Ax + By = C, where slope = −A/B When to use: Constraint problems like "a store sells x shirts at $12 and y hats at $8 for $240 total." The x-intercept is C/A and the y-intercept is C/B.
5. Parallel and perpendicular lines: Parallel lines have equal slopes. Perpendicular slopes are negative reciprocals, so their product is −1. For example, slope 3/4 → perpendicular slope −4/3.
6. Number of solutions to a linear system:
| Condition | Solutions |
|---|---|
| Different slopes | Exactly one |
| Same slope, different intercepts | None (parallel lines) |
| Same slope, same intercept | Infinitely many (same line) |
When to use: "For what value of k does the system have no solution?" Set the slopes equal, then check that the intercepts differ.
Desmos shortcut: Type both equations and click the intersection. For the "no solution" questions above, type the equations with k as a slider and drag it until the lines are parallel.
Advanced Math (about 35% of Math)
7. Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
8. Discriminant: D = b² − 4ac
| D | Real solutions | Graph |
|---|---|---|
| D > 0 | Two | Crosses the x-axis twice |
| D = 0 | One | Touches the x-axis at the vertex |
| D < 0 | None | Never touches the x-axis |
When to use: "How many solutions?" or "For what value of c does the equation have exactly one solution?" This is among the most-tested formulas not on the sheet.
9. Vertex x-coordinate: x = −b / 2a When to use: Maximum or minimum questions in standard form, like maximum height or maximum revenue. Plug x back into the equation to get the y-value.
10. Vertex form: y = a(x − h)² + k, with the vertex at (h, k) Watch the sign: y = 2(x − 3)² + 5 has its vertex at (3, 5), not (−3, 5).
11. Sum and product of roots: For ax² + bx + c = 0, the sum of the roots is −b/a and the product is c/a. When to use: "What is the sum of the solutions?" You get the answer without solving. It works even when the roots are ugly irrational numbers.
12. Factoring patterns: - Difference of squares: a² − b² = (a + b)(a − b) - Perfect square: a² ± 2ab + b² = (a ± b)²
When to use: Simplifying rational expressions, and spotting that an expression like 9x⁴ − 16 factors as (3x² + 4)(3x² − 4).
13. Exponent rules:
| Rule | Example |
|---|---|
| xᵃ · xᵇ = xᵃ⁺ᵇ | x³ · x⁴ = x⁷ |
| xᵃ / xᵇ = xᵃ⁻ᵇ | x⁸ / x² = x⁶ |
| (xᵃ)ᵇ = xᵃᵇ | (x²)⁵ = x¹⁰ |
| x⁻ᵃ = 1/xᵃ | 2⁻³ = 1/8 |
| x^(a/b) = ᵇ√(xᵃ) | 8^(2/3) = 4 |
14. Exponential growth and decay: y = a(1 + r)ᵗ for growth, y = a(1 − r)ᵗ for decay When to use: Any question about a fixed percent change per period. A 6% decrease per year gives a factor of 0.94, not 0.06. That mix-up is a common trap answer.
15. Compound growth with other time units: y = a(1 + r/n)ⁿᵗ, or a·b^(t/k) when something doubles or halves every k units Example: A population that doubles every 5 years: P = P₀ · 2^(t/5).
Linear vs. exponential test: A constant amount of change per step means linear. A constant percent of change per step means exponential.
Desmos shortcut: Type the quadratic, then click the gray points to see the roots and the vertex. For a sum-of-roots question with nice numbers, this can be faster than the formula. For messy coefficients, the formula wins.
Problem-Solving & Data Analysis (about 15% of Math)
16. Percent change: (new − old) / old × 100 Trap: Divide by the original value. A rise from 80 to 100 is a 25% increase. A drop from 100 to 80 is a 20% decrease.
17. Successive percents: Multiply the factors. A 20% increase followed by a 20% decrease gives 1.2 × 0.8 = 0.96, a 4% net decrease, not 0%.
18. Mean, median, range: - Mean = sum / count, so sum = mean × count. Use this when a value is added to or removed from a data set. - Median = the middle value of the sorted data - An outlier pulls the mean much more than the median.
19. Standard deviation (concept only): You won't calculate it. Know that data clustered near the mean has a smaller standard deviation, and data spread out has a larger one.
20. Probability: P = favorable outcomes / total outcomes When to use: Two-way tables. Read the condition carefully. "Given that the student is a junior" means the denominator is the juniors' total, not the grand total.
21. Unit conversions and rates: distance = rate × time. Chain conversion factors so the units cancel: 60 mi/hr × (5,280 ft/mi) × (1 hr/3,600 s) = 88 ft/s.
Desmos shortcut: Desmos runs regressions directly. Put data in a table and type y1 ~ mx1 + b for a line of best fit or y1 ~ ab^x1 for an exponential model. For mean and median, type mean([3,7,8,12]) or median([...]).
Geometry & Trigonometry (about 15% of Math)
22. Circle equation: (x − h)² + (y − k)² = r², with center (h, k) and radius r Hard-question version: You're given x² + y² + 6x − 4y − 12 = 0. Complete the square on both variables to get (x + 3)² + (y − 2)² = 25, so the center is (−3, 2) and r = 5. Shortcut: For x² + y² + Dx + Ey + F = 0, the center is (−D/2, −E/2).
