Complete SAT Math Formula Sheet: All Formulas You Need to Ace the Test

complete math formula sheet

Preparing for the SAT Math section? You’ve come to the right place! This comprehensive formula sheet contains every formula you need to know for SAT Math, organized by topic for easy reference.

Whether you’re tackling algebra, geometry, trigonometry, or statistics, this guide has you covered. We’ve included not just the formulas, but also clear descriptions and tips for when to use them.

Pro Tip: Bookmark this page and review it regularly during your SAT prep. The SAT doesn’t provide all formulas, so memorizing these is crucial for success!

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1. Algebra I – Linear Equations & Inequalities

Linear equations form the foundation of algebra. These formulas help you work with straight lines, slopes, and systems of equations. Master these first!

TopicFormulaDescription
Slope-Intercept Formy = mx + bm = slope, b = y-intercept
Point-Slope Formy – y₁ = m(x – x₁)Line through point (x₁, y₁) with slope m
Standard FormAx + By = CA, B, C are constants
Slope Formulam = (y₂ – y₁)/(x₂ – x₁)Slope between two points (rise over run)
Parallel Linesm₁ = m₂Same slope means parallel
Perpendicular Linesm₁ × m₂ = -1Negative reciprocal slopes
Distance Between Pointsd = √[(x₂ – x₁)² + (y₂ – y₁)²]Pythagorean theorem in coordinate plane
Midpoint FormulaM = ((x₁ + x₂)/2, (y₁ + y₂)/2)Point halfway between two points
💡 Quick Tip: When two lines are perpendicular, their slopes multiply to give -1. For example, if one line has slope 2, the perpendicular line has slope -1/2.

2. Algebra II – Quadratics & Polynomials

Quadratic equations are tested heavily on the SAT. You’ll need to factor, complete the square, and use the quadratic formula with confidence.

TopicFormulaDescription
Standard Formy = ax² + bx + ca ≠ 0, opens up if a > 0, down if a < 0
Vertex Formy = a(x – h)² + kVertex at (h, k)
Factored Formy = a(x – r₁)(x – r₂)r₁, r₂ are x-intercepts (roots)
Quadratic Formulax = [-b ± √(b² – 4ac)]/(2a)Solutions to ax² + bx + c = 0
DiscriminantΔ = b² – 4acΔ > 0: two real roots
Δ = 0: one real root
Δ < 0: no real roots
Vertex (x-coordinate)x = -b/(2a)x-coordinate of vertex/axis of symmetry
Vertex (y-coordinate)y = c – b²/(4a)y-coordinate of vertex
Axis of Symmetryx = -b/(2a)Vertical line through vertex
Sum of Rootsr₁ + r₂ = -b/aVieta’s formula
Product of Rootsr₁ × r₂ = c/aVieta’s formula
Completing the Squarex² + bx + (b/2)² = (x + b/2)²Add (b/2)² to both sides
Difference of Squaresa² – b² = (a + b)(a – b)Important factoring pattern
Perfect Square Trinomiala² ± 2ab + b² = (a ± b)²Factoring pattern
⚠️ Important: The quadratic formula is NOT provided on the SAT. You must memorize it! Many students sing it to the tune of “Pop Goes the Weasel” to remember it.

3. Exponentials & Logarithms

Exponential functions model growth and decay in real-world scenarios like compound interest, population growth, and radioactive decay.

TopicFormulaDescription
Exponential Growthy = a(1 + r)ᵗ or y = a·bᵗa = initial, r = rate, t = time
Exponential Decayy = a(1 – r)ᵗ or y = a·bᵗ0 < b < 1 for decay
Compound InterestA = P(1 + r/n)^(nt)P = principal, r = rate, n = times/year, t = years
Continuous CompoundA = Pe^(rt)e ≈ 2.718
Logarithm Definitionlog_a(x) = b ↔ a^b = xa is base
Natural Logln(x) = log_e(x)Base e
Common Loglog(x) = log₁₀(x)Base 10
Product Rulelog_a(xy) = log_a(x) + log_a(y)Log of product
Quotient Rulelog_a(x/y) = log_a(x) – log_a(y)Log of quotient
Power Rulelog_a(x^n) = n·log_a(x)Log of power
Change of Baselog_a(x) = log(x)/log(a) = ln(x)/ln(a)Convert to any base

4. Systems of Equations

Systems of equations appear frequently on the SAT, often in word problems. You’ll solve them using substitution or elimination.

