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SAT & ACT math formula sheet

The formulas you'll actually use on the test, organized by how often they show up.

The SAT provides a small reference sheet at the start of each math section; the ACT does not. In both cases, the formulas you actually need to have committed to memory are short. The list below is ordered roughly by how often each formula shows up on a typical test.

Algebra

  • Slope: m = (y₂ − y₁) / (x₂ − x₁)
  • Slope-intercept form: y = mx + b
  • Point-slope form: y − y₁ = m(x − x₁)
  • Distance between two points: d = √((x₂ − x₁)² + (y₂ − y₁)²)
  • Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
  • Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
  • Vertex of parabola: x = −b/(2a)
  • Sum of roots of ax² + bx + c: −b/a
  • Product of roots: c/a

Geometry

  • Pythagorean theorem: a² + b² = c²
  • Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
  • Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x
  • Area of triangle: ½ × base × height
  • Area of circle: πr²
  • Circumference of circle: 2πr
  • Arc length: (θ/360) × 2πr (θ in degrees) or rθ (θ in radians)
  • Sector area: (θ/360) × πr²
  • Surface area of rectangular prism: 2(lw + lh + wh)
  • Volume of rectangular prism: l × w × h
  • Volume of cylinder: πr²h
  • Volume of cone: ⅓ πr²h
  • Volume of sphere: 4/3 πr³

Trigonometry

  • SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
  • Pythagorean identity: sin²θ + cos²θ = 1
  • tan θ = sin θ / cos θ
  • Unit circle key values: sin 0°=0, sin 30°=½, sin 45°=√2/2, sin 60°=√3/2, sin 90°=1 (cos values are the mirror)
  • Radian conversion: π radians = 180°

Statistics

  • Mean: (sum of values) / (number of values)
  • Median: middle value of sorted list (or average of the two middle values)
  • Standard deviation: not formula-tested, only conceptual — bigger = more spread
  • Probability: favorable outcomes / total outcomes
  • Compound events (independent): P(A and B) = P(A) × P(B)

Exponents and logarithms

  • Product rule: aˣ · aʸ = aˣ⁺ʸ
  • Quotient rule: aˣ / aʸ = aˣ⁻ʸ
  • Power rule: (aˣ)ʸ = aˣʸ
  • Negative exponent: a⁻ˣ = 1/aˣ
  • Fractional exponent: a^(p/q) = ⁿ√(aᵖ) where n = q

Memorize, don't derive

On test day, every second you spend re-deriving a formula is a second you don't have to spend reading the next problem carefully. The list above is short for a reason: it's the floor of what you must have automatic. Drill it until you can recite each formula in under two seconds. Print this page, fold it into your notebook, and quiz yourself on it twice a week until you don't need to look.