SAT Math Formulas You Need to Know

# SAT Math Formulas You Need to Know

The Digital SAT provides a reference sheet with some formulas, but it does not include everything you need. Students who walk into the test relying entirely on the reference sheet will waste time on problems that require formulas from memory — and may not be able to solve some problems at all.

This guide covers every formula you need for the SAT math section: what is on the reference sheet, what is not, and how each formula is actually used on the test.

Formulas on the SAT Reference Sheet

The Digital SAT gives you access to a reference sheet during both math modules. You can open it at any time by clicking the reference button. Here is what it contains and how each formula typically appears on the test.

Circle Formulas

Area of a circle: A = πr²

Circumference of a circle: C = 2πr

These appear frequently. You might be given the area and asked to find the radius, given the circumference and asked to find the diameter, or asked to find the area of a sector (a fraction of the full circle). Even though the basic formulas are provided, you should have them memorized so you do not waste time opening the reference sheet for something this common.

Arc length and sector area are not explicitly on the reference sheet but follow directly from these formulas. The arc length of a sector with central angle θ (in degrees) is (θ/360) × 2πr, and the sector area is (θ/360) × πr². These come up regularly in [geometry and trigonometry](/sat-prep/geometry-and-trigonometry) questions.

Rectangle and Triangle Area

Area of a rectangle: A = lw

Area of a triangle: A = ½bh

The rectangle formula is straightforward. The triangle formula requires you to identify the base and the corresponding height (perpendicular to that base), which is where students sometimes make errors — especially when the triangle is oriented in an unusual way.

Pythagorean Theorem

a² + b² = c²

This is arguably the most-tested formula on the entire SAT. It appears directly in right triangle problems, but it also shows up indirectly in distance problems, coordinate geometry, and even some [algebra](/sat-prep/algebra) problems involving radicals. Know it cold.

Special Right Triangles

45-45-90 triangle: sides in ratio x : x : x√2

30-60-90 triangle: sides in ratio x : x√3 : 2x

The reference sheet includes diagrams of these triangles with their side ratios. They appear on the SAT when a problem involves an isosceles right triangle or an equilateral triangle split in half. Recognizing these patterns lets you skip the Pythagorean theorem entirely and write down the missing side immediately.

Common applications:

• A square's diagonal creates two 45-45-90 triangles. If the side is s, the diagonal is s√2.
• An equilateral triangle with side s has a height of (s√3)/2.
• Trigonometric values for 30°, 45°, and 60° come directly from these triangles.

Volume Formulas

The reference sheet provides volume formulas for several solids:

Rectangular prism (box): V = lwh

Cylinder: V = πr²h

Sphere: V = (4/3)πr³

Cone: V = (1/3)πr²h

Pyramid: V = (1/3)lwh

Cylinder and cone volumes appear most frequently on the SAT. Sphere problems are rarer but do show up. Pyramid problems are the least common.

A typical SAT volume problem gives you two of the three dimensions and asks for the third, or gives you the volume and asks for a dimension. Some harder problems involve comparing volumes — for instance, asking how the volume changes if the radius is doubled (it quadruples for a cylinder, since r is squared in the formula).

Formulas NOT on the Reference Sheet

These formulas will not be provided on the test. You must memorize them.

Linear Equations and Slope

Slope-intercept form: y = mx + b

Where m is the slope and b is the y-intercept.

Slope formula: m = (y₂ - y₁) / (x₂ - x₁)

Point-slope form: y - y₁ = m(x - x₁)

Standard form: Ax + By = C

You need to be able to convert between these forms fluently. The SAT frequently asks you to identify the slope or y-intercept from an equation in standard form, which requires rearranging to slope-intercept form. These are core [algebra](/sat-prep/algebra) skills.

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals (if one slope is m, the other is -1/m). These relationships appear in at least one or two questions on most tests.

Midpoint and Distance

Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Distance: d = √((x₂ - x₁)² + (y₂ - y₁)²)

The midpoint formula is just the average of the x-coordinates and the average of the y-coordinates. The distance formula is the Pythagorean theorem applied to the coordinate plane. If you forget the distance formula, you can always draw a right triangle between the two points and use a² + b² = c².

