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Technical calculators for engineering
Calculate, understand and verify your results.
Formulaxis brings together calculators for mechanics, fluid mechanics, electricity and thermal engineering with diagrams, units, formulas, numerical substitutions and clearly stated assumptions.
Choose your field
Open the complete library for each engineering field.
- Mathematics calculators18Useful mathematics for everyday calculations, science, technical studies and engineering: percentages, proportions, numbers, algebra, geometry, vectors, matrices and statistics.
- Physics37Classical mechanics: motion and energy.
- Mechanics41Section properties, transmissions, beams, strength of materials, stability, torsion, mass and machining.
- Electricity10Ohm’s law, power and initial circuit quantities.
- Thermal engineering12Thermal expansion, heat transfer and energy.
- Fluid mechanics17Flow, hydraulics, piping, pressure and wall pre-sizing.
Mathematics calculators
Useful mathematics for everyday calculations, science, technical studies and engineering: percentages, proportions, numbers, algebra, geometry, vectors, matrices and statistics.
Numbers & everyday maths
- Percentage calculatorAnswer the common percentage questions directly: percent of a value, percentage represented by a part, original total, or increase and decrease.
- Proportion calculatorChoose A, B, C or D as the unknown, then enter the three known values. The diagram substitutes your numbers directly into the two fractions.
- Fraction calculatorEnter two integer fractions, choose an operation and get the reduced exact fraction together with its decimal value.
- Scientific notationConvert a finite decimal number to scientific notation, or reconstruct the decimal value from a coefficient and base-10 exponent.
- Significant figuresEnter the number as written so trailing decimal zeros are preserved, then choose how many significant figures to keep.
Algebra
Geometry
- Volume calculatorChoose a common 3D shape, enter its dimensions in one consistent length unit and calculate the enclosed volume.
- Triangle solverSolve a triangle from SSS, SAS, ASA or AAS data. Find all sides and angles, area and perimeter; detect impossible triangles.
- Coordinate geometry calculatorFind distance, midpoint, slope and line equation between two 2D points, with explicit vertical-line and identical-point states.
- Circle calculatorStart from radius, diameter, circumference or area and calculate the other circle quantities on one page.
- Regular polygon calculatorCalculate area, perimeter, apothem, circumradius and angles from side count and side length of a regular polygon.
Vectors & matrices
- Vector calculatorCalculate norms, vector sums, differences, scalar products kA and unit vectors in 2D or 3D.
- Dot product and angle between vectors calculatorFind the dot product, vector norms, cosine and angle between two 2D or 3D vectors.
- Cross product calculatorCalculate A×B, its norm and the areas of the associated parallelogram and triangle from two 3D vectors.
- Matrix calculatorCalculate determinants, inverses, sums, differences and products of 2×2 or 3×3 matrices with structured matrix results.
Statistics
Physics
Classical mechanics: motion and energy.
Kinematics
Motion, free fall and trajectories.
- Braking and stopping distanceSeparate distance travelled during reaction from distance under constant deceleration. This is an ideal physics calculation, not a guaranteed real stopping distance or a driving rule.
- Speed–distance–timeUse the basic relation v = d/t for constant or average speed. Choose speed, distance or time as the unknown quantity.
- Constant acceleration motionFind final velocity and displacement from elapsed time, or use the time-free relation to find a positive final velocity or acceleration.
- Free fallCalculate vertical velocity and displacement after a time, or the fall time from a height when released from rest. The positive axis points downward.
- Projectile motionFind range, flight time and maximum height from launch speed and angle. Launch and landing must be at the same elevation; air resistance is neglected.
Forces & momentum
Forces, impulse and circular motion.
- Normal forceFind the surface reaction on a horizontal or inclined plane, including a signed vertical load on a horizontal surface. The normal force cannot pull a body toward the surface.
- Rope tensionCalculate tension for one hanging mass at rest, vertical acceleration, or two symmetric supporting ropes. In the two-rope mode the result is tension in each rope.
- Resultant forceAdd two, three or four forces in a plane. Enter each magnitude and counterclockwise angle from +x; inspect the resultant, its direction and signed components.
- Density calculatorUse ρ = m/V to calculate density, mass or volume. Choose the unknown quantity and enter the other two values.
- Inclined plane forcesResolve an object's weight into components parallel and perpendicular to an inclined plane. The normal reaction shown here assumes no other force or acceleration perpendicular to the plane.
