Algebra, geometry & advanced math calculators
Algebra, geometry and higher maths with the working shown. Angles, units and assumptions are always stated next to the answer.
Algebra
Algebra & polynomials
- Polynomial Roots CalculatorFind every real and complex root of a polynomial up to degree 20, with exact multiplicities for repeated roots and a residual check.
- Cubic Equation CalculatorSolve any cubic equation with the trigonometric method or Cardano’s formula, with the discriminant, depressed cubic and real or complex roots.
- Quartic Equation CalculatorSolve a fourth-degree equation with Ferrari’s method: the depressed quartic, its resolvent cubic and all four real or complex roots.
- Polynomial Long Division CalculatorDivide one polynomial by another and get the exact quotient and remainder, with each long-division step and fractional coefficients kept exact.
- Synthetic Division CalculatorDivide a polynomial by x − r using synthetic division: the full tableau, the quotient, the remainder and P(r) by the remainder theorem.
Analytic geometry
- Line Intersection CalculatorFind the point where two lines cross from slope-intercept or ax + by = c form, the angle between them, and whether they are parallel or the same line.
- Parabola Vertex CalculatorEnter y = ax² + bx + c to get the vertex, axis of symmetry, focus, directrix, vertex form and whether the vertex is a minimum or maximum.
- Ellipse Eccentricity CalculatorFind an ellipse’s eccentricity, focal distance, flattening, latus rectum, area and perimeter from its semi-major and semi-minor axes.
- Hyperbola Asymptotes CalculatorGet the two asymptote equations of a hyperbola in standard form, plus its foci, vertices, focal distance and eccentricity, for any centre.
- Slope-Intercept Form CalculatorWrite a line as y = mx + b from two points, a point and a slope, or standard form Ax + By = C, with fractions, the x-intercept and standard form.
- Distance from Point to Line CalculatorFind the shortest (perpendicular) distance from a point to a line in the plane, and the foot of the perpendicular, from ax + by + c = 0, two points or y = mx + b.
Circles
- Arc Length CalculatorFind the length of a circular arc from the radius and central angle in degrees or radians, with the chord, sector area and segment area.
- Sector Area CalculatorWork out the area of a circle sector from the radius and either the central angle or the arc length, with the angle, arc and perimeter.
Complex numbers
- Complex Number Addition CalculatorAdd or subtract two complex numbers a + bi and c + di, with the result in rectangular form, its modulus and its argument in degrees.
- Complex Number Multiplication CalculatorMultiply or divide two complex numbers, with the result in rectangular form, its modulus and argument, and a check in polar form.
- Complex Number Polar Form ConverterConvert a complex number between rectangular form a + bi, polar form r∠θ and exponential form re^(iθ), with the angle in degrees or radians.
- Complex Number Modulus CalculatorFind the modulus (absolute value) of a complex number as an exact simplified surd and a decimal, plus its conjugate, unit vector and argument.
Coordinate geometry
- Slope CalculatorFind the slope between two points as a decimal and a fraction, the line’s equation y = mx + b, its angle, the distance and the midpoint.
- Distance CalculatorFind the straight-line distance between two points from their coordinates in the plane or in space, plus the midpoint and slope.
Geodesy & coordinates
- Degrees Minutes Seconds ConverterConvert decimal degrees to degrees, minutes and seconds (DMS) or back, for coordinates and angles, with the degrees-decimal-minutes form too.
- Great Circle Distance CalculatorShortest distance between two latitude/longitude points over a spherical Earth by the haversine formula, in kilometres, miles and nautical miles.
- Bearing Between Coordinates CalculatorFind the initial and final great-circle bearing from one latitude/longitude to another, in degrees from true north, with the compass point.
- Coordinate Midpoint CalculatorFind the true midpoint of two latitude/longitude points on the globe, or the geographic centre of up to 200 points, without the errors of simple averaging.
Geometry
- Triangle Circumradius CalculatorFind the radius of the circle through a triangle’s three corners from its three sides, one side and the opposite angle, or vertex coordinates.
- Triangle CalculatorSolve any triangle from three sides, two sides and the included angle, or two angles and a side: all sides, angles, area, perimeter and type.
- Area CalculatorFind the area of a rectangle, square, triangle, circle, ellipse, trapezoid, parallelogram or sector, with the result in m² and ft² too.
- Volume CalculatorCalculate the volume and surface area of a box, cube, cylinder, cone, sphere or pyramid, with the capacity in litres and US gallons.
- Circle CalculatorEnter any one of a circle’s radius, diameter, circumference or area and get the other three, using π to full precision.
- Surface Area CalculatorTotal, lateral and base surface area of a sphere, hemisphere, cube, rectangular box, cylinder, cone or square pyramid, with the volume too.
