Spherical Cap Volume & Surface Area Calculator

Percent of, percent change, and percent add/subtract in one.

Enter 2–6 positive integers to get the greatest common divisor (HCF / GCD) and least common multiple (LCM), with the Euclidean step chain shown.

Enter a list of numbers to get the mean, median, mode, range plus standard deviation, variance and total.

Enter the coefficients of ax² + bx + c = 0 to find the real or complex roots, discriminant and vertex.

Estimate a password's bit entropy, brute-force time and strength tier. All computation happens in your browser.

Convert between standard decimal numbers and scientific notation, with significant figures and order of magnitude.

Compute permutations P(n,r), combinations C(n,r) and factorial n! — useful for probability problems, lottery odds and combinatorics homework.

Paste a list of numbers to compute mean, median, sample and population variance and standard deviation — with the working shown.

Solve a triangle from 3 sides, 2 sides + 1 angle, or 2 angles + 1 side — area, perimeter and remaining parts via the law of sines / cosines.

Given any two sides of a right triangle (two legs, or one leg plus the hypotenuse), instantly find the third side, area, perimeter and the two non-right angles.

Enter any one of radius, diameter, circumference or area to get the other three — useful for design, engineering and DIY.

Two-way conversion between Arabic numbers (1–3999) and Roman numerals (I, V, X, L, C, D, M) — handy for typesetting, chapter titles and homework.

Enter two points (x₁, y₁) and (x₂, y₂) to instantly get the slope, y-intercept, line equation y = mx + b, distance, and midpoint — a classroom staple for algebra and coordinate geometry.

Enter group size n to see the probability that at least two people share a birthday — the classic birthday problem.

Compute logₐ(x) for any base — natural log (ln), common log (log₁₀), binary log (log₂) and a custom base, with the change-of-base steps shown.

Enter a value, the mean and the standard deviation to compute the z-score and the corresponding normal-distribution percentile and probabilities.

Enter the screen resolution and diagonal size to get pixel density (PPI), real width/height, dot pitch and total pixel count.

Enter the number of tickets / selections and see the actual probability of hitting first, second … prizes in a Mark Six (6-of-49) draw.

Convert any decimal (including repeating decimals) to a simplified fraction and a mixed number.

Give a sphere any one of radius, diameter, surface area or volume and instantly get the other three — plus the great-circle circumference and area.

Enter the radius and height of a cylinder to get volume (π r²h), lateral surface, base area and total surface area.

Compute nPr — the number of ordered arrangements of r items chosen from n — alongside n!, r! and the related combination nCr.

Factor any integer from 2 up to 10¹² into primes, see the canonical exponent form, count and list all divisors.

Compute the n-th root of the product of n positive numbers — the right average for growth rates, returns and ratios — alongside the arithmetic mean for comparison.

Enter any integer n from 0 to 1500 to instantly compute F(n) and F(n−1) with BigInt precision, the consecutive-term ratio (converging to the golden ratio φ) and the first 30 terms of the sequence.

Pick the number of dice, sides (d4 / d6 / d8 / d10 / d12 / d20) and a target sum to compute the probability of rolling exactly that total, at least, or at most.

Enter first term a, common difference d and number of terms n to compute the nth term aₙ and the partial sum Sₙ = n/2·(2a + (n − 1)d).

Enter confidence level, margin of error, expected proportion (and optional population size) to compute the survey sample size you need.

Enter the first term a, common ratio r and number of terms n to find the sum of the first n terms of a geometric series — plus the infinite-series sum when |r| < 1.

Enter base radius and height to get the cone volume, slant height, lateral, base and total surface area.

Pick a note (C, C♯, D, …), octave and tuning reference A4 (440 Hz by default) and the tool returns the frequency in Hz, the wavelength in air and the MIDI note number via f = A4 × 2^((n − 69)/12).

Enter two known points (x₁, y₁) and (x₂, y₂) and a target x to instantly read off y via y = y₁ + (x − x₁)(y₂ − y₁)/(x₂ − x₁) — the tool flags whether your x sits inside the two points (interpolation) or outside (extrapolation).

Enter the two parallel sides and height of a trapezoid to get its area, midsegment length and perimeter when the legs are known.

Enter the number of trials n, the per-trial success probability p and a target number of successes k to get P(X = k), P(X ≤ k), P(X ≥ k) along with the distribution mean and standard deviation.

Paste two parallel data series (X and Y) and get the Pearson correlation coefficient r, r², the best-fit line slope and intercept, and the sample means and standard deviations.

Enter the sample mean, sample standard deviation, sample size and confidence level to compute a confidence interval for the population mean using the t (or z) distribution, with margin of error and standard error.

Enter the means, standard deviations and sample sizes of two groups to compute Cohen's d and Hedges' g effect sizes, classified per Cohen 1988 as trivial / small / medium / large.

Enter dividend a and divisor n to compute the quotient and remainder a mod n. The truncated (C / JavaScript %), floored (Python / mathematical) and Euclidean conventions are shown side by side so the negative-input differences are obvious.

Toggle read/write/execute for user, group and other to instantly get both octal (e.g. 755) and symbolic (e.g. rwxr-xr-x) notations.

Enter 2D or 3D vector components (x, y, z) to compute the magnitude, unit vector and direction angles.

Enter experimental and theoretical (accepted) values to get absolute percent error, signed relative error, and absolute error. Switch to percent-difference mode to compare two measurements where neither is the accepted value.

