Double and Half Angle Calculator
Last reviewed 7 September 2026
The ± in the half-angle formulas is resolved from the quadrant of θ/2 rather than of θ, which is the part most often dropped and the part that is actually wrong when it is.
Sum and Difference Identities Calculator
Last reviewed 7 September 2026
Points out that these are the foundation rather than a curiosity — setting B = A gives the double-angle formulas and rearranging those gives the half-angle ones.
Phase Shift Calculator
Last reviewed 7 September 2026
States the direction convention outright: a positive shift moves the graph RIGHT, even though the sign inside the bracket is a minus.
Distance Calculator
Last reviewed 7 September 2026
Squaring removes the sign, so the order of the two points never matters; that is stated rather than left for the reader to discover.
Midpoint Calculator
Last reviewed 7 September 2026
Notes that the midpoint of a diagonal is what proves a quadrilateral is a parallelogram, since both diagonals then share it.
Slope Calculator
Last reviewed 7 September 2026
Adds the angle of inclination and the gradient as a percentage, because a road sign and a roof pitch use those rather than the fraction.
Line Equation Calculator
Last reviewed 7 September 2026
Standard form is normalised the conventional way, with integer coefficients and A non-negative, so two people solving the same problem get the same equation.
Parallel and Perpendicular Lines Calculator
Last reviewed 7 September 2026
Separates parallel from identical, which is the distinction that decides whether a pair of simultaneous equations has no solution or infinitely many.
Vector Calculator
Last reviewed 7 September 2026
Says what each product measures — a dot product of zero is a right angle whatever the lengths, and a cross product of zero is parallelism — so the two are chosen on purpose rather than by memory.
Absolute Value Calculator
Last reviewed 7 September 2026
Values are parsed from the digits typed rather than through a double, so the input echoed back beside the answer is the number the reader entered.
Floor and Ceiling Calculator
Last reviewed 7 September 2026
Computed on exact rationals parsed from the digits typed, so flooring never depends on whether a decimal happens to be representable in binary.
Digital Root Calculator
Last reviewed 7 September 2026
Lets a reader feed it a wrong sum on purpose, so the one-in-nine failure mode is something you can watch rather than something you have to take on trust.
Place Value Calculator
Last reviewed 7 September 2026
Decomposed from the digit string rather than by dividing a floating-point value repeatedly by ten, which would lose the low decimal places long before the last one.
Consecutive Integers Calculator
Last reviewed 7 September 2026
The suite re-derives every case from 1 to 300 across run lengths 2 to 8 by exhaustive search rather than trusting the closed form.
Absolute Value Equation Calculator
Last reviewed 7 September 2026
Roots are kept as exact fractions, so |3x − 1| = 5 returns −4/3 rather than −1.3333333, which is the answer a marker wants.
Polar Coordinates Calculator
Last reviewed 7 September 2026
States its own conventions — 0° to 360° rather than −180° to 180°, and the mathematical rather than the physics reading of θ and φ — because a formula copied from the other tradition is silently wrong.
Linear Regression Calculator
Last reviewed 7 September 2026
Computed from centred sums rather than raw sums of squares, which is the numerically stable arrangement for data far from the origin.
Parabola Calculator
Last reviewed 7 September 2026
The suite checks each vertex a second way, by sampling the curve either side and confirming the reported point really is the extreme value rather than merely a point on the curve.
Polygon Area Calculator
Last reviewed 7 September 2026
Notes that on integer coordinates the area is always a whole number of halves, which is the fact Pick’s theorem rests on.
Intersection of Two Lines Calculator
Last reviewed 7 September 2026
Takes both lines in standard form, so a vertical line needs no special case — and tests perpendicularity on the dot product of the normals, which keeps working when one line has no slope.
Plane Equation Calculator
Last reviewed 7 September 2026
States that a plane equation is only fixed up to a scale factor, so two correct answers to the same problem can look different.
Matrix Calculator
Last reviewed 7 September 2026
Powers use binary exponentiation rather than repeated multiplication, so a twelfth power is four multiplications rather than eleven.
Determinant Calculator
Last reviewed 7 September 2026
Explains what the number measures — the factor by which area or volume is scaled, with a negative one reversing orientation — rather than presenting it as a rule to memorise.
