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Math Calculators

Percentages, factors, fractions, equations, vectors, sequences, triangles, circles, and a safe scientific expression parser.

Everyday and classroom math: percentages, greatest common factors and least common multiples, fractions, linear and quadratic equations, systems of equations, vectors and sequences, coordinate geometry, area and volume of shapes, trigonometry, logarithms, and expression evaluation with exact arithmetic where it matters.

416 free tools · each with formula, worked example & methodology notes

Category context

The mathematical structure behind the tools

Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.

Method and background

Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.

See the complete WorldCalculate collection →

Calculator directory

Choose the tool that matches your question

416 tools arranged by purpose, so you can scan the right group before opening a calculator.

94 tools

Arithmetic and number theory

Percentages, fractions, factors, primes, ratios, and properties of numbers.

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Fraction Calculator

Add, subtract, multiply, or divide two signed fractions and obtain an exact reduced fraction.

You enter: First numerator · First denominator · Operation · Second numerator · Second denominator

Example: 5/6 = 0.8333333333

Open Fraction Calculator →

Golden Ratio Calculator

Calculate the golden ratio and its complementary reciprocal relationships from a positive reference length.

You enter: Reference length

Example: φ≈1.618034; long segment≈16.18034; reciprocal-scale segment≈6.18034.

Open Golden Ratio Calculator →

Equivalent Ratio Calculator

Solve the missing denominator in a proportion of the form a:b = c:x and verify the resulting ratio.

You enter: First ratio numerator a · First ratio denominator b · Second ratio numerator c

Example: Missing denominator: 12; both ratios equal about 0.6667.

Open Equivalent Ratio Calculator →

Prisoner’s Dilemma Payoff Calculator

Test a symmetric two-player payoff set for the strict Prisoner’s Dilemma ordering and identify dominant defection and the one-shot Nash outcome.

You enter: Temptation payoff T (defect while other cooperates) · Reward payoff R (both cooperate) · Punishment payoff P (both defect) · Sucker payoff S (cooperate while other defects)

Example: T > R > P > S; defection is strictly dominant for both players; the one-shot Nash outcome is mutual defection with payoff 1 each.

Open Prisoner’s Dilemma Payoff Calculator →

Terminating Decimal Checker

Reduce a fraction, factor its denominator, and determine whether its base-10 decimal expansion terminates and how many places it needs.

You enter: Numerator · Denominator

Example: 3/40 reduces to 3/40 = 0.075; the denominator is 2³×5, so the decimal terminates in 3 places.

Open Terminating Decimal Checker →

Two’s Complement Converter

Encode a signed integer into a fixed-width two’s-complement bit pattern and decode the same pattern back to its signed and unsigned values.

You enter: Signed integer · Bit width

Example: −37 in 8-bit two’s complement is 11011011; its unsigned residue is 219, which decodes back to −37.

Open Two’s Complement Converter →

Adding and Subtracting Fractions

Add or subtract two fractions with a least-common-denominator walkthrough, reduced exact answer, mixed-number form, and decimal check.

You enter: Operation · First numerator · First denominator · Second numerator · Second denominator

Example: The LCD is 12: 5/6 = 10/12 and 1/4 = 3/12, so 5/6 − 1/4 = 7/12 ≈ 0.583333.

Open Adding and Subtracting Fractions →

Binary Subtraction Calculator

Subtract signed binary numbers, inspect the decimal check, see aligned operands, and encode the result in a chosen two’s-complement width when it fits.

You enter: Binary minuend · Binary subtrahend · Fixed-width display

Example: 1101₂ − 110₂ = 111₂ = 7₁₀; the 8-bit two’s-complement encoding is 00000111.

Open Binary Subtraction Calculator →

Doubling Time Calculator

Find how long a quantity takes to double under a constant percentage growth rate, or find the periodic rate needed to double in a chosen time.

You enter: Calculate · Periodic growth rate · Doubling time · Initial amount

Example: At 15% growth per period, the doubling time is about 4.960 periods; 1,000 becomes 2,000 after one doubling.

Open Doubling Time Calculator →

Average Percentage Calculator

Find both the ordinary mean and the sample-size-weighted mean of a list of percentages so grouped survey or test results are not averaged incorrectly.

You enter: Percentages · Matching sample sizes

Example: The ordinary mean is 60%; the sample-size-weighted mean is 72%.

Open Average Percentage Calculator →

Compatible Numbers Estimator

Round numbers to an easy place value for quick mental arithmetic, then compare the compatible-number estimate with the exact operation and its error.

You enter: Arithmetic operation · Numbers · Rounding unit

Example: 458+673 = 1,131 exactly; compatible numbers 460+670 give an estimate of 1,130 with an absolute error of 1.

