所有計算機與轉換工具

Find the determinant, trace, rank, eigenvalues and inverse of any 2×2 matrix in one step, with an animated transformation.

Calculate the straight-line Euclidean distance between two points on a plane, plus midpoint, slope and angle.

Calculate a triangle's area from all three side lengths using Heron's formula, plus angles, altitudes, inradius and circumradius.

Solve a 30-60-90 special right triangle from any one side. Instantly get the short leg, long leg, hypotenuse, area and perimeter.

Solve a 45-45-90 isosceles right triangle from either side. Get the two equal legs, hypotenuse, area, and perimeter instantly.

Add, subtract, multiply and divide complex numbers in a+bi form; also computes modulus and argument for each number.

Solve a triangle from all three angles (AAA). Optionally supply one side to scale the triangle and find all sides, area, perimeter, and radii.

Solve a triangle from two angles and a non-included side (AAS) — find all sides, angles, area, perimeter, inradius, and circumradius via law of sines.

Calculate the absolute change (net difference) and percentage change between two values instantly.

Solve absolute value inequalities of the form |ax + b| < c, ≤, >, ≥ and display the solution set on a number line.

Check if a triangle is acute and calculate its area, all angles, perimeter, inradius and circumradius from the three side lengths.

Add or subtract two polynomials up to degree 3 by entering their coefficients — shows the result and a step-by-step like-terms solution.

Add or subtract two fractions with different denominators — finds the LCD, converts, computes, and simplifies step by step.

Add two fractions with any denominators — automatically finds the LCD, converts both fractions, and simplifies the result step by step.

Add two or three numbers instantly and see the sum, the full arithmetic expression, and a number line showing every addend.

Compute the adjoint (adjugate) matrix and determinant for any 2×2 or 3×3 matrix, with step-by-step cofactor working shown.

求正弦函數 y = A sin(Bx + C) + D 的振幅、週期、相位移與垂直位移。

Calculate the bitwise AND of two integers and see the result in decimal, binary, and hexadecimal.

Calculate the antilogarithm (inverse log) of any value in any base: find b^x, 10^x (antilog₁₀), and e^x (antiln).

計算曲線 y = f(x) 在 x = a 至 x = b 之間的弧長。

Calculate arccos(x) — the inverse cosine of a value — in both degrees and radians. Input must be in [−1, 1].

Calculate arcsin(x) — the inverse sine of any value between −1 and 1 — in both degrees and radians.

Calculate arctan(x) — the inverse tangent of any real number — in degrees and radians, with the complementary arccot.

Calculate the inverse tangent (arctan) of any real number — returns the angle in radians, degrees and gradians.

計算兩個函數 f 和 g 在區間 [a,b] 上所圍成的面積。

Calculate the area, circumference and diameter of a circle from its radius — using the formula Area = π × r².

Calculate the area, perimeter and midline of a trapezoid from its two parallel sides and height using Area = ½ × (a + b) × h.

Calculate the area of an oblique (non-right) triangle from two sides and their included angle using ½ × a × b × sin(C).

Calculate the area and all angles of an obtuse triangle from its three side lengths using Heron's formula.

使用精確數值積分計算 a 和 b 之間曲線下的面積。

Solve a triangle from two angles and the included side (ASA). Calculates all sides, the third angle, area, perimeter and height.

Verify the associative property for addition or multiplication with any three numbers — shows both groupings step by step.

求任何一串數字的平均值，並附總和、計數、最小值與最大值。

Calculate the simple average (arithmetic mean) of up to five percentage values — useful for grades, survey results, growth rates and more.

計算函數在兩個 x 值 a 和 b 之間的平均變化率（割線斜率）。

使用積分均值定理求函數在區間 [a,b] 上的平均值。

Convert any decimal integer (1–215,999) to its Babylonian cuneiform numeral representation in base-60 with a step-by-step decomposition.

Find the base of a triangle from its area and height, or from two sides and the included angle using the law of cosines.

Compute Bessel functions of the first kind J₀(x), J₁(x) and J₂(x) using the power series, and plot all three curves from x = 0 to 15.

以精確的大整數精度對極大的整數進行加、減、乘、除、取模或冪運算。

Interpolate a value at any (x, y) point inside a 2D rectangular grid using the four corner values — step-by-step.

Add two binary numbers and see the sum in binary, decimal, and hex with a step-by-step carry trace.

Divide two binary numbers to find the quotient and remainder. See results in binary and decimal with the verification N = Q × D + R.

Convert decimal fractions to binary or binary fractions to decimal. Shows repeated ×2 steps and place-value bit contributions.

Multiply two binary numbers using the shift-and-add method. See partial products, the binary result, and the decimal equivalent.

Perform AND, OR, XOR, NOT, NAND, NOR, XNOR on binary number inputs. See results in binary, decimal, and hex with bit-by-bit steps.

Subtract two binary numbers using the two's complement method. See the result in binary, decimal, and hex with step-by-step working.

Calculate the binomial coefficient C(n, k) — the number of ways to choose k items from n — with Pascal's triangle row visualised.

在固定寬度內將整數向左或向右旋轉 k 位。移出一端的位元會繞回到另一端。

使用位元遮罩操作設定、清除、切換或測試整數中位置 k 的單個位元。

將整數邏輯或算術地左移或右移 k 位。以十進制、二進制和十六進制顯示移位值。

Perform bitwise AND, OR, XOR, NAND, NOR, XNOR and NOT on two integers within a 4, 8, 16, 32 or 64-bit register, with binary and hex output.

對兩個整數計算位元 AND、OR、XOR、NOT（以及 NAND/NOR/XNOR）。以十進制、二進制和十六進制查看結果。

為兩個布林變數在所選運算子（AND、OR、XOR、NAND、IMPLIES、IFF）下建立真值表。

Multiply two binomials (ax + b)(cx + d) using the box (grid) method with step-by-step workings.

Convert a Cartesian point (x, y) to polar coordinates (r, θ) — get the radius and angle in degrees and radians instantly.

計算第 n 個卡特蘭數，計數平衡括號、二叉樹、路徑和三角剖分的數量。

Calculate the arc length, catenary parameter, and shape of a hanging cable or chain from its span and midpoint sag.

Calculate the ceiling ⌈x⌉ of any number — the smallest integer ≥ x — plus floor, round and truncate for comparison.

Find the center (h, k) of any ellipse from its general conic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, including tilted ellipses.

Calculate the center of mass (x̄, ȳ) for a system of up to three point masses in 2D using x̄ = Σ(mᵢxᵢ)/Σmᵢ.

Calculate the central angle of a circle from radius and arc length. Get angle in degrees and radians, sector area, and chord length.

Find the centroid (geometric centre) of 2, 3 or 4 points in 2D space — the arithmetic mean of all x and y coordinates.

Find the centroid (intersection of medians) of a triangle from its three vertex coordinates, plus side lengths, area and median lengths.

Compute log_b(x) for any positive base b and argument x using the change-of-base formula: log_b(x) = ln(x)/ln(b).

Find the characteristic polynomial and eigenvalues of a 2×2 matrix. Enter a, b, c, d and get det(A − λI) = 0 with all roots shown.

Solve a system of three simultaneous congruences using the Chinese Remainder Theorem (CRT) with step-by-step working.

Compute the Cholesky decomposition L of a 2×2 symmetric positive-definite matrix A so that L × Lᵀ = A.

Find the center (h, k) and radius of a circle passing through any three non-collinear points.

Calculate circle diameter from radius, area, or circumference — plus area, circumference, and radius from any single input.

Write the equation of a circle in standard form (x−h)²+(y−k)²=r² and general form from its center and radius.

Apply all circle formulas at once: area, circumference, arc length, sector area, chord length, and segment area from radius and angle.

Calculate the arc length of a circle from its radius and central angle in degrees or radians.

Calculate the area, circumference, diameter and sector values of a circle from its radius in any unit.

Calculate the circumference, arc length, chord length and sector or segment perimeter of a circle.

由半徑和圓心角計算圓扇形面積，並顯示其在整個圓中的比例。

Apply circle theorems to find inscribed angles, arc length, chord, sector area and cyclic quadrilateral angles from a central angle.

Find the circumcenter coordinates and circumradius of a triangle — the centre and radius of the unique circle passing through all three vertices.

