Math Formula Reference — 100+ Formulas, Search & Filter

Roots of ax² + bx + c = 0 when a ≠ 0.

Δ > 0: two real roots; Δ = 0: one repeated root; Δ < 0: two complex roots.

Sum and product of roots of ax² + bx + c = 0.

Factor the difference of two perfect squares.

Expansion of the square of a sum.

Expansion of the square of a difference.

Expansion of the cube of a sum.

Expansion of the cube of a difference.

Factor the sum of two cubes.

Factor the difference of two cubes.

Expansion of (a + b)ⁿ using binomial coefficients C(n,k) = n! / (k!(n−k)!).

nth term of an arithmetic sequence with first term a₁ and common difference d.

Sum of the first n terms of an arithmetic sequence.

nth term with first term a₁ and common ratio q.

Sum of the first n terms when the common ratio q ≠ 1.

Sum to infinity exists only when |q| < 1.

Log of a product is the sum of logs.

Log of a quotient is the difference of logs.

Log of a power moves the exponent out front.

Convert between log bases — c is any new base.

When multiplying same bases, add exponents.

Absolute value of a sum is at most the sum of absolute values.

Rewrite a quadratic in vertex form to find its minimum or maximum.

Vertex of y = ax² + bx + c; the axis of symmetry is x = −b/2a.

Recognising the expanded cube of a difference in factoring problems.

A sum of squares factors over the complex numbers, with i² = −1.

When dividing same bases, subtract the exponents.

Raising a power to a power multiplies the exponents.

A product raised to a power distributes over the factors.

A negative exponent means reciprocal of the positive power.

A fractional exponent links powers and roots.

Two boundary values every log obeys.

Exponential and logarithm undo each other with the same base.

Closed form for the sum of the first n positive integers.

Closed form for the sum of the first n perfect squares.

The sum of the first n cubes equals the square of their sum.

Arithmetic mean is at least the geometric mean; equal when a = b.

A foundational inequality; equality when the vectors are proportional.

Distance of a complex number from the origin in the plane.

Multiplying by the conjugate gives a real number.

Powers of the imaginary unit cycle with period 4.

Determinant of a 2×2 matrix; zero means the matrix is singular.

Inverse of a 2×2 matrix when ad − bc ≠ 0.

Solve a 2×2 linear system using ratios of determinants, D ≠ 0.

Half base times height. Works for any triangle.

Area of a triangle from its three side lengths.

In a right triangle, the square of the hypotenuse equals the sum of squares of legs.

Area of an equilateral triangle with side a.

Length times width.

Twice the sum of length and width.

Side squared.

Base times perpendicular height.

Average of the two parallel sides times the height.

Half the product of the two diagonals.

π times radius squared.

Diameter times π, or 2π times radius.

Half radius squared times central angle in radians.

Radius times central angle in radians.

Semi-major axis a times semi-minor axis b times π.

Edge length cubed.

Six times the area of one face.

Product of the three edge lengths.

Twice the sum of the three pairwise face areas.

Base area times height.

Two circular bases plus lateral surface.

One-third base area times height.

Base plus lateral surface; ℓ is the slant height.

Four-thirds π times radius cubed.

Four π times radius squared.

Euclidean distance between (x₁, y₁) and (x₂, y₂).

Midpoint of the segment between two points.

Rise over run between two points.

The three interior angles of any triangle sum to a straight angle.

Sum of interior angles of a convex polygon with n sides.

Exterior angles of any convex polygon always sum to 360°.

n sides of length s, with apothem a (distance to a side midpoint).

One-third base area times perpendicular height.

Base area times height for any straight prism.

Volume of a truncated cone with base radii R and r.

Volume of a cap of height h cut from a sphere of radius R.

R is the center-to-tube distance, r the tube radius.

Surface area of a ring torus with radii R and r.

Euclidean distance between two points in space.

Equation of a line through (x₁, y₁) with slope k.

k is the slope, b the y-intercept.

Distance from point (x₀, y₀) to the line Ax + By + C = 0.

Circle centered at (a, b) with radius r.

Ellipse centered at the origin with semi-axes a and b.

Two non-vertical lines are perpendicular when slopes multiply to −1.

For similar figures with length ratio k, areas scale by k² and volumes by k³.

An inscribed angle is half the central angle on the same arc.

Area between a chord and its arc, central angle θ in radians.

Area of a regular hexagon with side a.

Volume of a regular tetrahedron with edge a.

Standard values at 30° (π/6).

Standard values at 45° (π/4).

Standard values at 60° (π/3).

Standard values at 90° (π/2). tan is undefined.