Open Compatible Numbers Estimator →

Powers of i

Reduce an integer power of the imaginary unit i using its four-step cycle and return its real and imaginary parts.

You enter: Integer exponent n

Example: 35 mod 4 = 3, so i³ = −i

Open Powers of i →

2×2 QR Decomposition Calculator

Factor a nonsingular real 2×2 matrix into an orthogonal Q matrix and an upper-triangular R matrix, then check reconstruction and orthogonality errors.

You enter: Matrix a₁₁ · Matrix a₁₂ · Matrix a₂₁ · Matrix a₂₂

Example: Q ≈ [[0.948683,−0.316228],[0.316228,0.948683]], R ≈ [[3.162278,1.581139],[0,1.581139]], with near-zero reconstruction and orthogonality errors.

Open 2×2 QR Decomposition Calculator →

Rationalize Denominator Calculator

Remove one square-root term from a numeric denominator by multiplying by its conjugate, then show the equivalent radical form and a numeric check.

You enter: Numerator · Rational denominator part b · Radical coefficient c · Positive non-square radicand d

Example: 5/(2+√3) becomes 10 − 5√3; the numeric value is approximately 1.3397459622.

Open Rationalize Denominator Calculator →

Complex Number Arithmetic

Adds, subtracts, multiplies, or divides two complex numbers and reports the magnitude.

You enter: Real part of first number · Imaginary part of first number · Real part of second number · Imaginary part of second number · Operation

Example: Result 4+6i; magnitude 7.2111.

Open Complex Number Arithmetic →

Sum of Digits

Add the decimal digits of a signed whole number without losing large integer digits.

You enter: Whole number

Example: The sum of the digits of 12345 is 15.

Open Sum of Digits →

Cofactor Matrix

Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33

Example: C(A) = [[-24, 20, -5], [18, -15, 4], [5, -4, 1]].

Open Cofactor Matrix →

Random Number Generator

Generate bounded whole numbers with optional uniqueness for simulations and classroom exercises.

You enter: Minimum inclusive · Maximum inclusive · Numbers to generate · Require unique values

Example: Five independent integers between 1 and 100 are returned.

Open Random Number Generator →

Dice Roller

Roll one or more equal-sided dice and show every roll, the total, and the average.

You enter: Sides per die · Number of dice

Example: Two six-sided dice produce two visible rolls, their total, and their average.

Open Dice Roller →

Ratio Calculator

Solve the missing fourth term in a:b = c:x and show the equivalent ratio.

You enter: First term a · Second term b · Known third term c

Example: 2:3 = 10:15, so the missing fourth term is 15.

Open Ratio Calculator →

79 tools

Algebra and equations

Solve equations, inequalities, polynomials, systems, exponents, and logarithms.

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2x2 Linear System Solver

Solve two equations in two unknowns with Cramer's rule.

You enter: a1 (x in equation 1) · b1 (y in equation 1) · c1 (right side, equation 1) · a2 (x in equation 2) · b2 (y in equation 2) · c2 (right side, equation 2)

Example: x = 2; y = 1.

Open 2x2 Linear System Solver →

Condense Logarithms Calculator

Combine three same-base logarithm terms into one equivalent logarithm and evaluate both forms numerically.

You enter: Logarithm base · Term 1 coefficient · Term 1 argument · Term 2 coefficient · Term 2 argument · Term 3 coefficient · Term 3 argument

Example: Condensed argument = 3²×5/2 = 22.5; both forms evaluate to log₁₀(22.5) ≈ 1.3522.

Open Condense Logarithms Calculator →

Expand Logarithms Calculator

Expand a logarithm of powered factors into a weighted sum of same-base logarithms and evaluate both forms.

You enter: Logarithm base · Factor 1 · Exponent 1 · Factor 2 · Exponent 2 · Factor 3 · Exponent 3

Example: Product argument = 4²×5/3 ≈ 26.667; both forms evaluate to log₁₀(26.667) ≈ 1.4265.

Open Expand Logarithms Calculator →

Mayan Numeral Converter

Convert a non-negative decimal integer into the place-value digits and readable dot-and-bar notation of the Maya vigesimal numeral system.

You enter: Decimal integer

Example: Digits from highest place to units: 8; 14; 3; 1; 12. The calendar-style expansion is 8×144,000 + 14×7,200 + 3×360 + 1×20 + 12.

Open Mayan Numeral Converter →

Rational Zeros Theorem Calculator

List rational-root candidates for an integer polynomial and test them exactly to identify the rational zeros of a linear, quadratic, or cubic polynomial.