Matrix Inverse Calculator
Last reviewed 7 September 2026
Argues that solving usually beats inverting: computing an inverse and multiplying is about three times the work of solving directly, and numerically worse.
Row Echelon Form Calculator
Last reviewed 7 September 2026
Reports the rank-nullity theorem with the reader's own numbers, and gives a null space basis the page verifies is sent to exactly zero.
Eigenvalue Calculator
Last reviewed 7 September 2026
Explains complex eigenvalues geometrically: a rotation has no real direction it leaves unturned, which is exactly what having none means.
Matrix Decomposition Calculator
Last reviewed 7 September 2026
Treats a Cholesky failure as a result rather than an error: reaching a non-positive value under the square root is exactly what 'not positive definite' means, which makes attempting one the cheapest test for that property.
Regular Polygon Calculator
Last reviewed 7 September 2026
Separates the two radii a polygon has, since the area formula wants the apothem and using the circumradius instead overstates the answer — by more the fewer sides there are.
Quadrilateral Calculator
Last reviewed 7 September 2026
Presents the trapezoid formula as an average, which contains the others — set the parallel sides equal and it becomes a parallelogram's, set one to zero and it becomes a triangle's.
Ellipse Calculator
Last reviewed 7 September 2026
Shows Ramanujan's second approximation being exact for a circle and degrading to about four parts in ten thousand for a 1000:1 ellipse, which is the range where a reader would otherwise be quietly misled.
Triangle Centers Calculator
Last reviewed 7 September 2026
Reports when the circumcentre and orthocentre leave the triangle, which happens for every obtuse one and surprises people who have only drawn acute examples.
Circular Segment Calculator
Last reviewed 7 September 2026
Handles major segments without a special case: past 180° the sine goes negative, so the triangle term correctly adds to the sector rather than subtracting.
Second Moment of Area Calculator
Last reviewed 7 September 2026
Quantifies why tubes and I-beams are shaped as they are: a tube keeps about a third of the stiffness for a fifth of the material, which is nearly twice the stiffness per unit of material.
Sphere Calculator
Last reviewed 7 September 2026
Shows the circumscribing cylinder alongside, so Archimedes' two-thirds result is visible in both volume and surface area rather than merely asserted.
Cylinder Calculator
Last reviewed 7 September 2026
Gives the least-material shape for the entered volume — height equal to diameter — and says why almost no real can is that shape: the ends are thicker gauge than the wall, so the true optimum shifts.
Cone Calculator
Last reviewed 7 September 2026
Derives πrl by unrolling the lateral surface into a sector, and reports the sector angle to cut, which turns the formula into something you can make a paper cone from.
Pyramid Calculator
Last reviewed 7 September 2026
Covers the regular tetrahedron with its arccos(1/3) dihedral angle, which is why five of them around an edge leave a 7.5° gap and tetrahedra do not tile space — an error of Aristotle's that stood for eighteen centuries.
Prism Calculator
Last reviewed 7 September 2026
Explains what makes a solid a prism — a cross-section that does not change along its length — which is also why anything tapering carries a factor of a third and a prism carries none.
Torus and Ellipsoid Calculator
Last reviewed 7 September 2026
Refuses a tube radius at or above the ring radius, since past that point the torus self-intersects and the Pappus formulas double-count the overlap rather than describing the solid.
Binary Arithmetic Calculator
Last reviewed 7 September 2026
Runs on arbitrary-precision integers, so a number far beyond a register width is handled exactly rather than pushed through a double.
Bitwise Calculator
Last reviewed 7 September 2026
Notes that XOR is its own inverse, which is why it underlies toggling, the one-time pad and RAID reconstruction.
Bit Shift Calculator
Last reviewed 7 September 2026
Flags when bits fall off the end, which is silent overflow in a fixed-width register — and notes that rotations lose nothing and are therefore reversible.
Twos Complement Calculator
Last reviewed 7 September 2026
Wraps out-of-range values rather than refusing them, since wrapping is what hardware does, and says explicitly that the wrap is silent and that signed overflow is undefined behaviour in C.