TopicFormulaDescription
Linear System (2×2)ax + by = e
cx + dy = f
Solve by substitution or elimination
Consistent SystemOne or infinite solutionsLines intersect or coincide
Inconsistent SystemNo solutionParallel lines
Cramer’s Rule (x)x = (ed – bf)/(ad – bc)For 2×2 systems
Cramer’s Rule (y)y = (af – ec)/(ad – bc)For 2×2 systems

5. Functions

Understanding function notation and properties is crucial for the SAT. You’ll work with domain, range, and function transformations.

TopicFormulaDescription
Function Notationf(x) = …Output for input x
DomainAll possible x-valuesInput values
RangeAll possible y-valuesOutput values
Composite Functions(f ∘ g)(x) = f(g(x))Function of a function
Inverse Functionf⁻¹(f(x)) = xReflects over y = x
Even Functionf(-x) = f(x)Symmetric about y-axis
Odd Functionf(-x) = -f(x)Symmetric about origin
Absolute Value|x| = x if x ≥ 0
|x| = -x if x < 0
Distance from zero
Piecewise Functionf(x) = {g(x) if condition 1
h(x) if condition 2}
Different rules for different domains

6. Polynomials & Rational Expressions

TopicFormulaDescription
FOIL Method(a + b)(c + d) = ac + ad + bc + bdFirst, Outer, Inner, Last
Difference of Cubesa³ – b³ = (a – b)(a² + ab + b²)Factoring pattern
Sum of Cubesa³ + b³ = (a + b)(a² – ab + b²)Factoring pattern
Rational ExpressionP(x)/Q(x), Q(x) ≠ 0Polynomial over polynomial
Vertical Asymptotex = a where Q(a) = 0Denominator = 0
Horizontal Asymptotey = lim(x→∞) f(x)Behavior as x → ∞

7. Radicals & Rational Exponents

Exponent rules are essential for simplifying expressions. These rules apply to both whole number and fractional exponents.

TopicFormulaDescription
Square Root√(x²) = |x|Principal square root
Rational Exponentx^(m/n) = ⁿ√(x^m) = (ⁿ√x)^mFractional exponent
Product Rulex^a · x^b = x^(a+b)Same base
Quotient Rulex^a / x^b = x^(a-b)Same base
Power Rule(x^a)^b = x^(ab)Power of a power
Negative Exponentx^(-n) = 1/x^nReciprocal
Zero Exponentx^0 = 1x ≠ 0
Distributive Power(xy)^n = x^n·y^nPower of product
Quotient Power(x/y)^n = x^n/y^nPower of quotient
Simplifying Radicals√(ab) = √a · √bProduct property
Rationalizing1/√a = √a/aMultiply by √a/√a

8. Geometry – Lines & Angles

Angle relationships are fundamental to geometry problems on the SAT. Know these cold!

TopicFormulaDescription
Supplementary Anglesα + β = 180°Angles on a straight line
Complementary Anglesα + β = 90°Angles in right angle
Vertical Anglesα = βOpposite angles are equal
Sum of Triangle Anglesα + β + γ = 180°Interior angles of triangle
Exterior AngleExterior = sum of remote interior anglesTriangle exterior angle theorem
Parallel Lines (Corresponding)Angles are equalSame position
Parallel Lines (Alternate Interior)Angles are equalZ-pattern
Parallel Lines (Co-interior)Angles sum to 180°C-pattern

9. Geometry – Triangles

Triangles are heavily tested on the SAT. Pay special attention to the special right triangles!

TopicFormulaDescription
AreaA = ½bhb = base, h = height
Area (Heron’s Formula)A = √[s(s-a)(s-b)(s-c)]s = (a+b+c)/2 (semi-perimeter)
PerimeterP = a + b + cSum of all sides
Pythagorean Theorema² + b² = c²Right triangle, c = hypotenuse
Pythagorean Triples3-4-5, 5-12-13, 8-15-17, 7-24-25Common right triangles
45-45-90 TriangleSides: x, x, x√2Isosceles right triangle
30-60-90 TriangleSides: x, x√3, 2xSpecial right triangle
Isosceles TriangleTwo sides equal, two angles equalBase angles are equal
Equilateral TriangleAll sides equal, all angles = 60°All sides equal
Area (Equilateral)A = (s²√3)/4s = side length
Similar Trianglesa₁/a₂ = b₁/b₂ = c₁/c₂Corresponding sides proportional
Triangle Inequalitya + b > cSum of two sides > third side
📐 Pythagorean Triples (Common Right Triangles):
  • 3-4-5 (and multiples: 6-8-10, 9-12-15, etc.)
  • 5-12-13 (and multiples: 10-24-26, etc.)
  • 8-15-17
  • 7-24-25

10. Geometry – Circles

Circle problems test your understanding of circumference, area, arcs, and the circle equation in coordinate geometry.

