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ACT Math Formulas to Memorize

The ACT does not give you a formula sheet. These are the formulas worth knowing cold, grouped by topic.

Algebra

  • Slope: m = (y₂ − y₁) ÷ (x₂ − x₁). Slope-intercept form: y = mx + b.
  • Distance: √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint: the average of the x-values and the average of the y-values.
  • Quadratic formula: x = [−b ± √(b² − 4ac)] ÷ 2a. The vertex of y = ax² + bx + c is at x = −b ÷ 2a.
  • Difference of squares: a² − b² = (a − b)(a + b). Perfect squares: (a + b)² = a² + 2ab + b².
  • Exponents: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1 ÷ aⁿ, and a^(m/n) is the nth root of aᵐ.
  • Circle: (x − h)² + (y − k)² = r², with center (h, k) and radius r.
  • Arithmetic sequence: aₙ = a₁ + (n − 1)d. Geometric sequence: aₙ = a₁ · rⁿ⁻¹.

Geometry

  • Triangle area = ½ × base × height. Pythagorean theorem: a² + b² = c².
  • 45-45-90 triangle: sides in the ratio 1 : 1 : √2. 30-60-90 triangle: sides in the ratio 1 : √3 : 2.
  • Sum of interior angles of a polygon with n sides: (n − 2) × 180°.
  • Circle: area = πr², circumference = 2πr. Arc length = (θ ÷ 360) × 2πr, and sector area = (θ ÷ 360) × πr².
  • Rectangular box volume = length × width × height. Cylinder volume = πr²h. Cone volume = ⅓πr²h. Sphere volume = ⁴⁄₃πr³.

Trigonometry

  • SOH-CAH-TOA: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.
  • sin²θ + cos²θ = 1.
  • Law of cosines: c² = a² + b² − 2ab·cos C.
  • Special values: sin 30° = ½, sin 45° = √2 ÷ 2, sin 60° = √3 ÷ 2. The cosine values are the same list in reverse order.

Statistics, probability and percent

  • Mean = sum ÷ number of values. The median is the middle value after sorting. The mode is the most frequent value.
  • Probability = favorable outcomes ÷ total outcomes.
  • Combinations: nCr = n! ÷ [r!(n − r)!]. Use them when order does not matter.
  • Percent change = (new − old) ÷ old × 100.
  • Distance = rate × time.

Test yourself

  1. In a 30-60-90 triangle, the shortest side has length 6. What is the length of the hypotenuse?

    • A. 6√2
    • B. 6√3
    • C. 12
    • D. 12√3
    • E. 18
    Show answer and solution

    Answer: C. In a 30-60-90 triangle the sides are in the ratio 1 : √3 : 2, and the shortest side is the 1. The hypotenuse is the 2, so it is 2 × 6 = 12.

  2. What is the distance between the points (1, 2) and (7, 10)?

    • A. 8
    • B. 10
    • C. 12
    • D. 14
    • E. √28
    Show answer and solution

    Answer: B. The horizontal change is 6 and the vertical change is 8. The distance is √(6² + 8²) = √(36 + 64) = √100 = 10.

Knowing a formula is only half the skill. Practice choosing the right one fast. Try the ACT Math practice questions.

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