Quadratic Equations

Standard form: ax² + bx + c = 0

Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Discriminant: D = b² - 4ac

The discriminant tells you the nature of the solutions:

• D > 0: two distinct real solutions
• D = 0: exactly one real solution (a repeated root)
• D < 0: no real solutions

The SAT tests the discriminant frequently, often asking how many solutions an equation has or what value of a constant would give exactly one solution. These are [advanced math](/sat-prep/advanced-math) staples.

Vertex form: y = a(x - h)² + k, where (h, k) is the vertex.

Vertex from standard form: The x-coordinate of the vertex is x = -b/(2a). Plug this back into the equation to find the y-coordinate.

Sum and product of roots: For ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. This shortcut can save significant time when the question asks for the sum or product without needing the individual roots.

Factoring Patterns

Difference of squares: a² - b² = (a + b)(a - b)

Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²

These patterns let you factor expressions instantly without trial and error. The difference of squares is particularly common on the SAT.

Exponent Rules

These are not formulas in the traditional sense, but they are rules you must know:

• x^a × x^b = x^(a+b)
• x^a / x^b = x^(a-b)
• (x^a)^b = x^(ab)
• x^0 = 1
• x^(-a) = 1/x^a
• x^(1/n) = ⁿ√x
• x^(m/n) = ⁿ√(x^m)

The SAT regularly tests whether you can manipulate expressions with exponents, especially fractional and negative exponents. These show up across both algebra and [advanced math](/sat-prep/advanced-math) domains.

Percent and Ratio Formulas

Percent change: ((new - old) / old) × 100

Percent of a number: (percent / 100) × total

Growth/decay: A = A₀(1 + r)^t (growth) or A = A₀(1 - r)^t (decay)

Where A₀ is the initial amount, r is the rate as a decimal, and t is the number of time periods.

The growth and decay formula appears in word problems involving population growth, compound interest, depreciation, and radioactive decay. The SAT loves to test whether you understand what each part of the formula represents — "what does the 1.03 represent in the equation P = 500(1.03)^t?" (Answer: a 3% growth rate per time period.)

Statistics Formulas

Mean (average): sum of values / number of values

Probability: favorable outcomes / total outcomes

These seem simple, but the SAT tests them in tricky ways. A common problem type gives you the mean of a data set and asks what value must be added to achieve a new mean. The key relationship is: sum = mean × count. If the mean of 5 numbers is 20, the sum is 100. If a sixth number is added and the new mean is 22, the new sum is 132, so the sixth number is 32.

Weighted average: (value₁ × weight₁ + value₂ × weight₂ + ...) / (weight₁ + weight₂ + ...)

Weighted averages appear when two groups with different sizes or frequencies are combined. These are classic [problem-solving and data analysis](/sat-prep/problem-solving-and-data-analysis) questions.

Standard deviation is tested conceptually, not computationally. You will never have to calculate a standard deviation on the SAT. But you do need to know that standard deviation measures how spread out data is from the mean. A data set with values clustered tightly around the mean has a low standard deviation; one with values spread far from the mean has a high standard deviation.

Absolute Value

|x| = a means x = a or x = -a

|x| < a means -a < x < a

|x| > a means x > a or x < -a

Absolute value equations and inequalities appear on the SAT regularly. The key insight is that absolute value creates two cases, and you must solve both.

Systems of Equations

A system of two linear equations can have:

• One solution: the lines intersect (different slopes)
• No solutions: the lines are parallel (same slope, different y-intercepts)
• Infinitely many solutions: the lines are identical (same slope, same y-intercept)

The SAT often asks you to determine the number of solutions or to find a constant that makes a system have no solutions or infinitely many solutions. These require you to compare the coefficients of the two equations.

Trigonometry

sin(θ) = opposite / hypotenuse

cos(θ) = adjacent / hypotenuse

tan(θ) = opposite / adjacent

The mnemonic SOH-CAH-TOA helps you remember these.

