Mathematics

Calculate the convective cloud base (lifted condensation level) in metres and feet from the surface temperature (°C) and dew point (°C) using Espy's 125 m/°C equation.

Calculate the coefficient of restitution (e = √(bounce height ÷ drop height)) from a simple ball-bounce test, on a 0 (perfectly inelastic) to 1 (perfectly elastic) scale.

Calculate the inductance of a long single-layer air-core solenoid from its number of turns, coil radius, and coil length — L = µ₀ × N² × A / l.

Find how many steps the Collatz sequence takes from any positive integer to 1 — halving even numbers and applying 3n + 1 to odd ones — plus the highest value reached along the way.

Solve the combined gas law (P₁V₁/T₁ = P₂V₂/T₂) for the final volume of a gas from its initial pressure, volume, and temperature and the new pressure and temperature.

Calculate the modulus |z| = √(a² + b²) and the argument φ = atan2(b, a) in degrees of a complex number z = a + bi from its real and imaginary parts.

Calculate the slant height, lateral surface area, and total surface area of a conical frustum from its bottom radius, top radius, and height — slant = √((R−r)²+h²), lateral = π(R+r)l, total = lateral + πR² + πr².

Calculate the volume of a conical frustum (a cone with the top sliced off) from its bottom radius, top radius, and height.

Calculate the total and lateral surface area of a cone from its base radius and slant height — total = πr(r + l), lateral = πrl.

Calculate the volume, slant height, and lateral surface area of a cone from its radius and height — volume = (1/3)πr²h, slant height = √(r²+h²), lateral surface = πrl.

Calculate the downstream velocity of an incompressible fluid with the continuity equation (A₁v₁ = A₂v₂) from the upstream area, velocity, and downstream area.

Calculate the area of a triangle from the (x, y) coordinates of its three vertices using the shoelace formula — works for any vertex order.

Calculate the electrostatic force (F = kₑ·|q₁·q₂|/r²) in newtons between two point charges from their charges and separation distance.

Calculate the magnitude of the cross product |A × B| of two 3D vectors from their x, y, and z components.

Calculate the volume, surface area, face diagonal, and space diagonal of a cube from its side length — volume = s³, surface area = 6s², face diagonal = s√2, space diagonal = s√3.

Calculate the surface area, volume, and space diagonal of a cuboid (rectangular box) from its length, width, and height — surface area = 2(lw+lh+wh), volume = lwh, diagonal = √(l²+w²+h²).

Calculate how a total current splits between two parallel resistors — I1 = I × R2 / (R1 + R2), I2 = I × R1 / (R1 + R2).

Calculate the total surface area, lateral surface area, and base area of a cylinder from its radius and height — total = 2πr² + 2πrh, lateral = 2πrh, base = πr².

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

Calculate the volume, cross-sectional area, and wall thickness of a hollow cylinder (pipe or tube) from its outer radius, inner radius, and height — volume = π·h·(R²−r²).

Convert digital storage between bits, bytes, kilobytes, megabytes, gigabytes, terabytes and their binary counterparts — KiB, MiB, GiB, TiB.

Convert data transfer rates between bit/s, kbit/s, Mbit/s, Gbit/s, B/s, kB/s, MB/s, and more — instantly, and see why a 100 Mbit/s line downloads at only 12.5 MB/s.

Convert a power level in dBm to milliwatts and watts instantly — useful for RF, Wi-Fi, cellular, and fibre engineers working with signal strength and transmitter power.

Convert a power level in dBW to watts and dBm instantly — essential for radar, satellite, and broadcast engineers working with transmitter power and link budgets.

Calculate the de Broglie (matter) wavelength λ = h / (m·v) of a moving particle in nanometres and metres from its mass and velocity.

Calculate the radioactive decay constant (λ = ln(2) / t½) and the mean lifetime of an isotope from its half-life.

Calculate the power ratio in decibels (dB = 10 · log₁₀(P₁/P₂)) from a measured power and a reference power.

Calculate voltage gain in decibels from an output and input voltage — gain (dB) = 20·log10(Vout/Vin). +20 dB is ×10 voltage, +6 dB ≈ ×2, and negative dB is attenuation.

Combine two sound pressure levels by incoherent addition and get the total level in decibels: 10 × log10(10^(L1/10) + 10^(L2/10)).

Turn a decimal into its simplest fraction, or a fraction into a decimal, in either direction.

Calculate density from mass and volume — density = mass ÷ volume, in kilograms per cubic metre (kg/m³).

Calculate the dew point in °C from the air temperature and relative humidity using the Magnus formula — the temperature at which air saturates and dew forms.

Work out how much concentrate and water to mix for a given dilution ratio like 1:10, for any target volume.

Calculate the final price after a discount and the amount you save from an original price and a discount percentage.

Calculate the straight-line (Euclidean) distance between two points (x1, y1) and (x2, y2) — distance = √((x2−x1)² + (y2−y1)²) — plus the horizontal and vertical legs.

Calculate the perpendicular distance from a point (x0, y0) to a line a·x + b·y + c = 0 in 2D using d = |a·x0 + b·y0 + c| / √(a² + b²).

Find how many positive divisors a number has and what they add up to — useful for number theory, identifying prime numbers, and checking whether a number is perfect or abundant.

Calculate the observed frequency for a moving sound source and a stationary observer — f′ = fv / (v − vₛ) when it approaches, f′ = fv / (v + vₛ) when it recedes — from the source frequency, speed of sound, source speed, and direction.

Calculate the dot product (scalar product) of two 3D vectors: A·B = ax·bx + ay·by + az·bz.

Calculate the doubling time (t = ln(2) / r) of anything growing at a constant rate, plus the quick rule-of-70 estimate, from a single growth rate.

Calculate the aerodynamic drag force (F_d = ½·ρ·v²·C_d·A) in newtons from the fluid density, velocity, drag coefficient, and cross-sectional area.

Calculate efficiency from useful output and total input — efficiency = (output ÷ input) × 100, as a percentage (%).

Calculate the electric charge (Q = I × t) in coulombs from the current and how long it flows, plus the equivalent number of electrons.

Calculate the uniform electric field strength (E = V / d) in volts per metre between two parallel plates from the voltage and their separation.

Work out what an appliance costs to run per day, month, and year from its wattage, daily run time, and your price per kilowatt-hour.

Calculate the area and circumference of an ellipse from its two semi-axes — area = πab, with Ramanujan's accurate perimeter approximation.

Calculate the eccentricity and foci of an ellipse from its semi-major and semi-minor axes — c = √(a²−b²), e = c/a. e = 0 is a circle and e → 1 is highly elongated.

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

Calculate the volume of an ellipsoid from its three semi-axes — volume = (4/3)·π·a·b·c. A sphere is the special case where a = b = c.

Convert energy between joule, kilojoule, megajoule, calorie, kilocalorie, watt-hour, kilowatt-hour, BTU, and foot-pound — instantly and with exact, joule-based conversion factors.