Physics Constants and Formula Sheet

34 physical constants at their current CODATA 2022 values, with units and uncertainties, and 66 core equations grouped by topic. Every formula that has a calculator on this site links straight to it. Free, no signup, and built to be printed or linked.

Constant values: NIST, Fundamental Physical Constants — Complete Listing (2022 CODATA adjustment), retrieved 2026-07-26.

Physical constants

Values are given exactly as NIST publishes them, digit grouping and all. The Copy button gives you the same number in the plain 1.602176634e-19 form that spreadsheets and code accept.

# The seven SI defining constants

Since the 2019 redefinition of the SI, these seven values are fixed by definition. They are not measured and they carry no uncertainty — the units are defined so that these numbers come out exactly right.

Quantity	Symbol	Value	Unit	Uncertainty
# Caesium-133 hyperfine transition frequencyDefines the second: one second is 9 192 631 770 periods of this transition.	ΔνCs	9 192 631 770	Hz	exact (by definition)
# Speed of light in vacuumDefines the metre. Light in a material always travels slower, by a factor equal to the refractive index.Used by: Wavelength, Snell's Law	c	299 792 458	m·s⁻¹	exact (by definition)
# Planck constantDefines the kilogram. Sets the energy of a photon of frequency f as E = hf.	h	6.626 070 15 × 10⁻³⁴	J·s	exact (by definition)
# Elementary chargeDefines the ampere. The charge of a proton; an electron carries −e.	e	1.602 176 634 × 10⁻¹⁹	C	exact (by definition)
# Boltzmann constantDefines the kelvin. The exchange rate between temperature and energy per particle. Often written kB.	k	1.380 649 × 10⁻²³	J·K⁻¹	exact (by definition)
# Avogadro constantDefines the mole: exactly this many entities.Used by: Ideal Gas Law	NA	6.022 140 76 × 10²³	mol⁻¹	exact (by definition)
# Luminous efficacy of 540 THz radiationDefines the candela, tying photometric units to radiant power.	Kcd	683	lm·W⁻¹	exact (by definition)

# Universal constants

Constants of nature that appear across mechanics, gravitation and electromagnetism. G is the least precisely known constant on this page by a wide margin — everything else here is known to at least nine significant figures.

Quantity	Symbol	Value	Unit	Uncertainty
# Reduced Planck constanth/2π. Exact by definition, but the decimal does not terminate — the trailing dots are NIST's, not a rounding.	ℏ	1.054 571 817... × 10⁻³⁴	J·s	exact (by definition)
# Newtonian constant of gravitationRelative uncertainty about 2 parts in 100 000 — poor for a fundamental constant, because gravity is extraordinarily weak and hard to isolate in the lab.	G	6.674 30 × 10⁻¹¹	m³·kg⁻¹·s⁻²	0.000 15 × 10⁻¹¹
# Standard acceleration of gravityA conventional agreed value, not a measurement: real surface gravity runs from about 9.78 m·s⁻² at the equator to about 9.83 m·s⁻² at the poles.Used by: Free Fall, Projectile Motion, Potential Energy, Tension, Terminal Velocity	gn	9.806 65	m·s⁻²	exact (by definition)
# Vacuum electric permittivityThe electric constant, 1/(μ₀c²). Sets the strength of the Coulomb force.	ε₀	8.854 187 8188 × 10⁻¹²	F·m⁻¹	0.000 000 0014 × 10⁻¹²
# Vacuum magnetic permeabilityBefore 2019 this was exactly 4π × 10⁻⁷ N·A⁻²; the redefinition of the ampere made it a measured quantity that still agrees with the old value to ten digits.	μ₀	1.256 637 061 27 × 10⁻⁶	N·A⁻²	0.000 000 000 20 × 10⁻⁶
# Characteristic impedance of vacuumμ₀c — the ratio of electric to magnetic field strength in a plane wave travelling through vacuum.	Z₀	376.730 313 412	Ω	0.000 000 059

# Atomic and electromagnetic constants

The constants that set the scale of atoms: how big they are, how tightly bound, and how strongly they couple to light.