Find the circumference and area of a circle from its radius, diameter, or area using C = 2πr and A = πr².

Calculate the circumference of a circle from its radius or diameter using C = 2πr.

Convert circumference to diameter using d = C ÷ π. Also finds the radius and enclosed area.

Find the radius of a circle from its circumference using r = C ÷ (2π).

Calculate the circumradius and circumcircle area of a regular polygon or any triangle.

Enter three side lengths to classify a triangle by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse), plus all angles and area.

Find the angle between the hour and minute hands of a clock at any time. Instant clock-hand angle calculator with animated clock face.

Compute the determinant of a 3×3 matrix by Laplace cofactor expansion along row 1, with full step-by-step working.

Find all six cofunction pairs — sin/cos, tan/cot, sec/csc — for any angle in degrees or radians.

Calculate how many times a rolling coin rotates when circling a fixed coin — including the paradoxical extra rotation.

為任意起始數生成科拉茲（3n+1）序列，並計算到達 1 的步數。

Find the column space (range) of a matrix: dimension, rank, nullity, and basis vectors — via row reduction for matrices up to 4×4.

計算組合數 nCr——從 n 個元素中選取 r 個不考慮順序的方式數。

計算多重集選取數：從 n 種類型中允許重複地選取 r 個，C(n+r-1, r)。

Find the least common denominator (LCD) of two fractions and convert both to equivalent fractions so they can be added or compared.

Find all common factors (divisors) of two numbers and the greatest common factor (GCF), with step-by-step Euclidean algorithm.

Find the least common multiple (LCM) of two numbers and list the first N common multiples — with formula, GCF, and animated sequence.

Compare two fractions and find which is greater, equal or smaller — with cross-multiplication steps, common denominators and decimal equivalents.

Find compatible numbers (friendly round numbers) for quick mental arithmetic and see how close the estimate is to the exact answer.

Find the complement of any angle (90° − θ), plus its supplement, explementary angle and all six trig cofunction identities.

Add, subtract, multiply or divide two complex numbers and find their modulus, argument and Argand-diagram representation.

Convert a complex number a + bi to polar form r∠θ — find the magnitude and argument instantly.

Convert polar form r∠θ to rectangular form a + bi using r·cos(θ) and r·sin(θ).

Convert a + bi to trigonometric form r(cos θ + i·sin θ) — find modulus and argument instantly.

Find all n complex roots of any complex number a + bi using De Moivre's theorem — roots plotted on the unit circle.

Condense a sum or difference of two logarithms into a single log using the power, product and quotient rules, with step-by-step working.

Compute the spectral and Frobenius condition numbers of a 2×2 matrix from its singular values, with an interpretation of numerical stability.

Calculate the lateral and total surface area of a cone from its base radius and height, including slant height and base area.

Calculate the volume of a cone from its base radius and height: V = (1/3)·π·r²·h, plus slant height, base area and surface area.

Explore properties of circles, ellipses, parabolas and hyperbolas — equations, eccentricity, foci, asymptotes and plotted curves.

Find the complex conjugate of z = a + bi: flip the sign of the imaginary part to get a − bi, with modulus, argument and modulus squared.

Find the sum, mean, range and product of k consecutive integers starting at n. Visualise the sequence on a number line.

Compute the discrete linear convolution of two finite sequences using the standard sum-of-products formula.

Calculate the Euclidean and Manhattan distance between two 2D points, plus midpoint, slope and Pythagorean diagram.

Enter two coordinate points and get slope, distance, midpoint, line equation, y-intercept and quadrant — with a live coordinate grid plot.

Find the corner point (vertex) where two constraint lines intersect using Cramer's rule — a key step in the linear programming corner-point method.

Calculate the cosine of any angle in degrees or radians — plus sine, tangent and cotangent, instantly.

Calculate cos(2θ) using all three double-angle identities, plus sin(2θ) and tan(2θ), for any angle in degrees or radians.

Calculate cos⁻¹(x) — the inverse cosine — in degrees and radians for any cosine value from −1 to 1.

Calculate the cosecant (csc) of any angle in degrees or radians. Shows csc(θ) = 1/sin(θ), all six trig functions, and a step-by-step solution.

Calculate cosh(x) — the hyperbolic cosine — plus sinh(x) and tanh(x), with a graph of the characteristic catenary curve.

Compute cosine similarity between two vectors of any dimension: dot product over product of magnitudes. Returns similarity, angle, and cosine distance.

Solve any triangle with the law of cosines. Enter two sides and the included angle (SAS) or all three sides (SSS) to find the missing side, all angles, area and perimeter.

Calculate the cotangent cot(θ) = cos(θ)/sin(θ) in degrees or radians — with sin, cos, tan and a graph of the cotangent function.

透過加減整數倍的完整旋轉，求任意角的正同終邊角和負同終邊角。

Calculate the area of a crescent (lune) formed by two overlapping circles of different radii and a given centre offset.

Solve a proportion a/b = c/x for the unknown x using cross multiplication. See every step of the working.

Calculate the cross-sectional area of circles, rectangles, triangles, ellipses, and hollow tubes — used in engineering and geometry.

Calculate the cube root of any number (positive, negative or zero) with step-by-step working and a y = ∛x graph.

Calculate the total surface area of a cube from its side length. Instantly get face area, perimeter and space diagonal.

Find the volume, surface area and diagonal of a cube from its side length. Covers space diagonal, face diagonal and total edge length.

Calculate the volume of a cube from its side length using V = s³. Also shows surface area and space diagonal instantly.

Calculate the volume of a cylinder in cubic feet from diameter and height. Also converts to cubic yards, cubic meters, liters and US gallons.

Calculate the volume of a cylinder in cubic inches from diameter and height. Converts instantly to cubic feet, liters and US gallons.

Calculate the volume of a cylinder in cubic yards from its diameter and depth in feet. Ideal for circular concrete pads, ponds and footings.

Calculate volume, total surface area, lateral surface area, space diagonal and face diagonals of a cuboid (rectangular box) from length, width and height.

Calculate the total and lateral surface area of a rectangular cuboid (box) from its length, width and height using SA = 2(lw + lh + wh).

Calculate the volume, surface area and space diagonal of a rectangular cuboid (box) from its length, width and height.

Calculate the arc length and area of a cycloid arch traced by a rolling circle of radius r. Shows parametric equations and animated curve plot.

Calculate McCabe's cyclomatic complexity (M = E − N + 2P) to measure code complexity and the minimum test paths needed.

Calculate a cylinder's base circumference (C = 2πr), diameter and lateral surface area from the radius and height.

Find a cylinder's diameter from its circumference (d = C ÷ π) or from its radius (d = 2r), with base area and radius shown.

Calculate the volume, base area, lateral surface area and total surface area of a cylinder from its radius and height.

Calculate the volume of a cylinder in litres, cm³ and m³ from its radius and height. Supports cm, mm, m, inches and feet.

Convert a cylinder's volume to US liquid gallons from dimensions in inches or centimetres — tanks, pools, pipes and drums.

Convert between cylindrical coordinates (ρ, θ, z) and Cartesian (x, y, z) in both directions. Instant results with angle in degrees.

Calculate the area, perimeter, circumradius, inradius and longest diagonal of a regular decagon (10-sided polygon) from its side length.

Calculate the area, perimeter, circumradius, inradius, and diagonals of a regular decagon (10-sided polygon) from any known measurement.

Convert fractions to decimals (including repeating decimal detection) and decimals to simplified fractions, with step-by-step working.

Convert decimal degrees to degrees-minutes-seconds (DMS) and back for GPS coordinates, mapping and surveying.

Convert any decimal to a simplified fraction and mixed number instantly, showing the step-by-step method.

Convert any decimal number to a percentage instantly — multiply by 100 and see the result with a visual proportion bar.

利用精確的反導函數，計算常見函數從 a 到 b 的定積分。

即時換算角度的度數與弧度，並以 π 的整潔倍數顯示結果。

Apply Descartes' rule of signs to count the possible numbers of positive and negative real roots of any polynomial.

Calculate the determinant of a 2×2 or 3×3 matrix with full step-by-step cofactor expansion and a geometric visualisation.

Find the diagonal of a rectangle from its length and width using the Pythagorean theorem, plus area and perimeter.

Find the eigenvalues, eigenvectors, and diagonal decomposition A = PDP⁻¹ of a 2×2 real matrix, with step-by-step working.