Linear algebra
- Matrix Eigenvalues CalculatorFind all eigenvalues of a square matrix up to 10 × 10, real and complex, with the exact characteristic polynomial and exact multiplicities.
- Gaussian Elimination CalculatorSolve a system of linear equations by Gaussian elimination with partial pivoting, showing every row operation, with exact fractional answers.
- Cramer’s Rule CalculatorSolve a square system of linear equations with Cramer’s rule: det(A), each replaced-column determinant and the exact solution, up to 6 unknowns.
- Matrix CalculatorFind the determinant, inverse, rank, trace or transpose of a matrix, or add, subtract and multiply two matrices, with exact fractions.
Logarithms
Number bases
Number theory
- Modular Exponentiation CalculatorCompute a to the power b modulo m exactly for whole numbers up to 300 digits, by square-and-multiply, including negative exponents via the modular inverse.
- Chinese Remainder Theorem CalculatorSolve a system of congruences x ≡ r (mod m) with the Chinese remainder theorem, including moduli that share factors, with exact big-integer arithmetic.
- Extended Euclidean Algorithm CalculatorFind gcd(a, b) and integers x, y with ax + by = gcd, the modular inverse when it exists, and every step of the extended Euclidean algorithm.
- Euler Totient CalculatorCompute Euler’s totient φ(n), the count of integers from 1 to n that are coprime to n, from the prime factorisation, for n up to 10¹⁵.
- Prime Sieve CalculatorList and count every prime between two limits up to 10 million with the sieve of Eratosthenes, with their sum, twin-prime pairs and density.
Plane figures
- Trapezoid Area CalculatorFind a trapezoid’s (trapezium’s) area from its parallel sides and height, or from all four side lengths when the height is unknown.
- Rhombus Area CalculatorFind a rhombus’s area, side, perimeter, height, both diagonals and angle from its diagonals, its side and height, or its side and an angle.
- Regular Polygon CalculatorFind the area and perimeter of any regular polygon (hexagon, octagon, n-gon) from its side, circumradius or apothem, plus its interior angle.
Solids
- Cylinder Surface Area CalculatorFind a cylinder’s curved (lateral) area, end area, total closed surface and open-top surface from its radius or diameter and height.
- Cone Surface Area CalculatorFind a cone’s lateral (curved) area, base area and total surface from the radius and either the vertical height or the slant height.
- Sphere Volume CalculatorFind the volume of a sphere or ball from its radius, diameter, circumference or surface area, with the hemisphere volume and capacity in litres.
- Torus Volume CalculatorFind the volume and surface area of a torus (ring or doughnut shape) from its major and tube radii, or from its inner and outer radii.
Triangles
- Heron’s Formula CalculatorFind a triangle’s area from its three side lengths with Heron’s formula, step by step, plus the semiperimeter, all three heights and the triangle type.
- Triangle Inradius CalculatorFind the radius of the largest circle that fits inside a triangle (the incircle) from its three sides or from its area and perimeter.
- Pythagorean Theorem CalculatorSolve a² + b² = c² for the missing side of a right triangle, with the working shown and a check for whole-number Pythagorean triples.
- Right Triangle CalculatorSolve a right triangle from two legs, a leg and the hypotenuse, or a side and an angle: all three sides, both angles, area, perimeter and altitude.
Trigonometry
- Law of Sines CalculatorSolve a triangle with the law of sines from two sides and a non-included angle (finding 0, 1 or 2 triangles) or from two angles and a side.
- Trigonometric Identity CheckerType both sides of a trigonometric identity and test them numerically at 400 angles to see whether they agree, with a counterexample if they do not.
- Radians to Degrees ConverterConvert radians to degrees or degrees to radians, with the angle as a multiple of π, in degrees-minutes-seconds, turns and gradians.
Vectors
- Vector Dot Product CalculatorFind the dot product of two vectors of any dimension, their magnitudes, the angle between them, and whether they are orthogonal or parallel.
- Vector Cross Product CalculatorFind the cross product of two 3D vectors, its magnitude, the parallelogram and triangle areas they span, and the unit normal vector.
- Vector Projection CalculatorProject one vector onto another: the vector projection, the scalar projection (component), and the perpendicular rejection, in any dimension.
- Vector Angle CalculatorFind the angle between two vectors in degrees and radians from their dot product and magnitudes, and see whether they are perpendicular or parallel.
- Vector Magnitude CalculatorWork out the magnitude (Euclidean length) of a vector from its components, plus its squared length, taxicab (L1) norm and maximum norm.
- Unit Vector CalculatorNormalise a vector by dividing each component by its magnitude to get the unit vector in the same direction, with the 2D direction angle.