You enter: Coefficient a (x³) · Coefficient b (x²) · Coefficient c (x) · Constant d

Example: The polynomial x³−6x²+11x−6 has rational zeros x = 1, 2, and 3.

Open Rational Zeros Theorem Calculator →

Cubic Equation Solver

Solve a real cubic equation for all three roots, including repeated and complex-conjugate roots, and classify the root pattern with the cubic discriminant.

You enter: Coefficient a of x³ · Coefficient b of x² · Coefficient c of x · Constant d

Example: x³−6x²+11x−6 = 0 has three distinct real roots: 1, 2, and 3; its discriminant is 4.

Open Cubic Equation Solver →

Interval Notation Converter

Build an interval and its matching inequality from finite or unbounded endpoints, endpoint inclusion, and an open or closed boundary choice.

You enter: Lower endpoint · Upper endpoint · Lower boundary · Upper boundary · Include lower endpoint · Include upper endpoint

Example: Interval = (-2, 5]; inequality = -2 < x ≤ 5; width = 7; midpoint = 1.5.

Open Interval Notation Converter →

FOIL Binomial Expansion Calculator

Expand two linear binomials, return the normalized quadratic coefficients, discriminant, and real or complex roots when the product is treated as an equation equal to zero.

You enter: First coefficient a · First constant b · Second coefficient c · Second constant d

Example: (2x+3)(x−4)=2x²−5x−12; discriminant = 121; roots = 4 and −1.5.

Open FOIL Binomial Expansion Calculator →

3×4 Reduced Row-Echelon Form Calculator

Transform a real 3×4 matrix into reduced row-echelon form with partial pivoting, then report rank and pivot columns without interpreting the matrix as a system solution.

You enter: Row 1, column 1 · Row 1, column 2 · Row 1, column 3 · Row 1, column 4 · Row 2, column 1 · Row 2, column 2 · Row 2, column 3 · Row 2, column 4 · Row 3, column 1 · Row 3, column 2 · Row 3, column 3 · Row 3, column 4

Example: The page returns the 3×4 reduced row-echelon matrix, its rank, and pivot-column positions after scaled row reduction.

Open 3×4 Reduced Row-Echelon Form Calculator →

Elimination Method Calculator

Solve a two-equation linear system by scaling and adding equations to eliminate the y variable, with the arithmetic shown.

You enter: Equation 1 x coefficient (a₁) · Equation 1 y coefficient (b₁) · Equation 1 right side (c₁) · Equation 2 x coefficient (a₂) · Equation 2 y coefficient (b₂) · Equation 2 right side (c₂)

Example: Scale and add to get x = 4, then substitute to get y = −1.

Open Elimination Method Calculator →

Substitution Method Calculator

Solve a two-equation linear system by isolating y in the first equation, substituting into the second, and checking the ordered pair.

You enter: Equation 1 x coefficient (a₁) · Equation 1 y coefficient (b₁) · Equation 1 right side (c₁) · Equation 2 x coefficient (a₂) · Equation 2 y coefficient (b₂) · Equation 2 right side (c₂)

Example: Equation 1 gives y = 7 − 2x; substitution gives x = 4 and y = −1.

Open Substitution Method Calculator →

Synthetic Division

Divide a polynomial by (x - c) in one row: quotient plus remainder.

You enter: Polynomial coefficients · Divisor root (c for x - c)

Example: Quotient 1, -5, 6; remainder 0, so (x - 1) is a factor.

Open Synthetic Division →

Linear Interpolation

Estimate y at an in-range x between two ordered points using a straight-line interpolation fraction.

You enter: First x0 · Second x1 · First y0 · Second y1 · Target x

Example: t = 0.25 and the interpolated y is 25.

Open Linear Interpolation →

Bilinear Interpolation

Interpolate a value inside an ordered rectangle from four finite corner values and an in-range x-y target.

You enter: Left x0 · Right x1 · Lower y0 · Upper y1 · Target x · Target y · Corner f00 · Corner f10 · Corner f01 · Corner f11

Example: s = 0.25, t = 0.25, and the interpolated value is 7.5.

Open Bilinear Interpolation →

3D Manhattan Distance

Calculate the axis-aligned Manhattan distance between two ordered points in a three-dimensional Cartesian coordinate system.

You enter: Point 1 x · Point 1 y · Point 1 z · Point 2 x · Point 2 y · Point 2 z

Example: Absolute differences are (3, 4, 12), so the Manhattan distance is 3 + 4 + 12 = 19 coordinate units.

Open 3D Manhattan Distance →

Absolute-Value Inequalities

Solve |a*x+b| in relation to a nonnegative threshold and report the interval or solution meaning explicitly.