Quantity	Symbol	Value	Unit	Uncertainty
# Fine-structure constantThe strength of the electromagnetic interaction. Being dimensionless, it has the same value in every unit system.	α	7.297 352 5643 × 10⁻³	(dimensionless)	0.000 000 0011 × 10⁻³
# Inverse fine-structure constantThe famous "roughly 1/137" — the dimensionless number that sets how strongly light couples to electric charge.	α⁻¹	137.035 999 177	(dimensionless)	0.000 000 021
# Rydberg constantFixes the wavelengths of the hydrogen spectral lines; one of the most precisely measured constants in physics.	R∞	10 973 731.568 157	m⁻¹	0.000 012
# Bohr radiusThe natural length scale of the hydrogen atom — roughly half an ångström.	a₀	5.291 772 105 44 × 10⁻¹¹	m	0.000 000 000 82 × 10⁻¹¹
# Compton wavelength (electron)h/(mec). The wavelength shift scale in Compton scattering.	λC	2.426 310 235 38 × 10⁻¹²	m	0.000 000 000 76 × 10⁻¹²
# Classical electron radiusA convenient length, not a physical size — the electron has no measured extent.	re	2.817 940 3205 × 10⁻¹⁵	m	0.000 000 0013 × 10⁻¹⁵
# Bohr magnetonThe natural unit of electron magnetic moment.	μB	9.274 010 0657 × 10⁻²⁴	J·T⁻¹	0.000 000 0029 × 10⁻²⁴
# Nuclear magnetonThe Bohr magneton's nuclear counterpart — smaller by roughly the proton-to-electron mass ratio.	μN	5.050 783 7393 × 10⁻²⁷	J·T⁻¹	0.000 000 0016 × 10⁻²⁷

# Particle and atomic masses

Rest masses in kilograms, with the energy equivalent (mc², via E = mc²) that particle physicists actually quote.

Quantity	Symbol	Value	Unit	Uncertainty
# Electron massEnergy equivalent 0.510 998 950 69(16) MeV — the number behind the 511 keV annihilation line.	me	9.109 383 7139 × 10⁻³¹	kg	0.000 000 0028 × 10⁻³¹
# Proton massEnergy equivalent 938.272 089 43(29) MeV.	mp	1.672 621 925 95 × 10⁻²⁷	kg	0.000 000 000 52 × 10⁻²⁷
# Neutron massEnergy equivalent 939.565 421 94(48) MeV — heavier than the proton, which is why a free neutron decays.	mn	1.674 927 500 56 × 10⁻²⁷	kg	0.000 000 000 85 × 10⁻²⁷
# Atomic mass constant (unified atomic mass unit)One twelfth of the mass of a carbon-12 atom; energy equivalent 931.494 103 72(29) MeV.	mu, u	1.660 539 068 92 × 10⁻²⁷	kg	0.000 000 000 52 × 10⁻²⁷
# Proton-to-electron mass ratioWhy almost all of an atom's mass sits in the nucleus.	mp/me	1836.152 673 426	(dimensionless)	0.000 000 032

# Thermodynamic and chemical constants

All of these are now exact: each is a fixed product of the SI defining constants above, so the decimals continue for ever rather than being uncertain.

Quantity	Symbol	Value	Unit	Uncertainty
# Molar gas constantNAk. The R in pV = nRT; exact since 2019 because both factors are exact.Used by: Ideal Gas Law	R	8.314 462 618...	J·mol⁻¹·K⁻¹	exact (by definition)
# Faraday constantNAe — the charge carried by one mole of electrons.	F	96 485.332 12...	C·mol⁻¹	exact (by definition)
# Stefan–Boltzmann constantA black body radiates σT⁴ watts per square metre. The fourth power is why a small temperature rise changes radiated power so sharply.	σ	5.670 374 419... × 10⁻⁸	W·m⁻²·K⁻⁴	exact (by definition)
# Wien wavelength displacement law constantλmax = b/T — the peak wavelength of a black body. At 5772 K (the Sun) it lands in visible green.	b	2.897 771 955... × 10⁻³	m·K	exact (by definition)
# Molar volume of an ideal gas (273.15 K, 101.325 kPa)22.414 litres per mole at 0 °C and one atmosphere. Note the conditions: at 25 °C and 1 bar the figure is different.Used by: Ideal Gas Law	Vm	22.413 969 54... × 10⁻³	m³·mol⁻¹	exact (by definition)

# Practical reference values

Defined conventions rather than constants of nature — but they show up in almost every applied calculation, and each is exact by agreement.

