Math & statistics
Circular sector area
Circular sector area: Sector area = π × radius² × central angle / 360 for angles from zero through 360 degrees.
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Math & statistics
Circular arc length
Circular arc length: Arc length = radius × angle in radians = radius × angle in degrees × π/180.
Math & statistics
Ellipse area
Ellipse area: Ellipse area = π × semi-major axis × semi-minor axis.
Math & statistics
Ellipsoid volume
Ellipsoid volume: Ellipsoid volume = (4/3) × π × a × b × c, where a, b and c are the three semi-axes.
Math & statistics
Circular frustum volume
Circular frustum volume: A circular cone frustum has volume πh(R² + Rr + r²)/3.
Math & statistics
Torus volume
Torus volume: For a ring torus, volume = 2π² × major radius × tube radius².
Math & statistics
Distance from a point to a line
Distance from a point to a line: For line ax + by + c = 0, perpendicular distance is |ax₀ + by₀ + c| / √(a² + b²).
Math & statistics
Three-dimensional dot product
Three-dimensional dot product: A·B = AxBx + AyBy + AzBz.
Math & statistics
Three-dimensional cross product
Three-dimensional cross product: A×B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx).
Math & statistics
Angle between two 2D vectors
Angle between two 2D vectors: Angle = arccos[(A·B)/(|A||B|)], returned in degrees from 0 to 180.
Math & statistics
Polar and Cartesian coordinate converter
Polar and Cartesian coordinate converter: Polar to Cartesian: x=r cos θ, y=r sin θ.
Math & statistics
Harmonic mean
Harmonic mean: Harmonic mean = count / Σ(1/value).
Math & statistics
Geometric mean
Geometric mean: Compute exp(mean(ln(value))) for positive observations.
Math & statistics
Coefficient of variation
Coefficient of variation: Sample coefficient of variation = sample standard deviation / arithmetic mean × 100.
Math & statistics
Mean absolute deviation about the mean
Mean absolute deviation about the mean: Find the arithmetic mean, then average the absolute distances of observations from that mean.
Math & statistics
Cohen’s d for independent samples
Cohen’s d for independent samples: Cohen’s d = difference of means divided by pooled sample SD; pooled variance weights each sample variance by n−1.
Math & statistics
Probability of at least one success
Probability of at least one success: For independent equal-probability trials, P(at least one success) = 1−(1−p)^n.
Math & statistics
Fractional odds implied probability
Fractional odds implied probability: Implied probability = denominator / (numerator + denominator) × 100.
Math & statistics
Finite population sample correction
Finite population sample correction: Corrected size = ceiling(n₀ / [1+(n₀−1)/N]).
Math & statistics
Polynomial synthetic division
Polynomial synthetic division: Synthetic division divides a polynomial by x−r.
Math & statistics
Modular exponentiation
Modular exponentiation: Compute base^exponent mod modulus using exponentiation by squaring with exact BigInt integer arithmetic.
Math & statistics
Round to significant figures
Round decimals or scientific notation to 1–30 significant figures, preserve meaningful trailing zeros and see the rounding convention.
Math & statistics
Conversion-rate sample size
Conversion-rate sample size: Approximate sample per variant = 2 × p̄(1−p̄) × (zα + zβ)² ÷ absolute difference².
Math & statistics
Proportion margin of error
Proportion margin of error: Margin of error in percentage points = z × √(p(1−p)/n) × 100.