You enter: Coefficient a · Constant b · Threshold c · Relation

Example: |2x-4| <= 6 gives center m=2 and half-width r=3, so the solution is -1 <= x <= 5, a closed interval.

Open Absolute-Value Inequalities →

Line Equation From Two Points

Construct a normalized A*x+B*y+C=0 line from two distinct two-dimensional Cartesian points, including vertical lines.

You enter: Point 1 x · Point 1 y · Point 2 x · Point 2 y

Example: The normalized coefficients are A=0.7071067812, B=-0.7071067812, C=0, giving 0.7071067812*x-0.7071067812*y=0, equivalent to x-y=0.

Open Line Equation From Two Points →

Circle Equation

Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients.

You enter: Center h · Center k · Radius r

Example: (x-2)^2+(y+3)^2=25; expanded form is x^2+y^2-4x+6y-12=0, so D=-4, E=6, and F=-12.

Open Circle Equation →

3x3 Matrix Determinant

Calculate the determinant of a bounded finite numeric 3x3 matrix in row-major order.

You enter: Matrix entry a (row 1, column 1) · Matrix entry b (row 1, column 2) · Matrix entry c (row 1, column 3) · Matrix entry d (row 2, column 1) · Matrix entry e (row 2, column 2) · Matrix entry f (row 2, column 3) · Matrix entry g (row 3, column 1) · Matrix entry h (row 3, column 2) · Matrix entry i (row 3, column 3)

Example: det([[1,2,3],[0,4,5],[1,0,6]]) = 1(24)-2(-5)+3(-4) = 22.

Open 3x3 Matrix Determinant →

Matrix Addition

Add two finite row-major 3x3 matrices entry by entry and return their deterministic matrix sum.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33 · B11 · B12 · B13 · B21 · B22 · B23 · B31 · B32 · B33

Example: A + B = [[3, 2, 4], [1, 4, 4], [5, 8, 1]].

Open Matrix Addition →

Matrix by Scalar

Multiply every entry of one finite row-major 3x3 matrix by one bounded scalar.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33 · Scalar

Example: 2A = [[2, 4, 6], [0, 2, 8], [10, 12, 0]].

Open Matrix by Scalar →

Matrix Multiplication

Multiply two finite row-major 3x3 matrices using ordered row-column dot products.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33 · B11 · B12 · B13 · B21 · B22 · B23 · B31 · B32 · B33

Example: A x B = [[4, 12, 4], [1, 11, 4], [16, 18, 5]].

Open Matrix Multiplication →

Matrix Transpose

Transpose a finite row-major 3x3 matrix by exchanging its row and column indices.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33

Example: A^T = [[1, 0, 5], [2, 1, 6], [3, 4, 0]].

Open Matrix Transpose →

Matrix Trace

Sum the three main-diagonal entries of a finite row-major 3x3 matrix.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33

Example: tr(A) = 1 + 1 + 0 = 2.

Open Matrix Trace →

Matrix Norm

Calculate the Frobenius norm of a finite row-major 3x3 matrix from all nine squared entries.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33

Example: ||A||F = sqrt(92) = 9.59166304663.

Open Matrix Norm →

Matrix Power

Raise a finite row-major 3x3 matrix to a bounded whole-number exponent from 0 through 12.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33 · Exponent

Example: A^2 = [[16, 22, 11], [20, 25, 4], [5, 16, 39]].

Open Matrix Power →

Matrix Rank

Estimate the numerical rank of a finite row-major 3x3 matrix with a documented scale-relative pivot tolerance.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33

Example: Partial-pivot elimination retains three pivots, so rank(A) = 3.

Open Matrix Rank →

Matrix Inverse

Calculate the inverse of a finite row-major 3x3 matrix through the determinant and adjugate identity with stability checks.

You enter: A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33

Example: det(A)=1, so A^-1 = [[-24, 18, 5], [20, -15, -4], [-5, 4, 1]].

Open Matrix Inverse →

11 tools

Calculus and analysis

Derivatives, integrals, limits, rates of change, series, and numerical methods.

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Riemann Sums

Left, right, and trapezoidal sums from evenly spaced heights.

You enter: Heights · Left endpoint a · Right endpoint b · Rule

Example: Left 1; right 5; trapezoid 3.

Open Riemann Sums →

Midpoint Riemann Sum

Estimates a definite integral from function values sampled at the midpoint of each equal subinterval.

You enter: Lower endpoint a · Upper endpoint b · Midpoint heights

Example: Integral estimate 4.000000 with width 1.000000 across 2 intervals.

Open Midpoint Riemann Sum →

137 tools

Geometry and measurement

Areas, volumes, triangles, circles, polygons, coordinates, and solid geometry.

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Trapezoid Area

Area of a trapezoid from its two parallel sides and height.