The fundamental trig identity, true for any θ.

Derived from sin²+cos²=1 by dividing by cos².

Derived from sin²+cos²=1 by dividing by sin².

tan equals sin divided by cos.

In any triangle, sides are proportional to sines of opposite angles. R is the circumradius.

Generalisation of the Pythagorean theorem to any triangle.

Area from two sides and the included angle.

Sine of a sum identity.

Sine of a difference identity.

Cosine of a sum identity.

Cosine of a difference identity.

Tangent of a sum identity.

Double-angle identity for sine.

Three equivalent forms of the cosine double-angle identity.

Double-angle identity for tangent.

Sign depends on the quadrant of θ/2.

Sign depends on the quadrant of θ/2.

Convert sum of sines to product.

Convert product of sin and cos to sum.

Convert between degrees and radians.

Standard values at 0°.

A function of an angle equals the cofunction of its complement.

Sine and tangent are odd; cosine is even.

Tangent of a difference identity.

Convert a difference of sines to a product.

Convert a sum of cosines to a product.

Convert a difference of cosines to a product.

Convert a product of cosines to a sum.

Convert a product of sines to a difference.

Two equivalent half-angle forms for tangent.

Reduce a squared sine to first power, handy for integration.

Reduce a squared cosine to first power.

Definitions of the three reciprocal trig functions.

Triple-angle identity for sine.

Triple-angle identity for cosine.

Area from the three sides and circumradius R.

Area equals inradius r times the semiperimeter s.

Limit definition of the derivative.

Derivative of x to the n.

A constant has derivative zero.

Derivative of sine is cosine.

Derivative of cosine is negative sine.

Derivative of tangent.

eˣ is its own derivative — that's what makes e special.

General exponential function.

Natural log differentiates to reciprocal.

General log derivative.

Derivative of a sum is the sum of derivatives.

Differentiation of a product.

Differentiation of a quotient.

Differentiation of a composite function.

Reverse power rule for integration. Add the integration constant C.

The exception to the power rule when n = −1.

eˣ is its own antiderivative too.

Antiderivative of sine.

Antiderivative of cosine.

Reverse of the product rule. Pick u and dv so the new integral is simpler.

Substitute an inner function to simplify the integral.

Connects differentiation and integration.

Square root differentiates to one over twice the root.

Reciprocal differentiates to negative one over x squared.

Derivative of cotangent.

Derivative of secant.

Derivative of inverse sine, |x| < 1.

Derivative of inverse tangent.

Constants factor out of differentiation.

Antiderivative of secant squared.

A standard integral yielding the arctangent.

A standard integral yielding the arcsine, |x| < 1.

Found via integration by parts.

Differentiate both sides treating y as a function of x.

For indeterminate 0/0 or ∞/∞, take the limit of the ratio of derivatives.

Expand a smooth function as a power series about a.

Power series of the exponential about 0, valid for all x.

Length of the graph of f from x = a to x = b.

Rotate y = f(x) about the x-axis from a to b.

Sum of all values divided by the count.

The middle value when sorted; average the two centre values if n is even.

The most frequent value; a dataset may have several modes or none.

Average squared deviation from the population mean μ.

Use n − 1 (Bessel correction) when estimating from a sample.

Square root of the variance — same units as the data.

Difference between the largest and smallest values.

Measures how two variables change together. Sign tells direction; magnitude is unit-dependent.

Unitless correlation in [−1, 1]. r = 1 perfect positive, r = −1 perfect negative.

How many standard deviations a value is from the mean.

The classic bell curve, fully characterized by μ and σ.

Approximate areas under the normal curve at 1/2/3 standard deviations.

Probability of exactly k successes in n independent trials with success probability p.

Expected value and variance of a binomial random variable.

Number of ordered arrangements of k items chosen from n.

Number of unordered selections of k items from n.

Weighted average of all possible outcomes by their probabilities.

Update probabilities given new evidence.

Average where each value carries a weight wᵢ.

nth root of the product; suited to rates and growth factors.

Reciprocal of the average of reciprocals; used for average rates.

Spread of the middle 50% of the data.

Relative dispersion; compares spread across different scales.

How much a sample mean is expected to vary from the true mean.

For 95% confidence with known σ, z ≈ 1.96.

Probability of k events when the mean rate is λ.

Probability the first success occurs on trial k.

Probability of A given that B has occurred.

For independent events, joint probability is the product.

Probability that A or B occurs, avoiding double counting.

Probability that A does not occur.

Average of absolute deviations from the mean.

Slope of the best-fit line in simple linear regression.

Fraction of variance in y explained by the model, in [0, 1].