Free interactive tools

Calculators, simulators and 3D models for maths, science, engineering, geography and business. Move the sliders and see the ideas work; every number is worked out from the formulas.

Civil and structural engineering

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Compaction test calculator: OMC and MDD
Enter Proctor test results to get dry density, the compaction curve, optimum moisture content (OMC), maximum dry density…
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Sieve analysis calculator: D10, D30, D60, Cu, Cc
Enter masses retained on each sieve to get % passing, the grading curve, D10, D30, D60, Cu and Cc, and whether the soil …
3D
Soil phase diagram calculator
From bulk density, moisture content and Gs get void ratio, porosity, degree of saturation and dry density, with the soil…
3D
Beam deflection calculator with SFD and BMD
Simply supported or cantilever beam with a point or uniform load: reactions, shear force and bending moment diagrams and…
3D
Reinforced concrete beam calculator (Eurocode 2)
Moment of resistance of a singly reinforced concrete beam: neutral axis depth, lever arm, x/d check. Eurocode 2 simplifi…
3D
Euler buckling calculator for columns
Critical buckling load of a steel column for four end conditions, with effective length and slenderness. Metric or imper…
3D
Truss calculator: method of joints
Load a Pratt, Howe or Warren truss and see the force in every member, tension or compression, and the reactions. Free on…
3D
Retaining wall earth pressure calculator (Rankine)
Rankine active earth pressure on a retaining wall: Ka, the pressure diagram, thrust and overturning moment, with surchar…
3D
Hydrostatic pressure and force on a wall
Water pressure on a dam or tank wall: pressure at depth, the resultant force and where it acts. Metric or imperial.
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Mohr's circle calculator
Enter σx, σy and τxy to draw Mohr's circle and get the principal stresses, maximum shear stress and principal angle. MPa…
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Stress–strain curves for steel, aluminium and concrete
Stress-strain curves: Young's modulus, yield strength, ultimate tensile strength and fracture for steel, aluminium and c…
3D
Orthographic projection: first and third angle
Turn a 3D block and see its front view, end view and plan in first or third angle projection, with hidden detail. Free e…

Mathematics and statistics

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Quadratic graph explorer: y = ax² + bx + c
Change a, b and c and see the parabola, turning point, roots and discriminant update. Free interactive quadratic graphin…
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Interactive unit circle and trig graphs
Turn the angle on the unit circle and see sin, cos and tan on their graphs, with exact values and the CAST rule.
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Derivative and tangent explorer
Move along a curve and see the tangent, the gradient and the gradient function. Find stationary points.
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Area under a curve: integration and the trapezium rule
Estimate the area under a curve with rectangles, the midpoint rule or the trapezium rule and compare with the exact inte…
3D
Volume, surface area and nets of 3D shapes
Change the size of a cube, cylinder, cone, pyramid, prism or sphere and see its volume, surface area, cross-sections and…
3D
3D vector calculator: dot and cross product
Set two 3D vectors and see their sum, dot product, cross product and the angle between them, drawn in 3D.
3D
3D surface plotter: z = f(x, y)
Plot bowls, saddles and hills in 3D and slice through them to see partial derivatives.
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Normal distribution calculator
Find the probability between two values of a normal distribution, with z-scores and the shaded area. Free interactive st…
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Correlation and line of best fit
See scatter graphs, Pearson's correlation coefficient r, r² and the least-squares line of best fit.