Floating Point Calculator
Last reviewed 7 September 2026
This is the page every 'computed on exact rationals' footnote elsewhere on the site points at — it shows why the linear algebra, shoelace and line-intersection engines refuse to use doubles for yes/no questions.
RSA Calculator
Last reviewed 7 September 2026
Explains why references disagree about d — the original paper uses Euler's φ(n) and modern implementations use Carmichael's λ(n), giving a smaller but equally valid key — which is why the suite verifies by round trip rather than by matching a published example.
Logarithm Rules Calculator
Last reviewed 7 September 2026
States outright that log(a + b) is not log a + log b, with a numeric counterexample, because it is the commonest error on the topic and there is no exponent law for a sum for it to mirror.
Partial Fraction Decomposition Calculator
Last reviewed 7 September 2026
Says why the technique exists at all — 1/((x−1)(x−2)) has no obvious antiderivative and splits into two logarithms — because otherwise it is a curious piece of algebra with no visible purpose.
Fuel Economy Converter
Last reviewed 7 September 2026
Conversion factors are exact — the 1959 mile and both statutory gallons — so all the uncertainty lives in the label figure, which is a test-cycle result rather than a measurement of driving.
Numeral Base Converter
Last reviewed 7 September 2026
Arbitrary-precision integers throughout, so a 20-digit hex constant keeps every digit past the point where ordinary numbers stop being exact.
Colour Converter
Last reviewed 7 September 2026
Also states two things a converter usually omits: RGB 128 is the byte midpoint but has a relative luminance near 0.216, so channel arithmetic is not brightness arithmetic; and greys have no hue at all, so the 0 every implementation reports is a convention rather than a measurement.
Coordinates Converter
Last reviewed 7 September 2026
Carries correctly at the 60 boundary, so a rounded second never prints as 60, and uses hemisphere letters rather than signs so that S with a negative is refused as ambiguous rather than doubly applied.
Octane Rating Converter
Last reviewed 7 September 2026
Verified by 50 automated cases, asserting that completing a triple from any of the three scales reproduces the same numbers, that the two conversion directions move oppositely with sensitivity, and that octane gain and energy content move in opposite directions across every ethanol blend.
Water Hardness Converter
Last reviewed 7 September 2026
Verified by 59 automated cases, asserting each factor against its molar-mass derivation rather than a decimal, that the ion factors 2.497 and 4.118 are separately correct, and that the classification bands tile the line with no gap or overlap.
Paper Weight Converter
Last reviewed 7 September 2026
Verified by 51 automated cases, asserting the spread across grades rather than any single conversion, the exact 5 gram A4 sheet at zero tolerance, and that caliper refuses without a bulk figure.
Viscosity Converter
Last reviewed 7 September 2026
Verified by 52 automated cases, asserting that every cross-family unit pair is refused, that the Saybolt ratio falls monotonically rather than being constant, and that the Saybolt round trip closes across four decades.
Watts to Amps Calculator
Last reviewed 7 September 2026
Verified by 50 automated cases, asserting that current refuses to compute without a positive voltage in every phase mode, that single-phase arithmetic overstates a three-phase current by exactly root three at every input, and that a poor power factor increases current rather than decreasing it.
Force Converter
Last reviewed 7 September 2026
Verified by 57 automated cases, asserting the exact factors as products of their defining constants rather than as copied decimals, that the three tons-force are three distinct values, and that mass refuses to become force without a positive gravity.
Illuminance Converter
Last reviewed 7 September 2026
Verified by 50 automated cases, asserting the refusals as hard as the arithmetic -- that no area, no beam angle and no reflectance each return nothing rather than a number -- alongside the cone formula at every whole degree and the inverse-square law across nine distances.
Exposure Stop Calculator
Last reviewed 7 September 2026
Verified by 73 automated cases, including that every marked aperture is a power of root two from f/1, that the printed rounding is real but under a tenth of a stop, and that exposure holds across all sixteen control trades.
Data Transfer Rate Converter
Last reviewed 7 September 2026
Verified by 66 automated cases, including that every unit's bits-or-bytes and decimal-or-binary flags match its actual arithmetic, and that overhead lengthens a transfer by exactly 1/(1 minus p).