Find the base diameter of a cone from its radius, plus slant height, volume, and surface areas in one step.

Convert between diameter and radius of a circle and get circumference and area instantly — works in both directions.

Find the two numbers that multiply to the top and add to the bottom of the diamond puzzle — instantly solves any diamond problem.

計算任意數字各位數之和，並透過反覆累加求其數字根。

Find the digital root of any whole number by repeatedly summing its digits until one digit remains.

Enter length and width to instantly get the area, perimeter, diagonal and aspect ratio of any rectangle.

計算函數 f(x,y) 在某點沿指定方向向量的方向導數。

Calculate the straight-line distance between two Cartesian points (x₁,y₁) and (x₂,y₂) using the distance formula.

Calculate the Euclidean (straight-line) and Manhattan distances between two points in 2D using the distance formula d = √(Δx² + Δy²).

Find the shortest (perpendicular) distance from a 3D point to a plane defined by ax + by + cz + d = 0.

Apply the distributive property to expand a(b+c) or use FOIL to expand (a+b)(c+d) with a step-by-step solution.

Divide two complex numbers (a+bi)÷(c+di) and get the real part, imaginary part and magnitude of the quotient.

Divide two fractions (a/b) ÷ (c/d) by multiplying by the reciprocal. Get the simplified fraction and decimal instantly.

Divide exponents with the same base using the quotient rule: a^m ÷ a^n = a^(m−n). See the step-by-step solution instantly.

Divide two square-root expressions (c₁√a) ÷ (c₂√b) using the quotient property. Get the rationalized form and step-by-step solution.

Test whether any integer is divisible by 2 through 11 instantly — with the shortcut rule shown for each divisor.

Find every divisor of any integer up to 100 million. See the count, sum of divisors σ(N), and whether the number is perfect, abundant, or deficient.

Find the area of a regular dodecagon from its side length, circumradius, or apothem. Includes perimeter and all key dimensions.

Calculate area, perimeter, interior angle, circumradius, apothem and diagonal of a regular dodecagon from its side length.

Calculate sin(2θ), cos(2θ) and tan(2θ) for any angle using the double-angle formulas. Accepts degrees or radians.

Apply the double-angle formulas step by step: evaluate sin(2θ), cos(2θ) and tan(2θ) with full working shown.

Evaluate all double-angle identities — sin(2θ), all three forms of cos(2θ), tan(2θ) — plus derived half-angle values at any angle.

Calculate how many periods it takes for a quantity to double at a given growth rate — exact formula and Rule of 72 compared.

Calculate e^x or b^x, see the Taylor series expansion term by term, the derivative, logarithm, and an interactive graph.

Decompose any positive fraction into a sum of distinct unit fractions (1/n) using the greedy algorithm, the method used in ancient Egyptian arithmetic.

Calculate the eigenvalues and eigenvectors of a 2×2 matrix, solve the characteristic polynomial, and visualise how the matrix transforms space.

Solve a system of two linear equations by elimination. Enter the six coefficients, get x and y with a step-by-step solution.

Calculate the area of an ellipse from its semi-major and semi-minor axes using the exact formula A = π × a × b.

Calculate the perimeter (circumference) of an ellipse from its semi-axes using Ramanujan's highly accurate approximation formula.

Enter center and semi-axes of an ellipse to get its standard equation, foci, eccentricity, area and a plot — all in one step.

Find the missing endpoint of a line segment given its midpoint and the other endpoint — instant coordinate geometry.

Convert any number to engineering notation — powers of 10 in multiples of 3, with matching SI prefix (k, M, G, m, μ, n, p…).

Find the standard-form equation of a circle given the two endpoints of its diameter — calculates center, radius, area and circumference.

Calculate the area of an equilateral triangle from its side length or height — includes perimeter and derived dimensions.

Calculate the area, perimeter, height, inradius and circumradius of an equilateral triangle from its side length.

Find equivalent fractions by simplifying to lowest terms and scaling by any factor — shows a fraction bar and a table of common equivalents.

Find equivalent ratios by simplifying a:b to its lowest terms and scaling by any integer — shows a ratio bar and a table of common equivalents.

Calculate the Gaussian error function erf(x), complementary error function erfc(x), and scaled erfc for any real x.

Find the GCD of two integers using the Euclidean algorithm — see every division step, from start to remainder zero.

Calculate the straight-line (Euclidean) distance between two points in 2D or 3D space using the Pythagorean theorem.

Verify or complete Euler's polyhedron formula V − E + F = 2. Enter any two of vertices, edges and faces to find the third.

計算歐拉 totient 函數 phi(n)：從 1 到 n 中與 n 互質的整數個數。

Get the exact radical form (e.g. √3/2) and decimal value of sin, cos, tan, csc, sec and cot for all standard angles 0°–360°.

Write any number in expanded form: break it into place-value addends (4523 = 4×1000 + 5×100 + 2×10 + 3) with powers of 10.

Expand logarithmic expressions using the product, quotient, and power rules. Shows the expanded symbolic form and computes the numeric result.

用此指數計算器將任何底數升至任何次方，支援負指數、分數指數與小數指數。

Convert any number to scientific notation (exponential form) and find its mantissa, exponent, log₁₀, and natural logarithm.

Evaluate f(x) = a × bˣ, determine growth or decay type, find the doubling time or half-life, and plot the exponential curve.

Calculate the final value, total growth and doubling time for discrete or continuous exponential growth.

Convert any number to scientific notation, engineering notation and E notation with a chosen number of significant figures.

Calculate all three exterior angles of a triangle from two interior angles and verify the exterior angle theorem.

Find the prime factorisation, all divisors, number of divisors and sum of divisors for any positive integer up to 1,000,000.

計算任何非負整數的階乘 n!，即從 1 到 n 所有整數的乘積。

Factor a quadratic trinomial ax² + bx + c into a(x − r₁)(x − r₂) using the quadratic formula, with step-by-step working and a parabola graph.

列出整數的所有因數，計算因數個數、求因數之和，並將每個因數與其配對因數顯示。

Compute a^(p−1) mod p to verify Fermat's little theorem, test primality and find the modular inverse a⁻¹ (mod p).

求第 n 個斐波那契數，列出前 n 項數列，並顯示累加和。

Calculate the fifth root of any real number instantly. Works for negative inputs, with a curve showing y = ⁵√x.

Calculate floor division ⌊a ÷ b⌋ and the true modulo remainder. Handles negative numbers correctly by rounding toward −∞, not toward zero.

Calculate the floor ⌊x⌋, ceiling ⌈x⌉, round, truncate and fractional part of any real number, with a number-line visualisation.

Find the foci, eccentricity, area and perimeter of an ellipse from its semi-major axis a and semi-minor axis b.

Expand (ax + b)(cx + d) using the FOIL method with full step-by-step working — First, Outside, Inside, Last — and get the simplified polynomial.

Calculate the fourth root of any non-negative number — the value that, raised to the power 4, equals x.

對兩個分數做加、減、乘或除，立即得到化簡後的結果及其小數值。

Convert any fraction to its decimal and percentage equivalents, simplify the fraction, and show the mixed-number form.

Convert any fraction to a percentage instantly. Enter the numerator and denominator to get the exact percentage, decimal equivalent and a visual bar.

Evaluate x^(p/q): raise any number to a fractional power and see the equivalent radical form plotted.

Multiply the number of choices for each independent event to find the total possible outcomes — up to 6 events.

Explore Galileo's paradox: count natural numbers vs perfect squares in [1, N] and see why infinite sets defy 'more' and 'fewer'.

Evaluate the Gamma function Γ(z) for any positive real number using the Lanczos approximation — accurate to ~15 significant digits.

Solve a 2×2 system of linear equations step by step using Gauss–Jordan elimination with full row-operation working.

使用歐幾里得演算法求一串整數的最大公因數（GCD/HCF）與最小公倍數（LCM）。

Find the greatest common divisor of two integers using the Euclidean algorithm, with every step shown.

Find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of two numbers using the Euclidean algorithm, with step-by-step working.

Find the greatest common factor (GCF) of two or more integers — lists all common factors and simplifies fractions.

Convert a circle's centre (h, k) and radius r to its general form x² + y² + Dx + Ey + F = 0.