Physics

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Projectile motion simulator
Set the launch speed, angle and height and see the trajectory, range, maximum height and time of flight. Metric or imper…
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Velocity–time graph: acceleration and distance
Build a velocity-time graph and get the acceleration from the slope and the distance from the area. Metric or imperial.
3D
Simple pendulum simulator
Change the length, amplitude and planet and see the period of a pendulum, solved exactly for large swings.
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Lens ray diagram simulator
Draw ray diagrams for converging and diverging lenses: image position, real or virtual, magnification. Uses 1/f = 1/u + …
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Refraction and Snell's law simulator
Shine light into or out of water, glass or diamond: Snell's law, the critical angle and total internal reflection.
3D
Electromagnetic wave simulation
Watch the electric and magnetic fields of a light wave travel, and change the wavelength to see frequency and photon ene…
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I–V characteristics: resistor, lamp, diode, thermistor
Compare current-voltage graphs for a resistor, filament lamp, diode and thermistor and read the resistance at any voltag…
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Capacitor charging and discharging: RC time constant
See a capacitor charge and discharge through a resistor, with the time constant τ = RC and the voltage at any time.
3D
Electric and magnetic field lines in 3D
See field lines around charges, dipoles and bar magnets in 3D, traced from Coulomb's law, with neutral points.
3D
The motor effect: F = BIL simulator
Change the field, current and length of wire and see the force on a conductor, with Fleming's left-hand rule.
3D
Electromagnetic induction simulator
Move a magnet into a coil and watch the meter: Faraday's law and Lenz's law, with more turns and faster motion.
3D
Transformer calculator: turns ratio
Change the turns and primary voltage to get the secondary voltage and currents of a step-up or step-down transformer.
3D
Wave interference simulator: ripple tank
Two-source interference in a ripple tank: lines of maximum, path difference and fringe spacing. Linked to Young's double…
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Radioactive decay and half-life calculator
See how many nuclei remain after any time for carbon-14, iodine-131, radon-222 and uranium-238, with the decay constant.
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Gas laws simulator: Boyle's and Charles's law
Explore Boyle's law, Charles's law and the pressure law with PV = nRT, isotherms and absolute zero. kPa or psi.

Chemistry

3D
3D molecule viewer: shapes and bond angles
Rotate molecules in 3D: VSEPR shapes with lone pairs and bond angles, isomers, benzene and glucose. Free interactive che…
3D
Atomic structure in 3D: the first 20 elements
See protons, neutrons and electron shells for hydrogen to calcium, with electron arrangement, group, period and the ion …
3D
Atomic orbitals in 3D: s, p and d shapes
See the shapes of s, p and d atomic orbitals in 3D, with their phases and nodes.
3D
Crystal lattices in 3D: NaCl, diamond, graphite, metals
Build sodium chloride, caesium chloride, diamond, graphite and metallic lattices in 3D and see the coordination number o…
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Titration curve simulator
Titration curves for strong and weak acids and bases, with the equivalence point, pKa and which indicator to use.
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Maxwell–Boltzmann distribution: rates of reaction
See how temperature and catalysts change the Maxwell-Boltzmann distribution and the fraction of molecules above the acti…

Biology

3D
DNA double helix 3D model
Type a DNA sequence and see the double helix in 3D, the complementary strand, base pairs and hydrogen bonds. Unzip it as…
3D
Cell membrane 3D model: the fluid mosaic model
Explore the fluid mosaic model of the cell membrane in 3D: phospholipid bilayer, channel and carrier proteins, glycoprot…
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Hardy–Weinberg calculator
Work out genotype frequencies p², 2pq and q² from allele frequency, and the share of carriers of a recessive condition.
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Population growth: exponential and logistic
Compare exponential (J-curve) and logistic (S-curve) population growth, carrying capacity and doubling time.

Geography and earth science

3D
Seasons and Earth's tilt simulator
Move Earth around the Sun and see where the Sun is overhead, day length and noon Sun height at your latitude through the…
3D
Phases of the Moon simulator
See why the Moon has phases: Sun, Earth and Moon in 3D, and the Moon as seen from Earth, with rise and set times.
3D
The solar system in 3D
The planets moving on their orbits at their real relative speeds, with distances, orbital periods and Kepler's third law…
3D
Plate boundaries in 3D
Divergent, convergent and transform plate boundaries in 3D: rift valleys like the East African Rift, volcanoes, fold mou…
3D
Contour lines and relief in 3D
See a landscape in 3D beside its contour map and change the contour interval. Spot hills, valleys and spurs. Metres or f…

Economics and business

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Interactive supply and demand graph
Shift supply and demand curves and watch equilibrium price, quantity, consumer and producer surplus change. Free interac…
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Break-even calculator and chart
Enter fixed costs, variable cost and price to get the break-even point, profit or loss and margin of safety, with a brea…
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Simple vs compound interest calculator
Compare simple and compound interest over time, yearly to continuous compounding, with doubling time and the rule of 72.

These models come inside StudyAI notes

Make notes on any topic you study and the right graph or 3D model appears in them, with quizzes and past-paper questions to go with it.

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