You enter: Parallel side a · Parallel side b · Height (distance between sides)

Example: 16 square units

Open Trapezoid Area →

Cone Volume Calculator

Calculate the volume and base area of a right circular cone from its radius and vertical height.

You enter: Base radius · Vertical height

Example: Volume: about 37.70 cubic units; base area: about 28.27 square units.

Open Cone Volume Calculator →

Regular Polygon Calculator

Calculate the perimeter, apothem, interior angle, and area of a regular polygon from its side count and side length.

You enter: Number of sides n · Side length

Example: Perimeter: 24 units; apothem: 3.464 units; area: 41.57 square units; interior angle: 120°.

Open Regular Polygon Calculator →

Möbius Strip Geometry Calculator

Estimate the exact-parametrization surface area and boundary length of a one-half-twist Möbius strip from its centerline radius and half-width.

You enter: Centerline radius · Half-width of strip

Example: The strip has centerline circumference 6.28319 m; the integrated surface area and one continuous boundary length are reported from the parametrized model.

Open Möbius Strip Geometry Calculator →

Regular Pentagon Calculator

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

You enter: Equal side length

Example: A regular pentagon with 7 length units per side has perimeter 35, apothem about 4.8173, circumradius about 5.9546, and area about 84.3034 square units.

Open Regular Pentagon Calculator →

Regular Hexagon Calculator

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

You enter: Equal side length

Example: A regular hexagon with 4 length units per side has perimeter 24, apothem about 3.4641, circumradius 4, and area about 41.5692 square units.

Open Regular Hexagon Calculator →

Regular Octagon Calculator

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

You enter: Equal side length

Example: A regular octagon with 13 length units per side has perimeter 104, apothem about 15.6924, circumradius about 16.9853, and area about 816.0042 square units.

Open Regular Octagon Calculator →

30-60-90 Triangle Calculator

Find every side, area, perimeter, and trigonometric ratio of a 30-60-90 right triangle from any one known side.

You enter: Known side length · Known side

Example: If the short leg is 6, the long leg is about 10.3923, the hypotenuse is 12, the area is about 31.1769, and the perimeter is about 28.3923 length units.

Open 30-60-90 Triangle Calculator →

Two-Circle Triangulation Calculator

Find the two possible planar intersection points from a baseline and two distances, exposing mirror-image ambiguity and the tangent case.

You enter: Distance between reference points · Distance from point A · Distance from point B

Example: x = 3.6; y = ±4.8; the two possible points are (3.6, 4.8) and (3.6, −4.8).

Open Two-Circle Triangulation Calculator →

Triangle Congruence by Side Lengths

Compare two triangles by their three side lengths, test SSS congruence, and compare perimeter and area.

You enter: Triangle 1 side a · Triangle 1 side b · Triangle 1 side c · Triangle 2 side a · Triangle 2 side b · Triangle 2 side c

Example: Congruent by side lengths; both perimeters =12 and both areas =6 square units.

Open Triangle Congruence by Side Lengths →

Triangle Scale Factor and Area Change

Calculate a triangle's linear scale factor and project how its perimeter and area change under a similar enlargement or reduction.

You enter: Source corresponding length · Target corresponding length · Source perimeter · Source area

Example: Scale factor =1.5; target perimeter =36; target area =67.5 square units; area change =125%.

Open Triangle Scale Factor and Area Change →

Triangle Inequality Calculator

Check whether three positive lengths can form a non-degenerate triangle, show every inequality slack, and calculate Heron area when valid.

You enter: Side a · Side b · Side c

Example: The sides form a valid triangle; the three slacks are 6, 4, and 2, and Heron area is 6 square units.

Open Triangle Inequality Calculator →

Ellipse Area

Area enclosed by an ellipse from its semi-major and semi-minor axes.

You enter: Semi-axis a · Semi-axis b

Example: Ellipse area 47.12.

Open Ellipse Area →

Hemisphere Geometry

Calculate a hemisphere's volume, curved surface area, and total surface area from its radius.

You enter: Radius

Example: Volume 56.548668; curved area 56.548668; total area 84.823002 square units.

Open Hemisphere Geometry →

Ellipsoid Volume

Calculate the volume of an ellipsoid from its three semi-axis lengths.

You enter: Semi-axis a · Semi-axis b · Semi-axis c

Example: Ellipsoid volume 100.530965 cubic units.

Open Ellipsoid Volume →

Pyramid Volume

Calculate the volume of a pyramid from its base area and perpendicular height.

You enter: Base area · Perpendicular height

Example: Pyramid volume 120 cubic units.

Open Pyramid Volume →

Conical Frustum Volume

Calculate the volume of a right circular frustum from its two radii and perpendicular height.

