📑 ACT reference library

Math formulas to know cold

Plan on every formula below coming out of your own memory during the section, because a formula you half-remember costs exactly what one you never learned costs. ACT does say that all of the mathematics problems can be solved without a calculator — and many are faster without one — so the formula, not the machine, is the part you cannot do without.

Lines & coordinate geometry

• Slope = (y₂ − y₁) / (x₂ − x₁)  — rise over run
• Slope-intercept form: y = mx + b  (m = slope, b = y-intercept)
• Point-slope form: y − y₁ = m(x − x₁)  (use when you have one point and the slope)
• Standard form: Ax + By = C  →  slope = −A/B, y-intercept = C/B
• Distance between two points: √[(x₂ − x₁)² + (y₂ − y₁)²]
• Midpoint: ((x₁ + x₂)/2 , (y₁ + y₂)/2)  — average the x's, average the y's
• Parallel lines have the same slope. Perpendicular slopes are negative reciprocals (m and −1/m).
• No solution = parallel lines (same slope, different intercept). Infinitely many solutions = the same line twice.

Quadratics

• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)  for ax² + bx + c = 0
• Discriminant = b² − 4ac.  Positive → two real solutions · zero → one · negative → no real solutions (two complex ones).
• Vertex: x = −b / (2a); in vertex form a(x − h)² + k the vertex is (h, k)
• Factored form a(x − p)(x − q): the solutions (x-intercepts) are p and q
• Sum of the roots = −b/a  ·  Product of the roots = c/a
• a > 0 opens up (vertex is the minimum); a < 0 opens down (vertex is the maximum)

Exponents & radicals

• xᵃ · xᵇ = xᵃ⁺ᵇ  ·  xᵃ / xᵇ = xᵃ⁻ᵇ  ·  (xᵃ)ᵇ = xᵃᵇ
• (xy)ᵃ = xᵃyᵃ  ·  (x/y)ᵃ = xᵃ/yᵃ
• x⁰ = 1 (for x ≠ 0)  ·  x⁻ᵃ = 1/xᵃ
• Fractional exponents: x^(1/2) = √x  ·  x^(a/b) = ᵇ√(xᵃ)
• Radicals: √(ab) = √a · √b  ·  √(a/b) = √a / √b
• √(a + b) is not √a + √b, and (x + y)² is not x² + y² — it is x² + 2xy + y².
• Difference of squares: x² − y² = (x + y)(x − y)

Area & perimeter

• Rectangle: area = lw, perimeter = 2(l + w)  ·  Square: area = s², perimeter = 4s
• Parallelogram: area = bh (h is the perpendicular height, not the slanted side)
• Triangle: area = ½ × base × height  ·  angles sum to 180°
• Trapezoid: area = ½(b₁ + b₂)h  — average the two parallel sides, times the height
• Any polygon: interior angles sum to 180(n − 2)°; each angle of a regular n-gon is 180(n − 2)/n

Volume & surface area

• Rectangular box: V = lwh  ·  SA = 2(lw + lh + wh)
• Cube: V = s³  ·  SA = 6s²
• Cylinder: V = πr²h  ·  SA = 2πr² + 2πrh
• Sphere: V = (4/3)πr³  ·  SA = 4πr²
• Cone: V = (1/3)πr²h  ·  SA = πr² + πrl (l = slant height)
• Pyramid: V = (1/3) × (area of the base) × height
• Any prism: V = (area of the base) × height

Circles

• Area = πr²  ·  Circumference = 2πr = πd
• Equation of a circle: (x − h)² + (y − k)² = r², center (h, k), radius r
• Arc length = (central angle / 360) × 2πr  — a fraction of the way around
• Sector area = (central angle / 360) × πr²  — the same fraction of the area
• In radians: arc length = rθ  ·  sector area = ½r²θ
• A full circle is 360° = 2π radians. An inscribed angle is half the central angle standing on the same arc.
• A radius drawn to the point of tangency is perpendicular to the tangent line.

Triangles

• Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Triples worth recognizing: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples: 6-8-10, 9-12-15).
• 45-45-90: sides are x, x, x√2 — the legs match, the hypotenuse is a leg times √2.
• 30-60-90: sides are x, x√3, 2x — the short leg is opposite 30°, the hypotenuse is twice it.
• Similar triangles: equal angles, and matching sides in proportion. Find the scale factor first; areas scale by the square of it.
• Area from two sides and the angle between them: area = ½ab·sin C
• The exterior angle equals the sum of the two remote interior angles.

Trigonometry

• SOH-CAH-TOA: sin = opposite/hypotenuse · cos = adjacent/hypotenuse · tan = opposite/adjacent
• Reciprocals: csc = 1/sin · sec = 1/cos · cot = 1/tan
• tan θ = sin θ / cos θ  ·  sin²θ + cos²θ = 1
• Degrees ↔ radians: 180° = π radians. Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
• Law of sines: a / sin A = b / sin B = c / sin C  — use it when you have a side and its opposite angle.
• Law of cosines: c² = a² + b² − 2ab·cos C  — use it for two sides and the included angle, or for all three sides. (With C = 90°, cos C = 0 and it collapses back to the Pythagorean theorem.)
• SOH-CAH-TOA only works in a right triangle. For any other triangle, reach for the law of sines or the law of cosines.

The unit circle at the common angles

On the unit circle a point is (cos θ, sin θ) — cosine is the x-coordinate, sine is the y-coordinate. That is why cos 90° = 0 and sin 90° = 1, and why tan is undefined wherever cos = 0.

Sequences

• Arithmetic (add the same amount d each time): aₙ = a₁ + (n − 1)d
• Sum of the first n terms: Sₙ = n(a₁ + aₙ)/2  — the number of terms times the average of the first and last
• Geometric (multiply by the same ratio r each time): aₙ = a₁ · r⁽ⁿ⁻¹⁾
• Sum of the first n terms: Sₙ = a₁(1 − rⁿ)/(1 − r), for r ≠ 1
• Infinite geometric sum, when |r| < 1: S = a₁/(1 − r)
• The (n − 1) is the usual slip: the first term has already happened, so you add d or multiply by r one time fewer than you expect.

Counting: permutations & combinations

• Order matters → permutation: ₙPᵣ = n! / (n − r)!
• Order does not matter → combination: ₙCᵣ = n! / (r!(n − r)!)
• Fundamental counting principle: multiply the number of choices at each step (4 shirts × 3 pants = 12 outfits).
• n! = n × (n − 1) × … × 2 × 1, and 0! = 1.
• Probability = favorable outcomes ÷ total outcomes. P(not A) = 1 − P(A).

Percent, interest & growth

• Percent change = (new − old) / old × 100  — always divide by the original.
• An increase of p% multiplies by (1 + p/100); a decrease multiplies by (1 − p/100). A 20% rise then a 20% fall does not return you to the start (1.20 × 0.80 = 0.96).
• Percent of: "is" over "of" — part / whole = percent / 100.
• Simple interest: I = Prt, so the balance is A = P(1 + rt)
• Compound interest: A = P(1 + r/n)⁽ⁿᵗ⁾  (P = principal, r = annual rate as a decimal, n = compoundings per year, t = years)
• Exponential growth or decay: final = start × (growth factor)^(time ÷ period), where the factor is 1 + r for growth and 1 − r for decay.
• Mean = sum ÷ count → sum = mean × count · Median = the middle value in order · Mode = the most frequent · Range = max − min
• Distance = rate × time · Average speed = total distance ÷ total time (not the average of the two speeds)