TopicFormulaDescription
CircumferenceC = 2πr = πdr = radius, d = diameter
AreaA = πr²r = radius
Arc Lengths = rθθ in radians
Arc Length (degrees)s = (θ/360°) × 2πrθ in degrees
Sector AreaA = ½r²θθ in radians
Sector Area (degrees)A = (θ/360°) × πr²θ in degrees
Chord Lengthc = 2r sin(θ/2)θ = central angle
Circle Equation (Standard)(x – h)² + (y – k)² = r²Center (h, k), radius r
Circle Equation (General)x² + y² + Dx + Ey + F = 0Expand to find center/radius
Inscribed Angleθ = ½ × central angleAngle on circumference
Tangent LinePerpendicular to radius at pointForms 90° with radius

11. Geometry – Quadrilaterals

TopicFormulaDescription
Rectangle AreaA = lwl = length, w = width
Rectangle PerimeterP = 2l + 2wSum of all sides
Square AreaA = s²s = side length
Square PerimeterP = 4sFour equal sides
Square Diagonald = s√2Diagonal of square
Parallelogram AreaA = bhb = base, h = height
Parallelogram (alt)A = ab sin(θ)a, b = sides, θ = angle
Trapezoid AreaA = ½(b₁ + b₂)hb₁, b₂ = parallel sides, h = height
Rhombus AreaA = ½d₁d₂d₁, d₂ = diagonals
Kite AreaA = ½d₁d₂d₁, d₂ = diagonals
Sum of Interior Angles(n – 2) × 180°n = number of sides

12. Solid Geometry (3D Shapes)

Three-dimensional geometry problems require you to calculate volume and surface area of common solids.

TopicFormulaDescription
Rectangular Prism VolumeV = lwhl = length, w = width, h = height
Rectangular Prism SASA = 2(lw + lh + wh)Surface area
Cube VolumeV = s³s = side length
Cube Surface AreaSA = 6s²Six square faces
Cylinder VolumeV = πr²hr = radius, h = height
Cylinder Surface AreaSA = 2πr² + 2πrh = 2πr(r + h)Two circles + lateral area
Cone VolumeV = ⅓πr²hr = radius, h = height
Cone Surface AreaSA = πr² + πrll = slant height
Sphere VolumeV = (4/3)πr³r = radius
Sphere Surface AreaSA = 4πr²r = radius
Pyramid VolumeV = ⅓BhB = base area, h = height

13. Trigonometry – Right Triangles

Remember: SOH-CAH-TOA! This mnemonic helps you remember the basic trig ratios.

TopicFormulaDescription
Sinesin(θ) = opposite/hypotenuseSOH
Cosinecos(θ) = adjacent/hypotenuseCAH
Tangenttan(θ) = opposite/adjacentTOA
Cosecantcsc(θ) = 1/sin(θ) = hypotenuse/oppositeReciprocal of sine
Secantsec(θ) = 1/cos(θ) = hypotenuse/adjacentReciprocal of cosine
Cotangentcot(θ) = 1/tan(θ) = adjacent/oppositeReciprocal of tangent

14. Trigonometry – Identities

TopicFormulaDescription
Pythagorean Identitysin²(θ) + cos²(θ) = 1Fundamental identity
Pythagorean (tan/sec)1 + tan²(θ) = sec²(θ)Divide by cos²(θ)
Pythagorean (cot/csc)1 + cot²(θ) = csc²(θ)Divide by sin²(θ)
Quotient Identity (tan)tan(θ) = sin(θ)/cos(θ)Definition
Quotient Identity (cot)cot(θ) = cos(θ)/sin(θ)Definition
Co-function (sine/cosine)sin(90° – θ) = cos(θ)Complementary angles
Co-function (tan/cot)tan(90° – θ) = cot(θ)Complementary angles

15. Special Angles & Unit Circle

Memorize these special angle values – they appear constantly on the SAT!