You enter: Larger radius R · Smaller radius r · Perpendicular height

Example: Frustum volume 205.250254 cubic units.

Open Conical Frustum Volume →

Kite Area

Calculate the area of a kite from the lengths of its perpendicular diagonals.

You enter: Diagonal 1 · Diagonal 2

Example: Kite area 30 square units.

Open Kite Area →

Coordinate Triangle Area

Calculate a triangle's area and orientation from the Cartesian coordinates of its three vertices.

You enter: Point 1 x · Point 1 y · Point 2 x · Point 2 y · Point 3 x · Point 3 y

Example: Triangle area 6 square units; orientation counterclockwise.

Open Coordinate Triangle Area →

Triangle Centroid

Find the centroid of a triangle from the Cartesian coordinates of its three vertices.

You enter: Point 1 x · Point 1 y · Point 2 x · Point 2 y · Point 3 x · Point 3 y

Example: Centroid (2, 2).

Open Triangle Centroid →

Volume of a Parallelepiped

Calculate the volume of a parallelepiped from three edge vectors.

You enter: Vector A x · Vector A y · Vector A z · Vector B x · Vector B y · Vector B z · Vector C x · Vector C y · Vector C z

Example: Parallelepiped volume is 24 cubic units.

Open Volume of a Parallelepiped →

Circumcenter of a 2D Triangle

Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices.

You enter: Vertex A x1 · Vertex A y1 · Vertex B x2 · Vertex B y2 · Vertex C x3 · Vertex C y3

Example: The circumcenter is (2, 1.5) and the circumradius is 2.5.

Open Circumcenter of a 2D Triangle →

3D Euclidean Distance

Calculate the straight-line Euclidean distance between two ordered points in a three-dimensional Cartesian coordinate system.

You enter: Point 1 x · Point 1 y · Point 1 z · Point 2 x · Point 2 y · Point 2 z

Example: Differences are (3, 4, 12), so the Euclidean distance is sqrt(9 + 16 + 144) = 13 coordinate units.

Open 3D Euclidean Distance →

Circle Chord Length

Find the straight chord length determined by a circle radius and a minor central angle measured in degrees.

You enter: Circle radius · Minor central angle

Example: A radius of 5 and a 60-degree central angle give c = 2 x 5 x sin(30 degrees) = 5 length units.

Open Circle Chord Length →

Great-Circle Distance

Calculate the shortest spherical-surface distance and central angle between two latitude-longitude points on a sphere.

You enter: Sphere radius · Point 1 latitude · Point 1 longitude · Point 2 latitude · Point 2 longitude

Example: The central angle is pi/2 radians and the quarter-sphere distance is 6371 x pi/2 = about 10007.543 distance units.

Open Great-Circle Distance →

Ring Torus Volume

Calculate the volume of a non-self-intersecting ring torus from its major radius and smaller tube radius.

You enter: Major radius R · Minor radius r

Example: V = 2 pi^2 x 5 x 2^2 = 40 pi^2, approximately 394.784 cubic length units.

Open Ring Torus Volume →

Convex Trapezoid Perimeter

Add the two parallel bases and two legs of a nondegenerate convex trapezoid after checking that the four lengths can form the shape.

You enter: Parallel base 1 · Parallel base 2 · Leg 1 · Leg 2

Example: The bases differ by 2, the legs satisfy the shape check, and P = 5 + 3 + 4 + 4 = 16 length units.

Open Convex Trapezoid Perimeter →

Minor Circular Segment Area

Calculate the area between a circle chord and its minor arc from radius and included central angle.

You enter: Circle radius · Minor central angle

Example: theta = pi/3 radians, so A = 1/2 x 25 x (pi/3 - sin(pi/3)) = about 2.26465 square length units.

Open Minor Circular Segment Area →

Ellipse Standard Form

Construct the standard axis-aligned ellipse equation from a center and positive horizontal and vertical semiaxes.

You enter: Center h · Center k · Horizontal semiaxis a · Vertical semiaxis b

Example: The standard equation is (x-2)^2/25 + (y+3)^2/4 = 1, with horizontal semiaxis 5 and vertical semiaxis 2.

Open Ellipse Standard Form →

Foci of an Ellipse

Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis.

You enter: Center h · Center k · Semimajor axis a · Semiminor axis b

Example: c=sqrt(25-16)=3, so the foci are (-1,3) and (5,3), focus separation is 6, and eccentricity is 0.6.

Open Foci of an Ellipse →

Cylindrical Coordinates

Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z.

You enter: Cartesian x · Cartesian y · Cartesian z

Example: rho=sqrt(1^2+1^2)=sqrt(2), azimuth=45 degrees, and cylindrical z remains 3 coordinate units.