Convert a circle from general form x² + y² + Dx + Ey + F = 0 to standard form (x−h)² + (y−k)² = r² by completing the square.

Calculate the geometric mean of any list of positive numbers — the n-th root of their product. Useful for growth rates, ratios and log-scale data.

Calculate both sides, area, perimeter and diagonal of a golden rectangle from one side, using the golden ratio φ ≈ 1.6180.

求二元函數在某點的梯度向量及其模長。

Apply the Gram-Schmidt process to two 2-D vectors and compute the orthonormal basis e₁, e₂ step by step.

計算 n 個頂點的完全圖的邊數：無向圖為握手數 n(n-1)/2，有向圖為 n(n-1)。

Solve and graph a linear inequality ax + b > c on a number line — shows boundary, interval notation, and open/closed endpoint.

Solve and graph ax² + bx + c > 0 (or ≥, <, ≤) — shows roots, vertex, solution set, and parabola graph.

Calculate the shortest distance between two points on Earth using the haversine formula: result in km, miles and nautical miles, plus the initial bearing.

Compare any two numbers and find whether A > B, A < B or A = B, plus the difference, ratio and percentage change between them.

Find the GCD of two denominators and the least common denominator (LCD) needed to add or subtract two fractions, with the simplified result.

Find the greatest common divisor (GCD) of two integers using the Euclidean algorithm, with full step-by-step working and the LCM as a bonus result.

Find the Greatest Common Factor (GCF) of two or more integers and list all their common factors, with Euclidean algorithm steps.

Compute the Hadamard (element-wise) product of two 2×2 matrices — multiply matching entries and see the step-by-step working.

Compute sin(θ/2), cos(θ/2) and tan(θ/2) from any angle in degrees or radians using the half-angle identities.

求兩個等長二進制（或十六進制）字串的漢明距離：位元不同的位置數。

Calculate the harmonic mean of a list of non-zero numbers — the correct average for rates, speeds, and prices per unit.

Compute the nth harmonic number H(n) = 1 + 1/2 + 1/3 + … + 1/n, its asymptotic approximation, and the Euler–Mascheroni constant.

Find the perpendicular height of a cone from its radius and slant height (h = √(l²−r²)), or from its radius and volume (h = 3V/πr²).

Find the height of a cylinder from its radius and volume (h = V/πr²) or from its radius and lateral surface area (h = A/2πr).

Find the perpendicular height of a square pyramid from its base side and slant height (h = √(l²−(a/2)²)), or from its base side and volume.

Calculate the curved surface area (CSA = 2πr²) and total surface area (TSA = 3πr²) of a hemisphere from its radius.

Calculate the area of a regular heptagon from its side length, circumradius or apothem, with all derived dimensions.

Calculate the area, perimeter, apothem, circumradius, interior angles and diagonals of a regular heptagon from its side length.

對兩個十六進制或二進制數進行加法或減法，同時以十六進制、二進制和十進制顯示結果。

Calculate the volume, total surface area, lateral area and slant height of a regular hexagonal pyramid from base side and height.

Find the total and lateral surface area of a regular hexagonal pyramid — enter base side and height, or base side and slant height.

Explore Hilbert's Hotel paradox: see how infinitely many rooms can always accommodate more guests by shifting room assignments.

Calculate the volume and surface area of a hollow cylinder (pipe or tube) from outer radius, inner radius and height.

Calculate all six hyperbolic functions (sinh, cosh, tanh, coth, sech, csch) and their inverses for any x.

Calculate the hypotenuse or a missing leg of a right triangle using a² + b² = c². Also computes angles, area, and perimeter.

Calculate powers of i (i^n = 1, i, −1, −i cycle), raise any complex number z = a+bi to the nth power, and find modulus, argument and conjugate.

Add, subtract, multiply or divide two complex numbers (a + bi) and find the result, modulus, argument and conjugate.

Convert any improper fraction (numerator ≥ denominator) to a simplified mixed number in one step, with the full working shown.

Convert any linear inequality (x > a, x ≤ a, a < x ≤ b …) to interval notation instantly, with a number-line diagram.

Calculate the inscribed angle from the central angle (arc) or vice versa. Applies the Inscribed Angle Theorem: inscribed angle = ½ × central angle.

Compute the GCD, LCM, quotient, remainder and basic arithmetic of any two integers instantly.

Find the exact (x, y) intersection point of two lines given in slope-intercept form y = mx + b, with a live graph.

Convert any inequality or set of real numbers to interval notation — with bracket type, set-builder form, and a number-line diagram.

Find the angle whose cosine equals a given value. Returns arccos(x) in degrees, radians or gradians for any x in [−1, 1].

Find the angle from a cosine value or from adjacent and hypotenuse sides of a right triangle. Outputs degrees, radians, and gradians.

Find the angle whose sine equals a given value. Returns arcsin(x) in degrees, radians or gradians for any x in [−1, 1].

Calculate arctan(x) or the quadrant-aware angle atan2(y, x) from coordinates — output in degrees and radians.

計算某值的 arcsin、arccos 或 arctan，以度數和弧度給出所得角度。

Calculate arcsin, arccos, arctan, arccsc, arcsec, or arccot of any valid input — angle in degrees and radians.

Calculate the gear involute function inv(α) = tan(α) − α and related gear parameters: base circle radius, module, addendum and dedendum.

Find the area and perimeter of any irregular polygon from its vertex coordinates using the Shoelace formula.

Calculate the area, perimeter, leg lengths and base angles of an irregular (general) trapezoid with two parallel sides and unequal legs.

Test whether (a mod b) mod c equals a mod (b mod c) for any three integers — and learn why modulo is not generally associative.

Calculate the area, height, perimeter, diagonal, and angles of an isosceles trapezoid from its two bases and equal leg.

Calculate the area, height, perimeter, diagonal and base angles of an isosceles trapezoid from its two parallel sides and leg length.

Find the equal leg (a) of an isosceles triangle from the base and one other known value: height, area, or base angle.

Calculate all three angles of an isosceles triangle from its sides, or find sides from the apex angle and base.

Calculate the area of an isosceles triangle from its equal side and base, or from the base and height directly.

Calculate the area, height, perimeter, angles, and medians of an isosceles triangle from its equal side and base.

Find the height (altitude) of an isosceles triangle from its equal side and base, from area and base, or from the equal side and apex angle.

Calculate the height, area, perimeter and angles of an isosceles triangle from its two equal legs and base.

Calculate the area and perimeter of a kite (quadrilateral) from its two diagonals. Formula: Area = (p × q) ÷ 2.

Calculate the Lagrange remainder bound for a Taylor polynomial: |Rₙ(x)| ≤ M · |x−a|^(n+1) / (n+1)!

Calculate the lateral (curved) surface area of a cone from radius and height or slant height: A = π·r·l, plus total surface area and volume.

Calculate the lateral surface area of a trapezoidal prism from the four sides of the trapezoid and the prism height.

Calculate the lateral (curved) surface area of a cylinder from radius and height, with total surface area and volume.

Calculate the latus rectum and semi-latus rectum of a parabola, ellipse or hyperbola from its defining parameters.

應用餘弦定理，由兩邊及夾角求三角形第三邊，或由三邊求角度。

使用正弦定理，由已知的邊角對求三角形的缺失邊或角。

Find the Least Common Denominator (LCD) of two or three fractions instantly, with step-by-step multipliers and a visual breakdown.

Find the Least Common Multiple (LCM) of two or three integers using the GCD / Euclidean algorithm, with step-by-step working.

Find the Least Common Denominator (LCD) of two or three fractions, convert them to equivalent fractions and compute their sum.

Find the least (smallest) factor greater than 1 that two integers share — the smallest prime factor of their GCF.

Find the Least Common Multiple (LCM) of up to four integers using the GCD method, with full step-by-step working shown.

Find the least-squares regression line (y = mx + b), correlation coefficient r, and R² for any set of x,y data points.

Find the width of a rectangle when the perimeter and length are known. Also computes area, half-perimeter, and diagonal.

Calculate the distance between two coordinate points using the distance formula d = √((x₂−x₁)²+(y₂−y₁)²).

Find the length of a rectangle from its width and either its area (l = A ÷ w) or its perimeter (l = P/2 − w).

透過數值探測兩側，估計函數在 x 趨近某值時的極限。

Find the parametric equation of the line where two planes intersect — direction vector via cross product and a point on the line.