- Friction forceEstimate the magnitude of dry friction at a contact. Choose sliding friction or the static limit: the latter is a maximum available force, not the force actually acting.
- Newton’s second lawFind net force, mass or acceleration along a chosen axis. Enter the resultant of all external forces, including friction when present.
- Momentum and impulseCalculate momentum, force impulse, average force or momentum change. Use signed components along one common axis.
- Perfectly inelastic collisionSolve a one-dimensional collision in which two bodies stick together after impact. Velocities are signed according to the positive direction shown in the diagram.
- Uniform circular motionConvert frequency, period, angular speed or tangential speed into the complete circular-motion quantities. Choose the mass mode to also calculate centripetal force.
- Center of massFind the center of mass of two positive masses on one signed coordinate axis. Both positions must refer to the same origin and positive direction.
- Gravitational forceCalculate the magnitude of the mutual gravitational force between two masses from their center-to-center separation using Newton’s law of universal gravitation.
Work & energy
Work, kinetic and gravitational potential energy.
- Mechanical powerUse average power P = W/Δt for work transferred over a duration, or P = Fv when the useful force and velocity are collinear and point in the same direction.
- Mechanical energyAdd translational kinetic energy and gravitational potential energy for one state, or use their conserved sum to find the speed at a second height. Both heights use the same zero reference.
- Work of a forceFind the work of a constant force from force magnitude, displacement length and their angle. Positive work supplies energy; negative work removes it.
- Kinetic energyFind translational kinetic energy from mass and speed, or solve for mass or speed from a known energy.
- Gravitational potential energyCalculate gravitational potential energy relative to a chosen zero level, or the energy change between two signed heights.
Rotation
Angular motion and rotational energy.
- Rotational dynamicsRelate net torque to the angular acceleration of a rigid body about a fixed axis. Choose the unknown quantity; positive means counterclockwise as viewed in the diagram.
- Rotational kinematicsFind final angular velocity and angular displacement under constant angular acceleration, or solve the time-free inverse relations.
- Rotational kinetic energyFind rotational energy, moment of inertia or angular-speed magnitude. Supply the moment of inertia about the actual axis of rotation.
- TorqueCalculate the magnitude of torque about a fixed axis from force, distance to the axis and their included angle, or solve for force or lever-arm distance.
- Angular momentumCalculate the axial angular momentum of a rigid body from its mass moment of inertia and angular speed, or solve either quantity inversely.
Optics
Lenses and refraction.
- Thin lensSolve the thin-lens equation for a real object and see the image distance and magnification. Distances are measured from the lens: f > 0 for convergence, d_i > 0 for a real image and d_i < 0 for a virtual image.
- Refraction — Snell’s lawEnter the refractive indices and the incident angle measured from the normal, not the surface. The calculator distinguishes refraction, grazing transmission and total internal reflection.
Acoustics
Sound intensity and level.
Oscillations & waves
Springs, pendulums and waves.
- Simple harmonic motionFind displacement, signed velocity and acceleration at a specified time in ideal harmonic motion. The phase convention is x = A cos(2πft): at t = 0 the body starts at +A with zero velocity.
- Doppler effectFind the observed sound frequency for a moving source and observer in a stationary medium. Positive source and observer velocities each mean motion toward the other.
- Frequency and periodConvert frequency and the duration of one cycle for any periodic signal, oscillation or rotation. Choose the quantity you know.
- Hooke’s lawCalculate force, stiffness or deformation of a linear spring using magnitudes. Displacement is measured from the unloaded rest position.
- Spring–mass natural frequencyFind undamped natural angular frequency, frequency and period from moving mass and spring stiffness. This describes oscillation about equilibrium.
- Simple pendulumEstimate period and frequency of a simple pendulum at small angular amplitude. Measure length from the pivot to the bob’s centre of mass.
- Wave speed, frequency and wavelengthFind wave speed, frequency, wavelength or period from v = fλ. This is the propagation speed of a periodic wave, not the velocity of the medium’s particles.
Mechanics
Section properties, transmissions, beams, strength of materials, stability, torsion, mass and machining.
Profiles and section properties
Calculate area, centroid, second moments of area, section moduli and radii of gyration.
- Choose a shape and calculate section modulusStarting point for solid, hollow and open-section shapes.
- Rectangular sectionArea, inertias, moduli and radii of gyration.
- Circular sectionSolid circle and centroidal section properties.
