Formula Library
A searchable library of formulas for algebra, geometry, trigonometry, statistics, probability, calculus and physics, each with an example.
A searchable library of formulas for algebra, geometry, trigonometry, statistics, probability, calculus and physics, each with an example.
54 formulas
Gives the solutions of a quadratic equation ax² + bx + c = 0.
a, b, cApplies when: a ≠ 0. Real solutions exist when b² − 4ac ≥ 0.
Example: x² − 5x + 6 = 0: x = (5 ± √1)/2, so x = 2 or 3.
Factors the difference of two squares into a product of a sum and a difference.
a, bApplies when: Always valid.
Example: x² − 9 = (x − 3)(x + 3).
Expands the square of a sum or difference.
a, bApplies when: Always valid.
Example: (x + 3)² = x² + 6x + 9.
Expands any whole-number power of a binomial.
\binom{n}{k}Applies when: n is a non-negative integer.
Example: (x + 1)³ = x³ + 3x² + 3x + 1.
Rules for combining powers with the same base.
m, nApplies when: a ≠ 0 for the quotient rule; a > 0 when exponents are not integers.
Example: 2³ · 2⁴ = 2⁷ = 128.
Logarithms turn products into sums and powers into multiples.
x, yApplies when: b > 0, b ≠ 1, x > 0, y > 0.
Example: log₁₀ 200 = log₁₀ 2 + log₁₀ 100 ≈ 0.301 + 2.
Computes a logarithm in any base from natural logarithms.
Applies when: b > 0, b ≠ 1, x > 0.
Example: log₂ 8 = ln 8 / ln 2 = 3.
Adds n terms that increase by a constant difference.
a_1a_nApplies when: n is a positive integer.
Example: 1 + 2 + … + 100 = 50 · 101 = 5050.
Adds terms that are multiplied by a constant ratio.
Applies when: r ≠ 1 for the finite sum; |r| < 1 for the infinite sum.
Example: 1 + ½ + ¼ + … = 1 / (1 − ½) = 2.
Describes a non-vertical line by its slope and where it crosses the y-axis.
Applies when: x₁ ≠ x₂ (the line is not vertical).
Example: Through (1, 2) and (3, 6): m = 2, b = 0, so y = 2x.
Relates the sides of a right triangle.
a, bApplies when: The triangle has a right angle.
Example: Legs 3 and 4 give c = 5.
Area of a triangle from its three sides.
a, b, cApplies when: The sides satisfy the triangle inequality.
Example: Sides 3, 4, 5: s = 6, A = √36 = 6.
The distance around a circle and the area it encloses.
Applies when: r > 0.
Example: r = 2: C ≈ 12.566, A ≈ 12.566.
The length of an arc and the area of the sector it bounds.
Applies when: θ must be in radians.
Example: r = 2, θ = π/2: s = π, A = π.
Average of the parallel sides times the distance between them.
a, bApplies when: a, b, h > 0.
Example: a = 3, b = 5, h = 4: A = 16.
The space inside a sphere and the area of its surface.
Applies when: r > 0.
Example: r = 3: V = 36π ≈ 113.1.
A cone holds one third of the cylinder with the same base and height.
Applies when: r, h > 0.
Example: r = 2, h = 5: cylinder 20π, cone 20π/3.
Straight-line distance between two points in the plane.
(x_1, y_1), (x_2, y_2)Applies when: Always valid.
Example: (0, 0) to (3, 4): d = 5.
The interior angles of any n-sided polygon add up to this total.
Applies when: n ≥ 3, simple polygon.
Example: Hexagon: 4 · 180° = 720°.
The fundamental relation between sine and cosine.
Applies when: Always valid.
Example: sin θ = 3/5 gives cos θ = ±4/5.
In any triangle, each side is proportional to the sine of the opposite angle.
a, b, cA, B, CApplies when: Any triangle.
Example: A = 30°, B = 90°, a = 1: b = sin 90° / sin 30° = 2.
Generalises the Pythagorean theorem to any triangle.
a, bApplies when: Any triangle.
Example: a = 7, b = 9, C = 60°: c² = 49 + 81 − 63 = 67.
Sine and cosine of a sum or difference of angles.
a, bApplies when: Always valid.
Example: sin 75° = sin 45° cos 30° + cos 45° sin 30° ≈ 0.9659.
Trigonometric functions of twice an angle.
Applies when: Always valid.
Example: sin 60° = 2 sin 30° cos 30° = √3/2.
Conversion between the two angle units.
radApplies when: Always valid.
Example: 60° = π/3 rad ≈ 1.0472.
The sum of the values divided by their count.
x_iApplies when: n ≥ 1.
Example: 2, 4, 6, 8: mean = 20/4 = 5.
Typical distance of sample values from their mean.
x_i\bar{x}Applies when: n ≥ 2. Use n in place of n − 1 for a whole population.
Example: 2, 4, 6, 8: s = √(20/3) ≈ 2.582.