Generate the output bit sequence of a Fibonacci LFSR with a maximal-length polynomial — choose stages (2–8), initial state, and steps.

Check whether three vectors in ℝ³ are linearly independent by computing the 3×3 determinant of the matrix they form.

Find an unknown y that lies on the straight line between two known (x, y) data points — the standard lerp formula.

Calculate the base-2 logarithm of any positive number, with log₁₀ and natural log shown alongside a plotted curve.

以任何底數計算任何正數的對數，並附自然對數（ln）與常用對數（log₁₀）。

Calculate the logarithm of any positive number in base 10, natural (e), base 2, or a custom base instantly.

Add two whole numbers using the long addition method, with a full column-by-column breakdown showing every carry.

Divide two whole numbers using the long division method, with a step-by-step breakdown showing each digit, subtraction and remainder.

Multiply two whole numbers using the long multiplication method, showing each partial product and the final addition step by step.

Subtract two whole numbers using the long subtraction method, with a full column-by-column breakdown showing every borrow.

Find the lowest common denominator (LCD) of 2–4 fractions instantly, with step-by-step GCD/LCM working and a visual number line.

Reduce any fraction to its lowest terms (simplest form) instantly using the Euclidean algorithm, with a full step-by-step GCD derivation.

Decompose a 3×3 matrix into lower (L) and upper (U) triangular factors using Doolittle elimination, with step-by-step working and determinant.

Validate any credit card, IMEI, or ID number using the Luhn algorithm checksum — with a step-by-step walkthrough.

Calculate the magic constant of an n×n magic square and display the actual grid for odd-order and doubly-even-order squares.

Calculate the Manhattan (taxicab/city-block) distance between two 2D points: |x₂ − x₁| + |y₂ − y₁|.

Calculate b raised to any power n — positive, negative, zero or fractional — with step-by-step working and a plotted power curve.

Multiply a 2×2 matrix by a scalar — every element is scaled by k — with the resulting matrix, determinant, and a geometric view of the transformation.

Calculate the Frobenius, 1-norm, ∞-norm and max-norm of a 2×2 matrix with step-by-step working and a comparison of all four norms.

在區間 [a,b] 上求函數的絕對最大值與最小值。

Convert any whole number to Mayan numerals — dots (1), bars (5), shell (0) — with a step-by-step base-20 decomposition and glyph display.

Calculate the arithmetic, geometric and harmonic mean of any list of numbers — enter values separated by commas.

使用中點法則以 n 個矩形近似曲線下的定積分。

Calculate all three midsegments of a triangle from its side lengths, plus the medial triangle area and perimeter.

Convert degrees, arc minutes and arc seconds (DMS) to decimal degrees — the format used in GPS, mapping, navigation and astronomy.

Calculate how often a "miraculous" rare event is statistically expected using Littlewood's Law — enter the probability and daily event rate.

Add, subtract, multiply or divide two mixed numbers — with full step-by-step working, simplification, and decimal result.

Convert any mixed number to an improper fraction instantly — with step-by-step working and simplification using GCD.

Calculate the surface area and edge length of a Möbius strip given its radius and strip width, with a 2-D boundary curve diagram.

使用大整數快速平方法高效計算 (base^exponent) mod m。

求數的模乘法逆元，即使 a*x 與 1 mod m 同餘的 x。

Find the modular multiplicative inverse of a modulo m using the extended Euclidean algorithm. Returns x where (a × x) mod m = 1.

求除法的餘數與商，支援截斷、向下取整和歐幾里得模運算以處理負數。

See how modulo behaves with negative dividends or divisors across the three conventions: truncated (JS/C), floored (Python), and Euclidean.

Explore how modulo (%) fits into PEMDAS/BODMAS precedence — same level as × and ÷ — with live worked examples and step-by-step evaluation.

Compute the Moore-Penrose pseudoinverse A† of any 2×2 matrix — handles full-rank, rank-1 and zero-matrix cases with verification.

Multiply any two numbers and see the product instantly. Works with decimals, negatives and large numbers.

Find the multiplicative inverse (reciprocal) of any non-zero number. 1/x — the number that when multiplied by x gives 1.

Find the multiplicative inverse of a number mod m — the x where a · x ≡ 1 (mod m), via the extended Euclidean algorithm.

Multiply two complex numbers (a + bi)(c + di) using FOIL — get the product, magnitude, and argument with an Argand-plane diagram.

Multiply two binomials (ax + b)(cx + d) using the FOIL method and get the expanded trinomial with step-by-step working.

Apply the product rule (xᵐ · xⁿ = x^(m+n)) or power rule ((xᵐ)ⁿ = x^(m·n)) to simplify and evaluate expressions with exponents.

Multiply two fractions (a/b × c/d) and instantly get the simplified result, decimal, and a step-by-step breakdown.

Multiply any two polynomials by entering comma-separated coefficients — get the expanded product and a step-by-step distributive breakdown.

Multiply radical expressions c₁·ⁿ√a × c₂·ⁿ√b and simplify: extracts the largest perfect nth-power factor to give the fully simplified form.

Calculate ln(x), log base 10, log base 2, eˣ, and the derivative 1/x for any positive number — with an animated ln curve.

Calculate the negative logarithm −log_b(x) in base 10, e, or 2. Used for pH, pKa, pOH, decibels, and information-theoretic measures.

從初始猜測出發，使用牛頓迭代求函數的根。

Calculate the bitwise NOR of two integers — NOT(A OR B) — and see the result in decimal, binary, and hexadecimal.

Find the null space (kernel) of a 2×2 matrix: compute its determinant, rank, nullity and a basis for the null space.

在二進制、八進制、十進制、十六進制及 2 到 36 的任意進制之間轉換數字。

在 2 到 36 進制之間轉換數字（包括二進制、八進制、十進制、十六進制等），完整支援所有數字。

將任何整數拼寫為英文文字，例如 1234 變為 one thousand two hundred thirty-four。

使用中央差分法求函數在某點的導數，無需符號代數。

Solve any oblique (non-right) triangle from SSS, SAS, AAS or SSA inputs — find all sides, angles, area, perimeter, inradius and circumradius.

Solve an obtuse triangle (one angle > 90°) — find all sides, angles, area, altitudes, inradius and circumradius from SSS or SAS inputs.

Calculate area, perimeter, circumradius, inradius, and all three diagonal lengths of a regular octagon from any known measurement.

在選定的位元寬度下，並排顯示有號整數的符號大小值、一補數和二補數編碼。

Compute the bitwise OR of two integers in decimal, binary or hex, with a full bit-by-bit breakdown.

Find the order of magnitude of any number: the integer exponent n so that |x| ≈ 10ⁿ. Also shows scientific notation and nearest powers of 10.

Find the orthocenter of any triangle given its three vertices. Shows coordinates, type (acute/right/obtuse), area and a plotted diagram.

Calculate the vertex, focus, directrix, axis of symmetry, x-intercepts and y-intercept of any parabola y = ax² + bx + c, with a live graph.

Calculate the area of a parallelogram from its base and perpendicular height. Optionally enter the side to get the perimeter too.

Calculate the perimeter of a parallelogram from its base and side length using the formula P = 2(a + b).

計算二進制字的偶數和奇數奇偶校驗位元，並計算其 1 位元個數以進行錯誤偵測。

計算二元函數 f(x,y) 在某點的數值偏導數。

Multiply two whole numbers using the old partial-products method — breaks each factor into place values and shows every multiplication step.

生成巴斯卡三角形的任意一列——k 從 0 到 n 的二項式係數 C(n,k)。

Calculate the area, perimeter, circumradius, inradius, and diagonals of a regular pentagon from side length, circumradius, or apothem.

計算舊值與新值之間的百分比增加或減少，以及絕對差值。

Calculate what percentage of a time period has elapsed and how much time remains, using any consistent time unit.

Calculate what percentage of a goal you have achieved, how much remains, and what percent is left — for any numeric target.

快速算出百分比：Y 的 X%、一數是另一數的百分之幾，以及百分比變化。

Calculate the percentage change between two values — finds the % increase or decrease from an original value to a new value.

Calculate the value after a percentage decrease, or find what % one value fell from another — simple formula, instant result.

Calculate the percentage difference between two values — symmetric formula using the average as the reference, not one specific starting point.