- Circular tubeAnnular section defined by diameters or thickness.
- Rectangular tubeRectangular hollow section with uniform thickness.
- I / H sectionFlanges, web, major and minor axes.
- Channel sectionCentroid and properties of an open channel.
- T sectionFlange and web calculated as a composite section.
- Angle sectionUnsymmetrical section with an offset centroid.
Transmission and rotation
Relate torque, power, speed and transmission ratio for simple rotating systems.
- Torque, power and speedCalculate one of the three quantities from the other two.
- Gear ratioRatio, output speed and output torque with efficiency.
- Spur gear forcesTangential, radial and normal forces from torque, pitch diameter and pressure angle.
- Gear pitch diameters and center distanceStandard pair geometry from module and tooth counts.
- Gear module and diametral pitchBidirectional conversion between millimetres and teeth per inch.
- Pulley and belt ratioRatio, output speed, torque and belt velocity.
- Belt length and center distancePitch length, center distance and wrap angles for two pulleys.
Bolts, preload and tightening
Estimate tightening torque, size preload and check a bolt under tensile load.
- Bolt tightening torqueEstimate required torque from preload, nominal diameter and nut factor.
- Bolt preload and clamp forceTarget preload from tensile stress area, proof stress and the selected ratio.
- Bolt tensile stress and proof loadStress, utilization and proof-load margin under total axial bolt force.
Bearings
Calculate equivalent dynamic load, basic L10 rating life and the static safety factor of a bearing.
Stress and deformation
Check axial loading, bending and combined stress states.
Beams and deflection
Calculate deflection, slope, reactions and maximum moment for standard beam cases.
Stability and buckling
Evaluate effective length, slenderness and critical load of an ideal compression member.
Shafts and torsion
Check an existing shaft or size a minimum diameter under torque.
Machining
Prepare drilling, tapping and cutting operations with a clear separation between geometry and workshop practice.
- Metric tap drill sizeTheoretical D − P diameter and common M1 to M48 drill sizes for cutting taps.
- Cutting speed, RPM and feed rateDrilling, turning and milling with metric or imperial units.
- Machining timeTime per cycle, total time and distance for one or more passes.
- Material removal rate and powerMaterial removal rate and theoretical power for milling, turning and drilling.
- Milling feed rate, chip load and spindle speedChoose the unknown quantity and enter the other three parameters. The core relation links table feed to spindle speed, active tooth count and chip load.
- Spindle RPM and cutting speedUse the effective diameter at the cutting point. The circumference relation gives the linear cutting speed directly.
- Tapping feed from pitch and spindle speedIn ideal synchronized tapping, the tool advances one thread pitch per revolution, directly linking pitch, spindle speed and axial feed.
- ISO metric thread basic dimensionsThis page calculates only the ISO metric basic-profile geometry. It does not replace tolerance limits for classes such as 6H or 6g.
- Bolt circle / PCD hole coordinatesDefine the circle diameter, hole count, first-hole angle and origin. Coordinates are calculated from the +X axis counter-clockwise.
Mass and materials
Convert section area and density into linear mass, total mass and weight.
Electricity
Ohm’s law, power and initial circuit quantities.
Essential electrical quantities
Connect voltage, current, resistance, power and energy with the fundamental relations.
- Ohm’s law and powerResistance, power and energy from U, I and t.
- Electricity cost calculator — power, time, kWh and costCalculate energy use from power and operating time, then apply your tariff per kWh. The cost result uses the same currency as the tariff you enter.
- Three-phase power, P–Q–S and currentCalculate P, Q and S from line-to-line voltage, line current and power factor, or solve current from kW or kVA.
- Single-phase power, P–Q–S and currentCalculate P, Q and S for a single-phase circuit from RMS voltage, RMS current and power factor, or solve current from kW or kVA.
- Power factor correction and reactive compensationFrom active power P and the initial and target power factors, calculate Q before and after correction, required compensation Qc, and the reduction in apparent power.
- Three-phase motor current — power, efficiency and power factorEstimate steady-state three-phase motor current from useful shaft power P₂, line-to-line voltage U, efficiency η and power factor.
- Transformer kVA, current and load sizingRelate transformer apparent power, voltage and current in single-phase or three-phase systems, or estimate required kVA from active load, power factor and a sizing margin.
- VFD frequency, motor speed and slipRelate electrical frequency, pole count, synchronous speed and shaft speed, with explicit slip for induction-motor operation.