23. Arc length and sector area: - Arc length = (θ/360) × 2πr - Sector area = (θ/360) × πr² - In radians: arc length = rθ
When to use: Any "fraction of a circle" problem. The central angle divided by 360 is the fraction.
24. Degree–radian conversion: radians = degrees × π/180. For example, 60° = π/3.
25. SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent
26. Complementary angle identity: sin(x) = cos(90° − x) Very commonly tested: "If sin(x°) = cos(32°), what is x?" The answer is 58.
27. Similar triangles and scale factors: If the side lengths scale by a factor of k, areas scale by k² and volumes by k³. Doubling the radius of a sphere multiplies its volume by 8.
28. Polygon interior angles: sum = (n − 2) × 180°. Each angle of a regular polygon = (n − 2) × 180° / n.
29. Distance and midpoint: - Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]. This is just the Pythagorean theorem. - Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
30. Circle angle theorems: An inscribed angle is half the central angle that intercepts the same arc. A radius drawn to a point of tangency is perpendicular to the tangent line.
Desmos shortcut: Desmos starts in radians. If a trig question uses degrees, open the wrench (settings) menu and switch to degrees before you calculate anything, or every trig value will be wrong. To find a circle's center and radius, type the equation and read the graph. To confirm, type (x − h)² + (y − k)² = r² as your guess and check that it lands exactly on top.
If you're counting, that's a little more than 25 items. Several entries pair related formulas, and a few (like 6 and 19) are rules rather than formulas. You need all of them.
Part 3: Formula Recall vs. Desmos — Which Is Faster?
Desmos is very useful, but it doesn't replace knowing the formulas. Use this table to decide which tool to reach for:
| Question type | Faster tool | Why |
|---|---|---|
| Sum of roots, messy coefficients | Formula (−b/a) | Exact answer in about 5 seconds |
| Number of solutions with a parameter k | Formula (discriminant) | Desmos sliders work but are slower |
| Intersection of two curves | Desmos | Click the point and you're done |
| Vertex of a quadratic in standard form | Either | −b/2a is quick; clicking the vertex is also quick |
| Circle from expanded form | Formula (−D/2, −E/2) | Desmos shows the circle but doesn't label the center |
| Interpreting "what does 0.94 mean" | Formula knowledge | Desmos can't interpret word problems for you |
| Linear or exponential regression | Desmos | Doing it by hand isn't realistic |
| sin(x) = cos(y) | Identity | Takes one line |
Whatever you choose, know the formula well enough to recognize when a question calls for it. Desmos solves equations you type in. It doesn't tell you which equation to write.
For a full breakdown of Desmos techniques, see our guide on how to use Desmos on the Digital SAT.
How to Actually Learn These (Without Flashcard Burnout)
Memorizing a list of 30 formulas in a weekend rarely sticks. Here's what works better:
Step 1: Find out which formulas you're already missing Most students already know about two-thirds of this list. The real question is which 8–10 they don't know, and those differ from student to student. One student may never miss a vertex-form question but lose every circle-equation question. Another may have the reverse problem.
A free MySatCoach diagnostic shows your accuracy by skill, so you can tell whether the problem is quadratics, exponential models, or circle geometry before you spend hours reviewing formulas you already know.
Step 2: Drill by formula family, not by formula Group related formulas and practice them together: - Quadratic family: discriminant, vertex, sum/product of roots, vertex form - Percent family: percent change, successive percents, exponential growth and decay - Circle family: circle equation, arc length, sector area, radians
Mixing related formulas in one session trains the skill that actually matters on test day: recognizing which formula a question needs.
Step 3: Practice in mixed sets Once each family is solid, switch to mixed sets where you don't know ahead of time which formula comes next. That matches the real test, where a discriminant question can follow a two-way table question.
MySatCoach's skill-targeted drills are free with an account. They adapt to the formulas you keep missing and keep sending those question types until your accuracy holds steady. Start with the diagnostic and the drills will focus on your weak spots.
Step 4: Write your own one-page sheet, then throw it away In the last week before the test, write every formula from this guide on one page from memory. Check it against this guide and fix what you got wrong. Do this three times. By the third attempt, most students are writing out the whole list in under 5 minutes.
Quick-Reference: The Ten Highest-Value Formulas Not on the Sheet
If you only have a few days, prioritize these. They show up most often, and the reference sheet won't help you with any of them:
- Slope and y = mx + b (including what each number means)
- Discriminant b² − 4ac
- Vertex x = −b/2a, and vertex form
- Sum of roots −b/a
- Exponential growth and decay a(1 ± r)ᵗ
- Percent change (new − old)/old
- Circle equation (x − h)² + (y − k)² = r²
- sin(x) = cos(90° − x)
- Arc length and sector area as a fraction of 360
- Scale factors: k for lengths, k² for areas, k³ for volumes
The Bottom Line
The Digital SAT reference sheet covers basic geometry. Most of the Math section tests formulas you're expected to bring with you: slopes, discriminants, exponential models, circle equations, and trig identities.
You don't need to memorize all of them with equal effort. Find the 8–10 you're missing, drill those as families, and use Desmos for intersections and regressions. Use recall for everything that's faster without it.
If you're not sure where to start, take the free MySatCoach diagnostic. It shows which formula families are costing you points, so your study time goes to the right places.