Calculate the value after a percentage increase, or find what % one value rose from another — quick formula, instant result.

Calculate the percentage point (pp) difference between two percentages and the relative % change — understand the key distinction between the two.

Check whether any integer is a perfect cube, find its exact integer cube root, and identify the nearest perfect cubes if it is not.

透過比較數字與其真因數之和，判斷它是完全數、盈數還是虧數。

Check whether any non-negative integer is a perfect square, find its exact integer square root, and see the nearest perfect squares if it is not.

Expand a perfect square trinomial (a + b)² or (a − b)² into a² ± 2ab + b² with step-by-step working.

Calculate the perimeter of a circle, square, rectangle, triangle or regular polygon by entering the shape dimensions.

Find the perimeter of any quadrilateral by entering all four side lengths — works for rectangles, parallelograms, rhombuses, and irregular shapes.

Find the perimeter of a rectangle when you know its area and one side — the calculator derives the other side and computes 2(l + w).

Calculate the perimeter of a circular sector from the radius and central angle in degrees or radians — perimeter = 2r + arc length.

Find the perimeter and area of a triangle from its three vertex coordinates using the distance formula.

Add fractional or mixed-number side lengths of a triangle to find the perimeter — supports entries like 3/4, 1 2/3, or 0.75.

Calculate the perimeter of a rectangle from its length and width, plus area, diagonal, and aspect ratio.

計算排列 nPr = n!/(n-r)!——從 n 個項目中選取 r 個的有序排列數。

計算允許重複時的排列數：從 n 個符號中選取 r 個位置的有序排列總數為 n^r。

Find the equation of a line perpendicular to a given line through a specific point, plus the foot of the perpendicular and the shortest distance.

Approximate π using Nilakantha, Gregory–Leibniz or Machin series — see convergence, error and correct decimal places for any number of terms.

應用鴿巢原理：計算最滿格子的保證最小數量，以及需要多少物品才能使某格達到目標數量。

Break any whole number into its place-value positions: ones, tens, hundreds, thousands and beyond.

Decompose a 2×2 matrix into an orthogonal factor Q (rotation/reflection) and a symmetric positive semidefinite factor S (stretch). Exact closed-form result.

Convert a complex number a + bi to polar form r∠θ and Euler form r·e^(iθ). Find the modulus and argument in degrees and radians.

Convert polar coordinates (r, θ) to Cartesian (x, y). Enter radius and angle in degrees or radians to get the rectangular point instantly.

將極座標 (r, θ) 轉換為直角座標 (x, y) 並繪製結果點。

Evaluate expressions written in Polish (prefix) notation. Operators come before their operands — no parentheses needed. Supports +, -, *, /, ^.

Calculate the interior angle, exterior angle, and angle sum of any regular polygon. Enter the number of sides to get all angle properties instantly.

Divide one polynomial by another using long division. Enter coefficients in descending degree order and get quotient, remainder, and full working steps.

Graph polynomial functions up to degree 4, evaluate f(x), find real roots, and identify the vertex of a parabola.

計算整數的置位位元數（漢明重量）和位元長度，並檢查是否為 2 的冪次。

Explore the Potato Paradox: calculate how much weight a batch loses when its water percentage drops slightly.

Evaluate the power function y = a·xⁿ for any real coefficient, exponent and x value, with a plotted curve.

Calculate (base^exponent) mod m — the remainder of a power divided by a modulus — using fast binary exponentiation.

Calculate 10 raised to any power (10ⁿ) and see the result in standard and scientific notation.

Calculate 2 raised to any power (2ⁿ) — instantly see memory sizes, bit depths and binary capacity values.

Apply power-reducing identities to express sin²(θ), cos²(θ) and tan²(θ) in terms of cos(2θ) — enter any angle.

計算 n 元素集合的冪集中子集數量：共 2^n 個，包括真子集和非空子集的計數。

Calculate any integer power of the imaginary unit i. i cycles through 1, i, −1, −i with period 4 — enter any exponent to get the result instantly.

將任意整數分解為質因數，並以指數形式表示，例如 360 = 2^3 x 3^2 x 5。

使用快速的試除法（試除至平方根）檢查任何整數是否為質數，並查看其質因數分解。

Enter the four payoff values of a prisoner's dilemma game to find the Nash equilibrium, Pareto-optimal outcome, and social cost of defection.

Calculate all key angles of a right square pyramid: face inclination, lateral edge angle, apex angle, and the dihedral angle between adjacent faces.

Calculate the volume, base area, lateral surface area, and total surface area of a square, rectangular, or equilateral-triangle-base pyramid.

Solve any right triangle with the Pythagorean theorem. Enter any two sides to find the third, plus all angles, area, perimeter, and altitude to the hypotenuse.

Generate Pythagorean triples (a, b, c) with a² + b² = c² using Euclid's parametric formula for integers m > n ≥ 1.

Factor a 2×2 matrix A into an orthogonal matrix Q and upper-triangular matrix R using Gram-Schmidt orthogonalisation.

Solve any quadratic equation ax² + bx + c = 0. Get both roots (real or complex), the discriminant, vertex, and a parabola plot.

Calculate the area of four common quadrilaterals: rectangle, parallelogram, trapezoid, and rhombus using exact formulas.

Calculate area, perimeter and diagonals for rectangles, parallelograms, isosceles trapezoids and rhombuses — select shape and enter dimensions.

Calculate the area of a quarter circle from its radius, or find the radius from a known area. Formula: A = πr²/4.

Calculate the area, arc length, perimeter and centroid of a quarter circle from its radius.

Calculate the perimeter of a quarter circle: two radii plus the curved arc. Formula: P = 2r + πr/2.

Add, subtract or multiply two quaternions and compute the magnitude, conjugate and inverse. Shows step-by-step Hamilton product.

Calculate M/M/c queue metrics: utilisation, average queue length, waiting time and service probability using the Erlang-C formula.

Calculate the integer quotient and remainder of any division — shows truncated, floored and exact decimal results.

Calculate the nth root (square root, cube root, or any radical) of a number — with step-by-step result.

Find the radius of a circle from its area, circumference, or diameter using simple inverse circle formulas.

Find the base radius of a right circular cone from its volume, lateral surface area, or total surface area — given the height.

Find the base radius of a right circular cylinder from its volume, lateral surface area, or total surface area — given the height.

Find the radius of a sphere from its volume or surface area, or compute volume, surface area and diameter from radius.

求解形如 a:b = c:d 的比例中缺漏的值，並將任何比率化簡至最簡形式。

Simplify the ratio of three numbers to lowest terms and show each number as a percentage of the combined total A + B + C.

Find the simplified ratio of two numbers A : B, convert to a unit ratio, and see each number as a percentage of the combined total.

Evaluate x raised to a fractional exponent p/q, with step-by-step root-then-power working and a curve showing where your value falls.

Find all possible rational zeros of a polynomial using the Rational Root Theorem: ±(factors of constant term) ÷ (factors of leading coefficient).

Rationalize the denominator of a/√b by multiplying by √b/√b, automatically simplifying perfect-square factors and reducing the resulting fraction.

Find the point that divides a line segment AB in a given ratio m:n — internally or externally — using the section formula.

Find the reciprocal (multiplicative inverse) of any non-zero number: 1 ÷ x. Decimals, fractions and negatives all supported.

Calculate the area, perimeter, diagonal and aspect ratio of a rectangle from its length and width in any unit.

Calculate the diagonal length and its angle in a rectangle from width and height. Uses the Pythagorean theorem and trigonometry.

Solve a rectangle from any two known values — length, width, area, or perimeter — and get all four measurements plus the diagonal.

Scale a rectangle by any factor: find new width, height, area, perimeter and diagonal. Area scales as k², linear dimensions scale as k.

Calculate the volume, surface area, space diagonal and face diagonals of a rectangular prism (cuboid) from its length, width and height.

Calculate the volume, total surface area, lateral surface area, space diagonal and base perimeter of a rectangular prism from length, width and height.

Calculate the volume, base area, lateral and total surface areas, slant heights and apex edge of a rectangular pyramid from its base dimensions and height.

使用模長和 atan2 角度將直角座標 (x, y) 轉換為極座標形式 (r, θ)。

計算任意角度的參考角，並顯示其所在象限，支援度數或弧度。

Calculate area, perimeter, apothem, long and short diagonals of a regular hexagon from side length, apothem, perimeter or area.