Open Cylindrical Coordinates →

Hollow Cylinder Volume

Calculate the material volume of a uniform hollow cylinder from outer radius, inner radius, and height.

You enter: Outer radius R · Inner radius r · Height h

Example: Annular area is pi(25-4)=21pi square units, so material volume is 210pi, approximately 659.734457 cubic units.

Open Hollow Cylinder Volume →

Torus Surface Area

Calculate the surface area of an ordinary ring torus from its major and minor radii.

You enter: Major radius R · Minor radius r

Example: S=4pi^2(5)(2)=40pi^2, approximately 394.784176 square units.

Open Torus Surface Area →

Trapezoid Height

Find the perpendicular height implied by two parallel bases and a positive trapezoid area.

You enter: Parallel base 1 b1 · Parallel base 2 b2 · Trapezoid area A

Example: h=2(39)/(8+5)=78/13=6 length units.

Open Trapezoid Height →

56 tools

Trigonometry and vectors

Angles, bearings, sine and cosine relationships, vectors, and rotations.

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AAS Triangle Solver

Solve the remaining angle and two sides of a triangle from two angles and one opposite side.

You enter: Angle A · Angle B · Side opposite angle A

Example: C = 105°; b ≈ 14.142; c ≈ 19.319 length units.

Open AAS Triangle Solver →

2×2 Null Space Calculator

Check a 2×2 matrix determinant and return a representative null vector when the matrix is singular.

You enter: Matrix a₁₁ · Matrix a₁₂ · Matrix a₂₁ · Matrix a₂₂

Example: Determinant = 0; a representative null vector is (−2, 1).

Open 2×2 Null Space Calculator →

ASA Triangle Solver

Solve an angle-side-angle triangle from two angles and the side opposite the first angle.

You enter: Angle A · Angle B · Side a opposite angle A

Example: Third angle: 70°; side b: about 11.29 units; side c: about 11.41 units.

Open ASA Triangle Solver →

Sinusoidal Phase-Shift Calculator

Analyze a transformed sine or cosine function to find amplitude, period, horizontal phase shift, midline, and the value at a chosen x-coordinate.

You enter: Waveform · Amplitude coefficient A · Frequency coefficient B · Phase coefficient C · Vertical shift D · Evaluate at x

Example: Amplitude = 3; period = 4; phase shift = 0.5 units right; midline y = 2; y(1) ≈ 4.12132.

Open Sinusoidal Phase-Shift Calculator →

Interior and Exterior Triangle Angles

Enter two interior angles of a triangle to find the missing interior angle, the three adjacent exterior angles, and both angle-sum checks.

You enter: Interior angle A · Interior angle B · Angle unit

Example: C = 70°, adjacent exterior angles are 135°, 115°, and 110°; the interior sum is 180° and the exterior sum is 360°.

Open Interior and Exterior Triangle Angles →

Reference Angle

Normalize an angle, identify its quadrant or axis, and calculate the reference angle with sine, cosine, and tangent checks.

You enter: Angle · Angle unit

Example: Quadrant III; reference angle 45°; sin −0.7071 and cos −0.7071

Open Reference Angle →

Quaternion Arithmetic and Rotation Calculator

Multiply two quaternions, inspect norms, conjugates, inverses, and rotate a 3D vector with a normalized quaternion.

You enter: Quaternion 1 scalar w · Quaternion 1 i component x · Quaternion 1 j component y · Quaternion 1 k component z · Quaternion 2 scalar w · Quaternion 2 i component x · Quaternion 2 j component y · Quaternion 2 k component z · Vector x · Vector y · Vector z

Example: The first quaternion is approximately a 90° rotation about z; it rotates (1, 0, 0) to approximately (0, 1, 0).

Open Quaternion Arithmetic and Rotation Calculator →

2×2 Polar Decomposition Calculator

Decompose an invertible real 2×2 matrix into an orthogonal factor Q and a symmetric positive-definite stretch factor H, with error checks for A = QH.

You enter: Matrix a₁₁ · Matrix a₁₂ · Matrix a₂₁ · Matrix a₂₂

Example: The tool returns an orthogonal Q and symmetric positive-definite H whose product reconstructs [[2,1],[0,3]] within floating-point residuals.

Open 2×2 Polar Decomposition Calculator →

2D Vector Tensor Product Calculator

Calculate the outer tensor product of two real two-dimensional vectors and report its four components, Frobenius norm, and rank-one consistency check.

You enter: First vector component a₁ · First vector component a₂ · Second vector component b₁ · Second vector component b₂

Example: a⊗b = [[6,8],[−3,−4]], Frobenius norm = √130, and determinant = 0.