Cake Pan Size Converter
Last reviewed 7 September 2026
Verified by 60 automated cases, including that the square-to-round ratio is size-independent across a seven-size sweep, that no linear comparison is offered between different shapes, and that every substitution is reciprocal.
Clothing Size Converter
Last reviewed 7 September 2026
Verified by 62 automated cases, including that the men's EU size equals the chest in cm halved at every size swept, that every women's result carries its convention flag, and that every size band resolves back to its own size at the midpoint.
Density Converter
Last reviewed 7 September 2026
Verified by 66 automated cases, including that API falls at every step as density rises, that all five water references give genuinely different answers, and that lb/in3 is exactly 1728 times lb/ft3.
Fractional Inch Converter
Last reviewed 7 September 2026
Verified by 65 automated cases, including that every rounding stays within half a division, that every fraction is returned in lowest terms across a full 128-step sweep, and that no round metric size below 100 mm lands on a binary fraction.
Dimensional Weight Calculator
Last reviewed 7 September 2026
Verified by 68 automated cases, including that the billable weight is never below the actual weight at any input, that the crossover flips exactly once at break-even, and that rounding never reduces a bill.
Hardness Converter
Last reviewed 7 September 2026
Verified by 80 automated cases, including each scale asserted from its own geometry, the refusal asserted for every material without a table, and the Brinell similarity law checked across genuinely different test setups.
Flow Rate Converter
Last reviewed 7 September 2026
Verified by 66 automated cases, including that the two gallon units genuinely differ, that standard and actual round-trip across 30 pressure and temperature combinations, and that mass flow refuses without a density.
Sheet Metal Gauge Converter
Last reviewed 7 September 2026
Verified by 72 automated cases, including that each series obeys its own defining rule, that aluminium alone is geometric while steel and stainless are arithmetic, and that every gauge round-trips through its own thickness.
Note to Frequency Converter
Last reviewed 7 September 2026
Verified by 90 automated cases, including the sign as well as the size of every temperament error, the comma asserted as integers rather than as a rounded cents value, and every MIDI note round-tripped at four reference pitches.
Textile Weight Converter
Last reviewed 7 September 2026
Verified by 76 automated cases, including the direction of all six systems swept across a 9-point range, the absence of any yarn-to-fabric conversion function, and the width factor checked at five bolt widths.
Bra Size Converter
Last reviewed 7 September 2026
Verified by 71 automated cases, including that the band methods differ by four inches at all 101 measurements swept and that the letters agree through DD and diverge at every step above it.
Wind Speed and Beaufort Converter
Last reviewed 7 September 2026
Verified by 75 automated cases, including that every fitted force midpoint falls inside its published mph band.
Indian Land Unit Converter
Last reviewed 7 September 2026
Verified by 55 automated cases, including that every variable unit refuses a null state and that the fixed units are exact fractions of an acre.
Oven Temperature Converter
Last reviewed 7 September 2026
Verified by 61 automated cases, including that the naive gas-mark formula works above mark 1 and demonstrably fails below it, and a 301-temperature sweep of the nearest-mark search.
Ring Size Converter
Last reviewed 7 September 2026
Verified by 93 automated cases, including that all 26 UK letters land on US quarter sizes and that whole US sizes 4 to 12 all land on UK half-letters.
Pixel Density Calculator
Last reviewed 7 September 2026
Verified by 55 automated cases, including that the PPI ranking and the sharpness ranking are genuinely different orderings, and that the reported retina distance yields exactly 60 PPD.
Slope and Grade Converter
Last reviewed 7 September 2026
Verified by 76 automated cases, including that all five entry points agree on the same slope and that every completed triangle satisfies Pythagoras.
Tyre Size Calculator
Last reviewed 7 September 2026
Verified by 77 automated cases including a 60-size formula sweep and assertions that the alternatives search never proposes an unfittable width.
Aspect Ratio Calculator
Last reviewed 7 September 2026
Verified by 67 automated cases, including a 45-combination sweep proving the letterbox-crop identity from an independently derived figure.