Calculate the relative change (dimensionless ratio) and percent change between a reference value and a new value.

Check whether two integers are relatively prime (coprime) — GCD = 1 — using the Euclidean algorithm with every step shown.

Find the quotient and remainder of integer division. Shows the full a = b × q + r breakdown step by step, including the modulo.

Factor any quadratic ax² + bx + c into (px + q)(rx + s) using the AC / grouping method with step-by-step working.

Calculate the area and perimeter of a rhombus using diagonals, base and height, or side length and interior angle — three input methods.

計算函數在區間上的左端點、右端點或中點黎曼和。

Calculate the volume, total surface area, lateral surface area and slant height of a right circular cone from radius and height.

Calculate the volume, total and lateral surface area, base area and space diagonal of a right circular cylinder from radius and height.

Calculate volume, total surface area, lateral surface area, slant heights and lateral edge of a right rectangular pyramid from base length, width and height.

Calculate volume, total surface area, lateral surface area, slant height and lateral edge of a right square pyramid from base side and height.

Calculate the area of a right trapezoid from its two parallel bases and perpendicular height, with a step-by-step formula walkthrough.

Calculate area, perimeter, slant side, diagonals and angles of a right trapezoid from its two parallel bases and perpendicular height.

由任意兩個已知值（直角邊、斜邊或銳角）求解直角三角形，包含面積和周長。

Calculate slope (rise ÷ run), angle in degrees and percent grade from rise/run values or two coordinate points.

求一數的平方根、立方根或任何 n 次方根，包含負值的實奇次方根。

Calculate the root mean square (RMS) of any set of numbers. Works with positive and negative values; shows sum of squares and arithmetic mean.

Rotate any 2D point (x, y) by any angle about the origin using the rotation matrix formula. Shows new coordinates, distance, and angles.

Round any dollar amount to the nearest cent ($0.01) — shows the rounded result, ceiling (always up), floor (always down) and the rounding difference.

Round any dollar amount to the nearest whole dollar — shows the rounded result, ceiling (always up), floor (always down) and the rounding difference.

Instantly round any number to the nearest 100. See the lower and upper hundreds and the rounding error.

Round any number to 2 decimal places (nearest 0.01). See the bounding hundredths and the exact rounding error.

Round any decimal to the nearest whole number. See floor, ceiling and rounding error instantly on a number line.

Round any monetary amount to the nearest cent or penny (2 decimal places). Essential for retail and financial calculations.

Round any number to the nearest multiple of 10. See the tens above and below and the rounding error instantly.

Round any number to 1 decimal place (nearest 0.1). See the tenths above and below and the exact rounding error.

Round any number to the nearest 1,000 instantly, with lower and upper bounding thousands and the rounding distance.

Round any number to three decimal places (nearest 0.001) with lower and upper bounding thousandths and the rounding error.

將任何數四捨五入到選定的小數位數，或最接近的 10、100 或 1000，並附下取整與上取整值。

Convert any matrix to row echelon form (REF) step by step using Gaussian elimination with partial pivoting — up to 6 × 7.

Generate RSA public and private keys from two primes, then encrypt and decrypt an integer message — step by step.

Given two sides and the included angle (SAS), find the third side, all angles, area and perimeter using the Law of Cosines.

Calculate the area, perimeter, heights and angles of a scalene triangle from its three side lengths using Heron's formula.

Calculate the area, perimeter, all three angles, inradius and circumradius of any triangle from its three side lengths.

Plot (x, y) data pairs on a scatter chart, compute Pearson correlation r, R², and fit a least-squares regression line.

在十進位數與科學記號之間互相轉換，顯示尾數、指數與標準式。

Calculate sec(θ) — the secant function — in degrees or radians, plus all six trig function values at the same angle.

求函數在某點的二階導數，並判斷曲線在該點是凹上還是凹下。

Apply the segment addition postulate AB + BC = AC to find any missing segment length instantly.

Calculate the area of a circular segment from the radius and central angle, plus arc length, chord, and sagitta.

Calculate the area, perimeter, arc length, and diameter of a semicircle given its radius.

Calculate the nth term and sum of an arithmetic or geometric sequence. Enter the first term, common difference or ratio, and n.

Enter two sets of numbers to compute their union, intersection, difference and symmetric difference — with a visual number line.

計算兩個輸入集合的聯集、交集、差集和對稱差，並提供元素計數。

Count the significant figures in any number and round it to a specified number of sig figs, with a step-by-step explanation of the rules.

Find missing sides of similar triangles. Enter all three sides of the reference triangle and one corresponding side to get the scale factor and remaining sides.

Simplify ∛N by extracting the largest perfect-cube factor and expressing the result as k∛m in simplest radical form.

Reduce any fraction to its lowest terms in one step using the GCD — shows the simplified form, decimal value, and a visual fraction bar.

Simplify √N by extracting the largest perfect-square factor and writing the result as k√m in simplest radical form.

Reduce a ratio A:B (or A:B:C) to its simplest whole-number form by dividing all parts by their GCD.

用辛普森法則（拋物線弧）近似定積分，獲得高精度結果。

Calculate sin(θ), cos(θ), tan(θ) and cot(θ) for any angle in degrees or radians instantly.

一步計算任意角度（度數或弧度）的正弦、餘弦、正切及其倒數。

Find the sine of any angle entered in degrees. Also shows cosine, tangent and the equivalent angle in radians.

Solve a triangle using the Law of Sines: enter two angles and one side to find all remaining sides, angles, perimeter and area.

Calculate sin, cos, tan, cot, sec and csc for any angle in degrees or radians — all six trigonometric ratios at once.

Calculate sin(2θ) using the double-angle identity 2 sin θ cos θ, plus cos(2θ) and tan(2θ), for any angle in degrees or radians.

Compute sin(θ) for any angle θ in degrees or radians, with the unit-circle coordinates, cos, tan and csc shown alongside.

Calculate sin⁻¹(x) — the angle whose sine equals x — in degrees and radians, for any x in [−1, 1].

Explore y = A·sin(Bx + C) + D: compute amplitude, period, frequency, phase shift and the function value at any x.

Compute the singular values σ₁ ≥ σ₂ of a 2×2 real matrix, plus condition number, nuclear norm and Frobenius norm.

Calculate sinh(x), cosh(x), tanh(x) and coth(x) for any real x. Verifies the identity cosh²−sinh²=1 and plots the curve.

Calculate the slant height of a square pyramid from perpendicular height and base side — plus lateral area, total area and volume.

Calculate the slant height of a cone from height and base radius — plus lateral area, total surface area, volume and apex half-angle.

Convert between slope percentage, angle (degrees), rise-over-run ratio and decimal slope in any direction. Useful for roads, ramps and roof pitches.

Find the slope-intercept equation y = mx + b from two points, a point and slope, or m and b directly. Evaluate y at any x and plot the line.

Find all sides of a similar triangle using the scale factor — enter the three sides of the original and one known side of the similar triangle.

Find the standard and general equation of a sphere in 3D from center coordinates and radius, plus volume and surface area.

Calculate the surface area of a sphere from its radius using 4πr². Also shows diameter, great-circle area and circumference.

Calculate the volume, surface area, diameter and circumference of a sphere from its radius or diameter with step-by-step working.

Convert between Cartesian (x, y, z) and spherical (r, θ, φ) coordinates in both directions, with radians and degrees.

Calculate the exact arc length of an Archimedean spiral from inner radius, outer radius, and number of turns.

Calculate the area, perimeter and diagonal of a square from its side length. Supports mm, cm, m, km, in, ft, yd and mi.

Calculate the diagonal of a square from its side length using d = s√2, plus area and perimeter.

Calculate the area of a rectangle in square feet from length and width in any unit — feet, inches, yards or metres.

Calculate the area of a circle in square inches from its diameter. Also converts to square feet and square centimetres.

Find the side, area and perimeter of the largest square fitting inside a circle, given the radius — or vice versa.

Expand (a + b)² using the binomial square identity a² + 2ab + b² — with step-by-step working.

Calculate the perimeter of a square from its side length using P = 4s, plus area and diagonal.

Calculate the volume, total surface area, lateral area, slant height, and lateral edge of a right square pyramid.