Quantity	Symbol	Value	Unit	Uncertainty
# Standard atmosphereExactly 101 325 pascals by definition. Sea-level pressure varies around it by a few percent with the weather.Used by: Pressure, Ideal Gas Law	atm	101 325	Pa	exact (by definition)
# Standard-state pressureExactly 1 bar — the thermodynamic reference pressure, and deliberately not the same thing as 1 atm.Used by: Pressure	p°	100 000	Pa	exact (by definition)
# Electron voltThe energy an electron gains crossing a potential difference of one volt; numerically identical to e because it is defined from it.Used by: Watts to Joules	eV	1.602 176 634 × 10⁻¹⁹	J	exact (by definition)

Formula sheet

Grouped by the same categories the calculators use. Rows marked with a calculator link are solved live on this site — click through, put your own numbers in, and the units are handled for you.

# Mechanicsall mechanics calculators →

Quantity	Equation	Notes & calculator
Newton's Second Law	`F = m × a`	Solve it: Newton's Second Law calculator →
Acceleration	`a = (v₁ − v₀) / t`	Solve it: Acceleration calculator →
Momentum	`p = m × v`	Solve it: Momentum calculator →
Projectile Motion	`R = v₀² · sin(2θ) / g`	Solve it: Projectile Motion calculator →
Free Fall	`t = √(2h/g), v = √(2gh)`	Solve it: Free Fall calculator →
Torque	`τ = r × F × sin(θ)`	Solve it: Torque calculator →
Terminal Velocity	`v = √(2mg / (ρ·A·Cd))`	Solve it: Terminal Velocity calculator →
Tension	`T = m × (g + a)`	Solve it: Tension calculator →
Atwood Machine	`a = (m₁ − m₂)g / (m₁ + m₂), T = 2·m₁·m₂·g / (m₁ + m₂)`	Solve it: Atwood Machine calculator →
Incline Tension	`T = m·g·(sin θ + μ·cos θ)`	Solve it: Incline Tension calculator →
Two-Rope Tension	`T = m·g / (2·sin θ)`	Solve it: Two-Rope Tension calculator →
Impact Force	`F = m·v² / (2·d)`	Solve it: Impact Force calculator →
Horizontal Projectile	`R = v·√(2h/g)`	Solve it: Horizontal Projectile calculator →
Hooke's Law	`F = k × x`	Solve it: Hooke's Law calculator →
Velocity under constant acceleration	`v = v₀ + a·t`	v₀ = initial velocity (m/s), a = constant acceleration (m/s²), t = time (s).Closest calculator →
Displacement under constant acceleration	`s = v₀·t + ½·a·t²`	s = displacement (m). Valid only while a is constant; with v₀ = 0 and a = g this is the free-fall distance.Closest calculator →
Velocity from displacement (no time)	`v² = v₀² + 2·a·s`	The kinematic equation to use when you know distance but not elapsed time. Constant acceleration only.Closest calculator →
Newton's law of universal gravitation	`F = G·m₁·m₂ / r²`	G = 6.674 30 × 10⁻¹¹ m³·kg⁻¹·s⁻², r = separation of the two centres of mass (m). Point masses or spherically symmetric bodies.
Weight	`W = m·g`	The gravitational force on a mass at rest on Earth's surface, with g ≈ 9.806 65 m/s². Weight is a force in newtons; mass is in kilograms.Closest calculator →
Centripetal acceleration and force	`a꜀ = v² / r, F꜀ = m·v² / r`	v = tangential speed (m/s), r = radius of the circular path (m). The force points to the centre; nothing supplies it unless something physically pulls or pushes inward.
Dry friction (Coulomb model)	`f = μ·N`	N = normal force (N), μ = coefficient of friction (static or kinetic). An approximation: it ignores contact area and speed.Closest calculator →
Impulse and momentum	`F·Δt = Δp = m·Δv`	Impulse (N·s) equals the change in momentum (kg·m/s). Spreading a collision over more time lowers the force — the whole principle behind crumple zones and airbags.Closest calculator →
Angular and linear speed	`v = ω·r, ω = 2π·f`	ω = angular velocity (rad/s), f = rotation frequency (Hz), r = radius (m).Closest calculator →