Open 2D Vector Tensor Product Calculator →

3D Vector Cross Product

Calculate the cross product and magnitude of two three-dimensional Cartesian vectors.

You enter: Vector A x · Vector A y · Vector A z · Vector B x · Vector B y · Vector B z

Example: A x B = (-3, 6, -3); magnitude 7.34847.

Open 3D Vector Cross Product →

3D Dot Product

Calculate the scalar dot product of two three-dimensional Cartesian vectors from their six ordered components.

You enter: Vector A x · Vector A y · Vector A z · Vector B x · Vector B y · Vector B z

Example: A dot B = 1 x 4 + 2 x 5 + 3 x 6 = 32.

Open 3D Dot Product →

3D Vector Addition

Add two three-dimensional Cartesian vectors component by component and return the summed x, y, and z values.

You enter: Vector A x · Vector A y · Vector A z · Vector B x · Vector B y · Vector B z

Example: A + B = (5, 7, 9).

Open 3D Vector Addition →

3D Unit Vector

Normalize a nonzero three-dimensional Cartesian vector to a vector of Euclidean length one.

You enter: Vector x · Vector y · Vector z

Example: The unit vector is (0.6, 0.8, 0).

Open 3D Unit Vector →

Catenary Curve Height

Evaluate the height of the canonical symmetric catenary y = a cosh(x/a) at a selected horizontal coordinate.

You enter: Catenary parameter a · Horizontal coordinate x

Example: The argument is 3/2 and y = 2 cosh(1.5), approximately 4.70482 length units.

Open Catenary Curve Height →

Line of Intersection of Two Planes

Find one finite point and a direction vector for the intersection line of two nonparallel planes.

You enter: Plane 1 coefficient a · Plane 1 coefficient b · Plane 1 coefficient c · Plane 1 constant d · Plane 2 coefficient a · Plane 2 coefficient b · Plane 2 coefficient c · Plane 2 constant d

Example: The planes x=1 and y=2 intersect in the line (1,2,0)+t*(0,0,1); the returned point is the closest point to the origin.

Open Line of Intersection of Two Planes →

10 tools

Sequences and discrete math

Sequences, counting, permutations, combinations, graph ideas, and discrete structures.

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29 tools

More Math Calculators tools

Additional tools in this category that do not yet need a narrower navigation label.

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M/M/1 Queueing Calculator

Calculate utilization, idle probability, average queue lengths, and waiting times for a stable single-server M/M/1 queue.

You enter: Arrival rate λ · Service rate μ

Example: Utilization = 60%; average system size L = 1.5; average queue size Lq = 0.9; mean system time W = 0.25 time units; mean queue wait Wq = 0.15 time units.

Open M/M/1 Queueing Calculator →

Logic Truth Table Calculator

Evaluate a selected two-input Boolean connective for a chosen row and display the complete truth table for AND, OR, XOR, NAND, NOR, XNOR, or implication.

You enter: Logical connective · Input A · Input B

Example: For A = True and B = False, AND is False; the complete AND table has only the True/True row as True.

Open Logic Truth Table Calculator →

Hilbert’s Hotel Mapping Calculator

Explore finite guest shifts and the classic countably infinite room mappings used to explain Hilbert’s hotel thought experiment.

You enter: Mapping scenario · Existing guests in finite demonstration · New guests in finite scenarios

Example: For one new guest, each existing guest shifts from room n to n+1; the new guest takes room 1 and rooms 1–21 are assigned in the finite demonstration.

Open Hilbert’s Hotel Mapping Calculator →

Variation Solver

Evaluates direct, inverse, and joint variation models from the constant k.

You enter: Variation mode · Constant k · Value x · Value z (joint only)

Example: y = 20 (direct: y = kx).

Open Variation Solver →

Distance from a Point to a Plane

Calculate the unsigned and signed Cartesian distance from a point to a plane given by ax + by + cz + d = 0.

You enter: Plane a · Plane b · Plane c · Plane d · Point x · Point y · Point z

Example: The unsigned distance is 1/sqrt(14), approximately 0.267261, and the signed distance is positive.

Open Distance from a Point to a Plane →

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How to use these tools

Each calculator shows its formula, assumptions, and a worked example. Enter your values, choose Calculate, and copy results for reports or planning. Financial and health tools are planning estimates with stated limits, not professional advice.

Frequently asked questions

How are expressions evaluated?

Parentheses and functions first, then right-associative powers, unary signs, multiplication and division, then addition and subtraction. Multiplication must be explicit.

Are fraction results exact?

Yes. Fractions use integer arithmetic and are reduced to lowest terms; the decimal shown is only an approximation.