How many standard deviations a value lies from the mean.
Applies when: σ > 0.
Example: x = 85, μ = 70, σ = 10: z = 1.5.
Strength and direction of a linear relationship, from −1 to 1.
x_i, y_iApplies when: Neither variable is constant.
Example: x = 1, 2, 3 and y = 2, 4, 6 give r = 1.
The line that minimises the sum of squared vertical errors.
Applies when: The x values are not all equal.
Example: x = 1…5, y = 2, 4, 5, 4, 5: ŷ = 2.2 + 0.6x.
The bell curve with mean μ and standard deviation σ.
Applies when: σ > 0.
Example: Standard normal at 0: f(0) ≈ 0.3989.
The probability that an event does not happen.
A^{c}Applies when: Always valid.
Example: P(no six on one roll) = 1 − 1/6 = 5/6.
Probability that at least one of two events happens.
A, BApplies when: Always valid; the last term is 0 for mutually exclusive events.
Example: King or heart: 4/52 + 13/52 − 1/52 = 16/52.
Probability of A given that B has occurred.
A, BApplies when: P(B) > 0.
Example: Two dice, sum is 8 given first is 3: 1/6.
Reverses a conditional probability.
P(A)P(B \mid A)P(A \mid B)Applies when: P(B) > 0.
Example: Prior 1%, sensitivity 90%, false-positive rate 5%: posterior ≈ 15.4%.
Ordered selections of r items from n without repetition.
Applies when: 0 ≤ r ≤ n, integers.
Example: P(5, 2) = 20.
Unordered selections of r items from n without repetition.
Applies when: 0 ≤ r ≤ n, integers.
Example: C(52, 5) = 2,598,960 poker hands.
Probability of exactly k successes in n independent trials.
Applies when: Independent trials with constant p.
Example: n = 10, p = 0.5, k = 3: ≈ 0.1172.
The long-run average of a random quantity.
x_iP(X = x_i)Applies when: The sum converges.
Example: Fair die: (1 + 2 + … + 6)/6 = 3.5.
The instantaneous rate of change of f at x.
Applies when: The limit exists.
Example: f(x) = x²: ((x + h)² − x²)/h → 2x.
Derivative of a power of x.
Applies when: x > 0 if n is not an integer.
Example: d/dx x³ = 3x².
Derivatives of products and quotients of functions.
f, gApplies when: g ≠ 0 for the quotient rule.
Example: (x sin x)' = sin x + x cos x.
Derivative of a composition of functions.
Applies when: Both functions are differentiable.
Example: d/dx sin(x²) = 2x cos(x²).
Evaluates a definite integral with any antiderivative F of f.
Applies when: f is continuous on [a, b].
Example: ∫₀³ x² dx = 9 − 0 = 9.
The product rule in reverse; trades one integral for another.
Applies when: u and v are differentiable.
Example: ∫ x eˣ dx = x eˣ − eˣ + C.
Represents a function as an infinite polynomial around a point.
Applies when: f is infinitely differentiable and x lies within the radius of convergence.
Example: eˣ = 1 + x + x²/2! + x³/3! + …
Evaluates limits of indeterminate quotients.
f, gApplies when: The original limit has the form 0/0 or ∞/∞ and the right-hand limit exists.
Example: lim sin x / x as x → 0 = lim cos x / 1 = 1.
Position and velocity under constant acceleration.
Applies when: Acceleration is constant.
Example: From rest with a = 2 m/s² for 5 s: s = 25 m, v = 10 m/s.
Net force equals mass times acceleration.
Applies when: Constant mass, inertial frame.
Example: m = 3 kg, a = 2 m/s²: F = 6 N.
Energy of motion.
Applies when: Speeds much lower than the speed of light.
Example: m = 2 kg, v = 3 m/s: 9 J.
Relates voltage, current, resistance and power.
Applies when: Ohmic conductor at constant temperature.
Example: I = 2 A through 5 Ω: V = 10 V, P = 20 W.
Attractive force between two masses: an inverse-square law.
m_1, m_2Applies when: Point masses or spherically symmetric bodies.
Example: Doubling the distance divides the force by 4.
Speed of a wave from its frequency and wavelength.
Applies when: Any periodic wave.
Example: f = 440 Hz, λ = 0.78 m: v ≈ 343 m/s.
A quantity whose rate of change is proportional to its size.
N_0t_{1/2}Applies when: The rate of change is proportional to the amount present.
Example: Half-life 10 years: k = −0.0693 per year.
A reference collection of 54 formulas across algebra, geometry, trigonometry, statistics, probability, calculus and physics-related mathematics. Every entry states what the formula computes, what each symbol means, the conditions under which it applies, and a worked numeric example.
Formulas can be copied as LaTeX, and most link directly to a tool that applies them.
Example
Search “heron”
→ A = √(s(s − a)(s − b)(s − c)) with its variables, conditions and an example