Anglesincostan
010
30°1/2√3/2√3/3 = 1/√3
45°√2/2√2/21
60°√3/21/2√3
90°10undefined
🎯 Memory Trick: For sine values at 0°, 30°, 45°, 60°, 90°, remember: √0/2, √1/2, √2/2, √3/2, √4/2
TopicFormulaDescription
Radian Conversionradians = (π/180°) × degreesDegree to radian
Degree Conversiondegrees = (180°/π) × radiansRadian to degree
Common Conversionsπ rad = 180°, π/2 = 90°, π/3 = 60°, π/4 = 45°, π/6 = 30°Key angles
Arc Lengths = rθθ in radians
Unit Circlex² + y² = 1Radius = 1
Coordinates(cos(θ), sin(θ))Point on unit circle

16. Coordinate Geometry

TopicFormulaDescription
Distance Formulad = √[(x₂ – x₁)² + (y₂ – y₁)²]Distance between points
Midpoint FormulaM = ((x₁ + x₂)/2, (y₁ + y₂)/2)Midpoint between points
Section Formula (Internal)x = (mx₂ + nx₁)/(m + n)
y = (my₂ + ny₁)/(m + n)
Divides segment m:n internally
Slope Formulam = (y₂ – y₁)/(x₂ – x₁)Rise over run
Collinear PointsSlope AB = Slope BCThree points on same line
Area of TriangleA = ½|x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)|Using coordinates

17. Statistics – Measures of Central Tendency

TopicFormulaDescription
Mean (Average)x̄ = (Σx)/nSum divided by count
Weighted Meanx̄ = (Σwx)/(Σw)w = weights
MedianMiddle value when ordered50th percentile
ModeMost frequent valueCan have multiple modes
RangeRange = Max – MinSpread of data

18. Statistics – Measures of Spread

TopicFormulaDescription
Variance (Population)σ² = Σ(x – μ)²/NN = population size
Variance (Sample)s² = Σ(x – x̄)²/(n – 1)n = sample size
Standard Deviation (Pop)σ = √[Σ(x – μ)²/N]Square root of variance
Standard Deviation (Sample)s = √[Σ(x – x̄)²/(n – 1)]Square root of variance
Interquartile RangeIQR = Q₃ – Q₁Middle 50% spread
Outlier (Lower)x < Q₁ - 1.5(IQR)Below lower fence
Outlier (Upper)x > Q₃ + 1.5(IQR)Above upper fence

19. Probability

TopicFormulaDescription
Basic ProbabilityP(A) = (favorable outcomes)/(total outcomes)0 ≤ P(A) ≤ 1
Complement RuleP(A’) = 1 – P(A)Probability of NOT A
Addition Rule (Mutually Exclusive)P(A or B) = P(A) + P(B)Events cannot occur together
Addition Rule (General)P(A or B) = P(A) + P(B) – P(A and B)Avoid double counting
Multiplication Rule (Independent)P(A and B) = P(A) × P(B)Events don’t affect each other
Conditional ProbabilityP(A|B) = P(A and B)/P(B)Probability of A given B
PermutationsₙPᵣ = n!/(n – r)!Order matters
CombinationsₙCᵣ = n!/[r!(n – r)!]Order doesn’t matter
Factorialn! = n × (n-1) × (n-2) × … × 10! = 1 by definition

20. Sequences & Series

TopicFormulaDescription
Arithmetic Sequenceaₙ = a₁ + (n – 1)dd = common difference
Arithmetic SumSₙ = n(a₁ + aₙ)/2Sum of n terms
Arithmetic Sum (alt)Sₙ = n[2a₁ + (n – 1)d]/2Alternative formula
Geometric Sequenceaₙ = a₁ · r^(n-1)r = common ratio
Geometric Sum (finite)Sₙ = a₁(1 – rⁿ)/(1 – r)r ≠ 1
Geometric Sum (infinite)S = a₁/(1 – r)|r| < 1 only

21. Complex Numbers (Advanced)

TopicFormulaDescription
Imaginary Uniti = √(-1), i² = -1Definition
Complex Numberz = a + bia = real part, b = imaginary part
Complex Conjugatez̄ = a – biConjugate of z
Addition(a + bi) + (c + di) = (a + c) + (b + d)iAdd real and imaginary parts
Multiplication(a + bi)(c + di) = (ac – bd) + (ad + bc)iFOIL and use i² = -1
Modulus|z| = √(a² + b²)Distance from origin

22. Matrices (Advanced)

TopicFormulaDescription
Matrix Addition[aᵢⱼ] + [bᵢⱼ] = [aᵢⱼ + bᵢⱼ]Add corresponding elements
Scalar Multiplicationk[aᵢⱼ] = [k·aᵢⱼ]Multiply each element by k
Matrix Multiplication(AB)ᵢⱼ = Σ(aᵢₖ·bₖⱼ)Row × column
Identity MatrixI = [1 0; 0 1] for 2×2AI = IA = A
Determinant (2×2)det(A) = ad – bcFor A = [a b; c d]