Wire Gauge Converter
Last reviewed 7 September 2026
Verified by 79 automated cases including the exact defining endpoints, the published area table, and a monotonicity sweep across the whole range.
Shoe Size Converter
Last reviewed 7 September 2026
Formulas derived from the barleycorn (one third of an inch) and the Paris point (two thirds of a centimetre) rather than transcribed from a chart. Verified by 67 automated cases including a 605-case rounding sweep and exact round-trips in every system.
Cooking Converter
Last reviewed 7 September 2026
Every volume is derived from a defining constant -- the 231-cubic-inch US gallon and the 4.54609-litre imperial gallon -- rather than typed in. Verified by 52 automated cases covering the defining volumes, the regional spreads, range width by ingredient class, linear scaling, and refusal on unknown measures.
Text Case Converter
Last reviewed 7 September 2026
A test caught a real internationalisation bug during the build: the naive title case used a word-boundary regex, and since that boundary is ASCII it capitalised the wrong letter in accented words, turning eire into eIre. Now matched on an explicit non-letter instead.
Percentage Difference Calculator
Last reviewed 6 September 2026
Both percentage changes for the same pair are shown beside the difference, so the distinction between a symmetric comparison and a based one is visible on the reader’s own numbers rather than argued.
Percent Error Calculator
Last reviewed 6 September 2026
Added to the Math category’s Statistics group alongside the Standard Deviation Calculator; the symmetric no-reference comparison is linked to the new Percentage Difference Calculator rather than duplicated here.
Doubling Time Calculator
Last reviewed 6 September 2026
A falling quantity is given a halving time and no doubling time at all, because ln(2) divided by a negative logarithm returns a negative number that is not a duration.
Slope Percentage Calculator
Last reviewed 6 September 2026
Added to the Math category as its first Geometry tool; roof and drainage slopes convert here but the quantities to buy stay with the DIY & Materials calculators.
Quadratic Formula Calculator
Last reviewed 6 September 2026
Added to the Math category as the first tool in a new Algebra subcategory, alongside the discriminant, completing-the-square and factoring calculators published the same day.
Discriminant Calculator
Last reviewed 6 September 2026
Maddrey’s discriminant function, a clinical index that shares the name, is explicitly named as out of scope rather than silently omitted.
Completing the Square Calculator
Last reviewed 6 September 2026
Carries the full derivation of the quadratic formula from the method, which the Quadratic Formula Calculator links to rather than repeating.
Factoring Trinomials Calculator
Last reviewed 6 September 2026
A trinomial with no integer factorisation is reported as prime with the reason, rather than falling back to decimal approximations of irrational factors.
System of Equations Calculator
Last reviewed 6 September 2026
Carries Cramer’s rule with its precondition stated: a square system and a non-zero determinant, and an explicit note that a zero determinant does not distinguish "no solution" from "infinitely many".
Elimination Method Calculator
Last reviewed 6 September 2026
Answers are substituted back into the ORIGINAL equations rather than the transformed ones, so an error in the scaling step cannot survive the check.
Substitution Method Calculator
Last reviewed 6 September 2026
Handles an equation already written as y = something, which is the shape a word problem usually produces and the case substitution suits best.
FOIL Calculator
Last reviewed 6 September 2026
Recognises the difference of squares and the perfect square, and explains where the vanishing or doubled middle term comes from.
Multiplying Polynomials Calculator
Last reviewed 6 September 2026
Offers the grid, the vertical layout and the plain distributive list, and reports the degree check that catches a lost leading term.
Adding and Subtracting Polynomials Calculator
Last reviewed 6 September 2026
Multivariate throughout, because x²y and xy² are not like terms and that is the case the strict definition exists for.
Polynomial Division Calculator
Last reviewed 6 September 2026
Reports the remainder theorem and the factor theorem against the division itself, so P(c) and the remainder are shown agreeing rather than asserted to agree.
Cubic Equation Calculator
Last reviewed 6 September 2026
Vieta's relations are computed independently from the coefficients and shown beside the roots, so the answer carries its own check.
Rational Zeros Calculator
Last reviewed 6 September 2026
Treats a zero constant term as the theorem's own limit and factors x out first, rather than returning an empty candidate list that reads as no rational roots.