Key identity: sin²(θ) + cos²(θ) = 1

Complementary angle relationship: sin(x) = cos(90° - x)

The complementary angle relationship is a favorite SAT question type. If sin(30°) = 0.5, then cos(60°) = 0.5 as well, because 30° and 60° are complementary. The SAT will present this in algebraic form — "if sin(x) = cos(y), what is x + y?" — and the answer is always 90 (assuming x and y are acute angles).

Radians: 180° = π radians. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.

These conversions appear in [geometry and trigonometry](/sat-prep/geometry-and-trigonometry) questions involving arc length and sector area when the angle is given in radians.

Circle Equations

Standard form: (x - h)² + (y - k)² = r²

Where (h, k) is the center and r is the radius.

The SAT often gives a circle equation in expanded form — x² + y² + Dx + Ey + F = 0 — and asks for the center or radius. You need to complete the square to convert it to standard form. This is one of the harder skills tested, but it follows a mechanical process once you learn it.

How to Memorize These Formulas

Memorizing a list of formulas through repetition is inefficient. The formulas stick when you understand them and use them repeatedly in context.

Practice-Based Memorization

The best way to memorize a formula is to solve 10 problems that require it. After the tenth problem, you will not need to look it up anymore. Use the [Grind1600 question bank](/question-bank) to filter by domain and drill specific formula applications.

Flashcard Strategy

For formulas you keep forgetting, make a flashcard with the formula on one side and a sample problem on the other. Review these cards for five minutes each day during the week before your test. Do not make flashcards for every formula — only the ones that you consistently fail to recall during practice.

Grouping by Domain

Rather than memorizing formulas as a disconnected list, group them by the domain where they appear:

• [Algebra](/sat-prep/algebra): slope, linear equations, systems of equations, absolute value, exponent rules
• [Advanced Math](/sat-prep/advanced-math): quadratic formula, discriminant, vertex form, factoring patterns, exponent rules, polynomial operations
• [Problem-Solving & Data Analysis](/sat-prep/problem-solving-and-data-analysis): mean, probability, percent change, growth/decay, weighted average, standard deviation concepts
• [Geometry & Trigonometry](/sat-prep/geometry-and-trigonometry): Pythagorean theorem, special right triangles, circle equations, SOH-CAH-TOA, arc length, sector area, volume formulas

This grouping makes the formulas easier to recall because you associate each one with a specific problem type rather than trying to search through a mental list of forty formulas.

Common Formula Mistakes on the SAT

Confusing Area and Circumference

Students mix up A = πr² and C = 2πr, especially under time pressure. One uses r², the other uses 2r. If you substitute the radius into the wrong formula, your answer will be among the choices — it will just be the wrong choice. The SAT designs answer traps around exactly this kind of error.

Using Diameter Instead of Radius

Many problems give you the diameter rather than the radius. If a circle has a diameter of 10, the radius is 5 — but students who plug 10 into πr² get 100π instead of 25π. Always check whether the given measurement is a radius or a diameter before applying any circle formula.

Forgetting the Negative in the Quadratic Formula

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. The -b at the front is easy to forget, especially when b is already negative. If b = -6, then -b = 6. Write out each substitution carefully.

Applying Percent Change Incorrectly

Percent change is (new - old) / old, not (new - old) / new. The denominator is always the original value. Getting this backward is one of the most common errors on [problem-solving and data analysis](/sat-prep/problem-solving-and-data-analysis) questions.

Misidentifying the Hypotenuse

In the Pythagorean theorem, c is always the hypotenuse — the longest side, opposite the right angle. Students sometimes plug the hypotenuse into a or b, which gives a wrong answer. Before applying a² + b² = c², identify which side is across from the 90° angle.

Final Advice

You do not need to memorize every formula before you start practicing. Start with the formulas in the domain you are weakest in, drill problems in that domain until the formulas become automatic, and then move to the next domain.

The goal is not to have a mental cheat sheet that you consult during the test. The goal is to see a problem and immediately know which formula applies, without any conscious recall. That level of fluency comes from practice, not from reading a list. Open the [question bank](/question-bank), pick a domain, and start solving.