23. Ratio, Proportion & Variation

TopicFormulaDescription
Ratioa:b or a/bComparison of quantities
Proportiona/b = c/d or a:b = c:dEqual ratios
Cross MultiplicationIf a/b = c/d, then ad = bcSolving proportions
Direct Variationy = kx or y/x = kk = constant of variation
Inverse Variationy = k/x or xy = kProduct is constant
Joint Variationz = kxyz varies with both x and y
Combined Variationz = kx/yDirect with x, inverse with y
Scale FactorNew/Original = kFor similar figures

24. Percent, Percent Change & Applications

TopicFormulaDescription
PercentPercent = (Part/Whole) × 100%Convert to percentage
Percent of a NumberPart = (Percent/100) × WholeFind the part
Percent Change% Change = [(New – Old)/Old] × 100%Increase or decrease
Percent IncreaseNew = Old × (1 + r/100)r = percent increase
Percent DecreaseNew = Old × (1 – r/100)r = percent decrease
Simple InterestI = PrtP = principal, r = rate, t = time
Total with InterestA = P(1 + rt)Simple interest
MarkupSelling Price = Cost × (1 + markup%)Retail pricing
DiscountSale Price = Original × (1 – discount%)Reduced price

25. Rate, Work & Mixture Problems

TopicFormulaDescription
Distance = Rate × Timed = rtUniform motion
Average SpeedAverage = Total Distance/Total TimeNOT average of speeds
Work RateWork = Rate × TimeW = rt
Combined Work1/t = 1/t₁ + 1/t₂Two workers together
Mixture FormulaAmount × Concentration = Pure quantityFor solutions

26. Important Constants & Values

ConstantValueDescription
π (Pi)≈ 3.14159 or 22/7Circle constant
e (Euler’s number)≈ 2.71828Natural exponential base
√2≈ 1.414Square root of 2
√3≈ 1.732Square root of 3
Golden Ratio (φ)≈ 1.618(1 + √5)/2

27. Useful Calculation Shortcuts

TopicFormulaDescription
Sum of First n Integers1 + 2 + 3 + … + n = n(n + 1)/2Quick sum
Sum of First n Odd1 + 3 + 5 + … + (2n-1) = n²Odd numbers
Sum of First n Even2 + 4 + 6 + … + 2n = n(n + 1)Even numbers
Sum of Squares1² + 2² + … + n² = n(n + 1)(2n + 1)/6Squared integers
Difference of Squaresa² – b² = (a + b)(a – b)Quick factoring

🎯 Top 10 Tips for SAT Math Success

  1. Memorize special right triangles: The 30-60-90 and 45-45-90 triangles appear frequently
  2. Know Pythagorean triples: Recognize 3-4-5, 5-12-13, 8-15-17 instantly
  3. Master the quadratic formula: It’s NOT given on the test
  4. Practice unit circle values: Know sin, cos, tan for 0°, 30°, 45°, 60°, 90°
  5. Understand slope concepts: Parallel lines (same slope), perpendicular lines (negative reciprocal)
  6. Master percent change: Growth/decay problems are common
  7. Know circle equations: Both standard (x-h)²+(y-k)²=r² and general form
  8. Be comfortable with functions: Function notation, composition, transformations
  9. Review exponent rules: These appear in multiple contexts
  10. Practice word problems: Learn to translate English to equations efficiently

Final Thoughts

This comprehensive formula sheet covers all the mathematical concepts tested on the SAT. While memorizing formulas is important, understanding when and how to apply them is even more crucial.

Here are some final study tips:

  • Practice regularly: Review these formulas daily, not just the night before the test
  • Create flashcards: Make cards for formulas you struggle with most
  • Apply them: Work through practice problems to see these formulas in action
  • Understand, don’t just memorize: Know WHY formulas work, not just WHAT they are
  • Take timed practice tests: Speed and accuracy both matter on the SAT

Remember, the SAT provides some formulas at the beginning of each math section (like area formulas for basic shapes), but most of these formulas you’ll need to know by heart. Focus especially on the quadratic formula, special right triangles, and trigonometric ratios.

💡 Pro Tip: Print this formula sheet and keep it by your study desk. Review it for 10-15 minutes each day, and you’ll have all these formulas memorized in no time!

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