Enter the mean event rate λ and an event count k to get P(X = k), P(X ≤ k), P(X ≥ k), the mean, variance and standard deviation — useful for queueing, call-centre staffing and rare-event modelling.

Enter the prior P(A), the sensitivity P(B|A) and the false-positive rate P(B|¬A); the tool applies Bayes’ theorem to give the posterior P(A|B) — handy for medical tests, spam filters and any classification decision.

Enter a circle’s radius and central angle (degrees or radians) to compute the arc length s = r·θ, sector area A = ½·r²·θ and chord length — useful for geometry, machining and architectural layout.

Enter the entries of a 2×2 or 3×3 matrix and instantly get the determinant det(A) via ad − bc or cofactor expansion along the first row, with an invertibility check and the step-by-step 2×2 minors shown for linear-algebra study.

Enter two 2D or 3D vectors and instantly compute the dot product, included angle, scalar projection and vector projection — with hints for orthogonal, parallel and anti-parallel cases.

Enter two numeric vectors of any dimension and compute cos θ = (a·b) / (|a|·|b|), plus the angular distance — the staple similarity metric in NLP, recommender systems and document-similarity scoring.

Enter two values and compute the symmetric percent difference |a − b| / ((|a| + |b|) / 2) alongside the directional percent change (b − a)/a — a common source of confusion in lab reports and news headlines.

Enter two 3D vectors a and b, compute the cross product a × b together with its magnitude (the area of the parallelogram they span) and sin θ — the workhorse of physics torque, computational geometry and 3D graphics.

Enter two points in the plane or in 3D space and the tool returns the straight-line Euclidean distance √Σ(Δᵢ)², each axis delta and the midpoint — handy for geometry homework, CAD measurements and planar GIS work.

Enter a dataset and the tool returns the coefficient of variation CV = σ / μ × 100% (standard deviation divided by mean) — the standard yardstick for comparing dispersion across datasets with different units or magnitudes (lab repeatability, portfolio risk).

Enter a dataset and the tool sorts it and returns the median, first and third quartiles, the inter-quartile range (IQR) and Tukey 1.5×IQR outlier fences — the five-number summary behind every box plot.

Enter a set of positive numbers and compute the harmonic mean HM = n / Σ(1/xᵢ) alongside the arithmetic and geometric means — the right average when the data behaves like rates or denominators (speeds, P/E ratios, flow rates).

Enter a list of class probabilities or frequencies and compute Shannon entropy H = −Σ pᵢ log₂ pᵢ (bits) alongside the maximum entropy and the normalised entropy — the foundation of information theory, decision-tree splits and password analysis.

Enter the side count n and side length s of a regular polygon and the tool returns the interior and exterior angles, the apothem and circumradius, the area and the perimeter in one shot — works for triangles, squares, pentagons, hexagons and beyond.

Enter n and the tool returns the n-th Catalan number Cₙ = (2n)! / ((n+1)! · n!) — the textbook count for balanced parentheses, binary tree shapes, Dyck paths and many other combinatorics problems.

Enter a column of actual values and a column of predicted values; the tool computes MAE, MSE, RMSE, R² and MAPE side-by-side — the staple set of regression-error metrics in machine learning, forecasting and lab-vs-model comparisons — and explains how each one weights outliers differently.

Enter the 6 coefficients of ax + by = e and cx + dy = f. The tool applies Cramer's rule to return x, y, the determinant D and the system's classification (unique solution / no solution / infinitely many), showing the working out so it doubles as a homework check.

Enter a per-trial probability p and a trial count n; the tool returns the probability of at least one success 1 − (1 − p)ⁿ, the expected count np, and the trials needed to reach a target probability — the maths behind raffles, gacha pulls, A/B tests and redundancy design.

Enter observed and expected counts; the tool computes the chi-square statistic χ² = Σ (O − E)² / E, degrees of freedom and the p-value to test whether the observed distribution deviates significantly from theory — the standard test for die fairness, Mendelian ratios and categorical A/B counts.

Enter outcome values x and matching probabilities p; the tool returns the expected value E[X] = Σ p·x, variance Var(X) and standard deviation σ — the foundational metric for casino games, insurance pricing, investment decisions and A/B-test analysis.

Enter sample size n, sample proportion p (or sample standard deviation) and a confidence level (90 / 95 / 99 %); the tool computes the margin of error MOE = z·√(p(1−p)/n) and the matching confidence interval — the core statistic behind political polls, A/B tests and market research.

Enter a list of numbers; the tool returns the mean absolute deviation (MAD = average of |xᵢ − mean|) — a more intuitive spread measure than standard deviation, used in school statistics and demand-forecasting accuracy.

Enter base dimensions and height; the tool returns volume, base area, slant height, lateral surface area and total surface area for square, rectangular, equilateral-triangular or regular n-gon based right pyramids using V = ⅓ × base area × height.

Enter base and height, or two side lengths and the included angle; the tool returns the parallelogram area, perimeter and both diagonals via A = b · h or A = a · b · sin θ.

Enter the semi-major axis a and semi-minor axis b; the tool returns the area A = π a b and uses Ramanujan's second approximation for the perimeter — relative error below 4 × 10⁻⁵ across all eccentricities.

Enter a Z-score; the tool returns the one-sided (left / right) and two-sided p-values, and flags significance against α = 0.05 / 0.01 / 0.001 — the missing step of nearly every hypothesis test.

Paste a list of values; the tool scans the dataset and reports the most-frequent value(s), handling unimodal, bimodal and multimodal cases, with the full frequency table — the standard intro-statistics tool for categorical and discrete data summaries.