- Single-phase / three-phase AC voltage dropCalculate AC cable voltage drop from the conductor linear resistance and reactance for single-phase or balanced three-phase circuits.
- Cable size from maximum voltage dropDetermine the conductor cross-section required to stay below a maximum voltage drop, then verify the drop with the selected preferred section.
Thermal engineering
Thermal expansion, heat transfer and energy.
Material expansion
Evaluate dimensional changes caused by a temperature variation.
Heat transfer · conduction
Calculate steady heat conduction through a homogeneous plane wall, or solve for the required thickness or thermal conductivity.
- Heat conduction calculator — Fourier's lawCalculate steady heat conduction through a homogeneous plane wall, or solve for the required thickness or thermal conductivity.
- Multilayer wall calculator — thermal resistanceCalculate steady conduction through 2 to 4 plane layers in series and see each layer's resistance and temperature drop.
- Cylindrical conduction calculator — pipe heat lossCalculate steady conduction through one homogeneous cylindrical layer from its inner diameter, thickness, conductivity and the temperature difference between its surfaces.
- Overall heat-transfer coefficient through a wallCalculate hot-fluid → wall → cold-fluid heat transfer by combining both convection resistances with conduction through one plane wall.
- Heat exchanger LMTD — duty, area and UCalculate the logarithmic mean temperature difference from four terminal temperatures, apply correction factor F, then solve Q̇, A or U.
Thermodynamics
Ideal gases and thermal energy without phase change.
- Latent heatFind the energy magnitude, transformed mass or specific latent heat in Q = mL. The model isolates a phase change and does not include heating or cooling before or after the transition.
- Thermal equilibrium temperatureFind the common final temperature of two isolated bodies with known masses, specific heats and initial temperatures. Signed heat values show which body gains or loses energy.
- Sensible heatCalculate Q = mcΔT to heat or cool a mass without a phase change. Enter specific heat and initial and final temperatures to find signed thermal energy.
- Ideal gas lawFind one unknown in PV = nRT from the other three quantities. Use absolute pressure and select your temperature unit: Celsius and Fahrenheit are converted to kelvins before calculation.
Fluid mechanics
Flow, hydraulics, piping, pressure and wall pre-sizing.
Flow rate and pipe flow
Relate area, flow and velocity, then evaluate Reynolds number and major pressure losses.
Hydraulics
Calculate essential relations between pressure, force, area, flow, speed, volume and power.
- Complete hydraulic cylinder sizingBore, indicative rod size, forces and required flow.
- Pressure, force and areaCalculate the missing quantity using F = P·A.
- Hydraulic cylinder forceExtension, retraction and piston areas.
- Hydraulic cylinder speedExtension and retraction speeds from flow rate.
- Hydraulic cylinder stroke timeOil volumes and theoretical travel times.
- Hydraulic powerPower from pressure, flow rate and efficiency.
Piping and pressure
Pre-size a wall and estimate membrane stresses in a pressurized pipe.
Hydrostatics
Upward buoyancy from displaced fluid.
- Hydrostatic pressureCalculate the pressure increase below a fluid surface, or infer depth from that pressure increase. Δp = ρgh is relative to the surface pressure; it is not absolute pressure.
- Buoyant forceCalculate upward buoyancy F_b = ρgV from fluid density, actual displaced volume and gravity. Displaced volume may differ from the total volume of the object.
Fluid dynamics
Continuity, ideal pressure balance and flow speed.
- Fluid continuity equationCompare two sections of the same incompressible stream with A₁v₁ = A₂v₂. Solve one area or speed from the other three values and obtain the common volume flow.
- Bernoulli equationCalculate pressure at either of two points from pressure, speed and elevation at the other point. The balance includes static pressure, kinetic energy per volume and gravitational energy per volume.
- Drag forceEstimate drag magnitude from fluid density, relative speed, reference area and a drag coefficient you supply. The secondary output is the mechanical power needed to overcome that drag at the same speed.
- Mach numberDivide relative speed by the local speed of sound, or recover speed from Mach number. Supply the sound speed for the actual fluid and thermodynamic state.
Technical guides
Understand before calculating
The first guides explain fundamental concepts and then lead to the appropriate calculator.
Formulaxis method
A useful result must be verifiable.
Each calculator exposes enough information for checking and reuse without turning the main interface into a theoretical course.
General formula, numerical substitution and detailed result.
Assumptions, limits and references stated under every tool.