Calculate the volume of a square pyramid from its base side and height using V = ⅓a²h, with step-by-step working.

Calculate the square root of any non-negative number, plus reciprocal, cube root, and a visual y = √x curve.

Solve a triangle from two sides and a non-included angle (SSA) — handles the ambiguous case where 0, 1, or 2 triangles exist.

Solve a triangle from all three side lengths (SSS) — find all angles, area, perimeter, inradius, circumradius and altitudes.

Write the standard (centre-radius) equation (x−h)²+(y−k)²=r² for any circle and find its area, circumference and axis intercepts.

Convert any number to standard form (scientific notation) a × 10ⁿ instantly, with the mantissa, exponent and step-by-step working.

Convert a linear equation from standard form Ax + By = C to slope-intercept form y = mx + b, with step-by-step working and a line plot.

Convert a number from scientific notation (a × 10ⁿ) to standard (decimal) notation with step-by-step decimal-place shifting.

Convert a circle equation between standard form (x−h)²+(y−k)²=r² and general form x²+y²+Dx+Ey+F=0 in both directions.

Calculate the area, perimeter, side length, and tip angle of a regular n-pointed star polygon given its outer and inner radii.

計算第二類斯特靈數 S(n,k)：將 n 個物品分成 k 個非空子集的分割數。

Solve the rope-around-the-Earth puzzle: how high does a rope rise when extra length is added to a loop around any sphere? Answer: Δr = ΔC ÷ 2π.

Count the subsets of a set with n elements: total 2ⁿ subsets, proper subsets, non-empty subsets and C(n,k) subsets of any size k.

Solve a 2-variable linear system by substitution — enter the six coefficients and get x, y, a graph of the intersection, and every working step.

Subtract two fractions with different denominators — finds the LCD, converts both fractions, subtracts, and simplifies to lowest terms with every step shown.

Subtract any two numbers (integers or decimals) and see minuend, subtrahend, and difference on an animated number line.

Calculate sin, cos, and tan of the sum (A+B) and difference (A−B) of two angles using the trigonometric sum and difference identities.

Calculate the sum of an arithmetic (linear) sequence given the first term, common difference and number of terms.

Find the partial sum of an arithmetic or geometric series; shows the infinite-sum limit for convergent geometric series with |r| < 1.

Calculate SUMPRODUCT = Σ(aᵢ × bᵢ): multiply matching elements of two arrays then sum all the products — identical to a vector dot product.

Find the supplementary angle (180° − θ) for any angle, plus the key trig identities sin(180°−θ) = sin θ and cos(180°−θ) = −cos θ.

Calculate the total and lateral surface area of a cylinder from radius and height — TSA = 2πr(r + h).

Calculate the area, perimeter and diagonal of a rectangle from its length and width using A = l × w.

Calculate the total surface area of a rectangular prism (cuboid) from length, width and height — TSA = 2(lw + lh + wh).

Calculate the total and lateral surface area of a rectangular pyramid from base length, base width and height.

Calculate the total surface area of a square pyramid from its base side length and height — TSA = b² + 2bs.

Calculate the total surface area and volume of a triangular prism from its three base sides and prism length.

計算曲線繞 x 軸旋轉一圈，在 a 到 b 之間所生成的表面積。

Calculate the SA:V ratio for a sphere, cube, cylinder or cuboid — essential in biology, chemistry and engineering.

Compute singular values, condition number, rank, Frobenius norm and spectral norm of a 2×2 matrix via SVD.

Calculate tan(θ) for any angle in degrees or radians — plus sin, cos, cot, sec and csc, with a plotted tangent curve.

Find the angle of a right triangle from its opposite and adjacent sides using the inverse tangent — with sin, cos, hypotenuse and a triangle diagram.

Calculate tan⁻¹(x) — the inverse tangent of any real number — in degrees or radians, with the matching right-triangle diagram.

Calculate the tangent of any angle in degrees or radians, with sin, cos, cot, sec, and a live plot of y = tan(x).

求函數在指定點 x = a 處切線的方程式。

Calculate the tangent ratio, missing side, or angle of a right triangle. Instantly find opposite, adjacent, hypotenuse, sin, and cos.

Find the tangent length from an external point to a circle, the tangent angle, power of a point, and triangle area using L = √(d² − r²).

Calculate tanh(x) — the hyperbolic tangent — plus sinh(x), cosh(x) and the derivative sech²(x), with the characteristic S-curve plot.

計算以 c 為中心的函數泰勒多項式，並與真實值進行比較。

Compute the tensor (outer) product of two 2-D vectors: returns the 2×2 result matrix with all entries, Frobenius norm, trace, and determinant.

Determine whether a fraction terminates or has a repeating decimal, and show the full decimal expansion with the repeating cycle in parentheses.

Calculate the volume, surface area, height, inradius and circumradius of a regular tetrahedron from its edge length.

Calculate sin, cos, tan, cot, sec and csc for any angle θ in degrees or radians, with quadrant and reference angle.

Calculate the surface area and volume of a torus (donut shape) from its major and minor radii using the Pappus theorem formulas.

Calculate the volume and surface area of a torus (donut shape) from its major and minor radii using V = 2π²Rr².

Find all four interior angles of an isosceles trapezoid from the two parallel sides and the leg length.

Calculate the perpendicular height of a trapezoid from its area and parallel sides, or from the equal leg length of an isosceles trapezoid.

Find the midsegment of a trapezoid from its two bases, find a missing base from the midsegment, or calculate the area using the midsegment and height.

Calculate the perimeter of a trapezoid from its four sides: two parallel bases and two non-parallel legs.

Find a missing side of a trapezoid from the perimeter and three known sides, or from the area, height, and one base.

Calculate the volume and surface area of a trapezoidal prism from the two parallel bases, the trapezoid height, and the prism length.

使用梯形法則以 n 個子區間估算曲線下的面積。

Find all three angles of a triangle from three sides (SSS) or two sides and the included angle (SAS) using the law of cosines.

Find the third interior angle of a triangle and all three exterior angles — with triangle classification (acute, right, obtuse, scalene, isosceles, equilateral).

Calculate a triangle's area from base and height, two sides and an angle (SAS), or all three sides with Heron's formula.

Find the area of a triangle from its three vertex coordinates (x₁,y₁), (x₂,y₂), (x₃,y₃) using the shoelace formula.

Calculate the area of a triangle in square feet from its base and height in any unit — feet, inches, metres, or more.

Find the area of a triangle from two sides and the included angle (SAS) using A = ½ab·sin(C). Also derives the third side and all angles.

Solve a triangle from its three sides (SSS): find all angles via the law of cosines and the area, perimeter, inradius, and circumradius via Heron's formula.

Check if two triangles are congruent or similar. Enter three sides for each triangle and see which congruence criteria (SSS, SAS, ASA, AAS, HL) apply.

Find the missing interior angle of a triangle in degrees. Enter two known angles; the third is computed as 180° minus their sum.

Find all three altitudes (heights) of a triangle from its three side lengths using the area from Heron's formula.

Check whether three side lengths satisfy the triangle inequality and can form a valid triangle. Shows all three conditions.

Calculate the perimeter, area, all three angles, inradius, and circumradius of any triangle from its three side lengths.

Apply the Triangle Proportionality Theorem: if a line is parallel to one side of a triangle, find the missing segment using AD/DB = AE/EC.

Find the actual side lengths of a triangle whose sides are in a given ratio r₁:r₂:r₃, using the known perimeter.

Find the scale factor between two triangles, verify similarity, and calculate scaled sides, area ratio and perimeter ratio.

Find the missing side of a triangle (SAS) or all angles from three sides (SSS) using the Law of Cosines.

Find the missing side of any triangle using the Law of Cosines (SAS) or the Law of Sines (AAS), plus all angles and area.

Find all sides and angles of a triangle from two angles and one side (AAS or ASA) using the Law of Sines.

Determine whether two triangles are similar using the SSS side-ratio test or the AA angle-equality test. Shows scale factor and area ratio.

Enter the three vertices of a triangle in the coordinate plane to find the slope, length and angle of each side, plus area and perimeter.

Use the triangle angle-sum theorem (A + B + C = 180°) to find any missing angle. Enter two angles and get the third instantly.

Enter three vertex coordinates to instantly find a triangle's area, side lengths, angles, perimeter, centroid and type.