# Energyall energy calculators →

Quantity	Equation	Notes & calculator
Kinetic Energy	`KE = ½ × m × v²`	Solve it: Kinetic Energy calculator →
Potential Energy	`PE = m × g × h`	Solve it: Potential Energy calculator →
Work	`W = F × d × cos(θ)`	Solve it: Work calculator →
Power	`P = W / t`	Solve it: Power calculator →
Watts to Joules	`E = P × t`	Solve it: Watts to Joules calculator →
Work–energy theorem	`W_net = ΔKE = ½·m·v² − ½·m·v₀²`	The net work done on an object equals its change in kinetic energy. Holds for any force, constant or not.Closest calculator →
Elastic potential energy	`PE = ½·k·x²`	k = spring constant (N/m), x = extension or compression from the natural length (m). Valid while the spring obeys Hooke's law.Closest calculator →
Efficiency	`η = E_useful / E_in`	A dimensionless ratio, usually quoted as a percentage. Always below 1 for a real machine — the rest leaves as heat.Closest calculator →
Mass–energy equivalence	`E = m·c²`	Rest energy of a mass m, with c = 299 792 458 m/s. One gram of matter carries about 9 × 10¹³ J.
Photon energy	`E = h·f = h·c / λ`	h = 6.626 070 15 × 10⁻³⁴ J·s, f = frequency (Hz), λ = wavelength (m). Across the visible range (about 700 nm down to 400 nm) that is roughly 1.8 to 3.1 eV per photon.Closest calculator →

# Electricity & Wavesall electricity & waves calculators →

Quantity	Equation	Notes & calculator
Ohm's Law	`I = V / R, P = V² / R`	Solve it: Ohm's Law calculator →
Wavelength	`λ = v / f`	Solve it: Wavelength calculator →
Lens Equation	`1/f = 1/dₒ + 1/dᵢ`	Solve it: Lens Equation calculator →
Pressure	`P = F / A`	Solve it: Pressure calculator →
Watts to Amps	`I = P / V`	Solve it: Watts to Amps calculator →
Charge from current	`Q = I·t`	Q = charge (C), I = current (A), t = time (s). One amp is one coulomb per second.
Electrical power, three ways	`P = V·I = I²·R = V² / R`	The three forms are algebraically identical given Ohm's law; use whichever two quantities you know.Closest calculator →
Resistors in series and in parallel	`R_series = R₁ + R₂ + … 1/R_parallel = 1/R₁ + 1/R₂ + …`	Series resistance always exceeds the largest member; parallel resistance is always below the smallest.Closest calculator →
Capacitance and stored energy	`C = Q / V, E = ½·C·V²`	C = capacitance (F), Q = stored charge (C), V = voltage across the plates (V), E = stored energy (J).
Period and frequency	`T = 1 / f`	T = period in seconds, f = frequency in hertz. True of any periodic process.Closest calculator →
Coulomb's law	`F = k_e·q₁·q₂ / r², k_e = 1/(4πε₀) ≈ 8.988 × 10⁹ N·m²·C⁻²`	q = charges (C), r = separation (m). Same inverse-square shape as gravity, but roughly 10³⁶ times stronger between two protons.
Simple pendulum, small swing	`T = 2π·√(L / g)`	L = length to the centre of mass (m). Accurate to about 1% only for swings under roughly 20°; the mass of the bob does not appear.
Mass on a spring	`T = 2π·√(m / k)`	m = mass (kg), k = spring constant (N/m). Ideal massless spring, no damping.Closest calculator →

# Thermodynamicsall thermodynamics calculators →

Quantity	Equation	Notes & calculator
Specific Heat	`Q = m × c × ΔT`	Solve it: Specific Heat calculator →
Thermal Expansion	`ΔL = α × L₀ × ΔT`	Solve it: Thermal Expansion calculator →
Ideal Gas Law	`P = n·R·T / V`	Solve it: Ideal Gas Law calculator →
Latent heat (phase change)	`Q = m·L`	L = specific latent heat of fusion or vaporisation (J/kg). During a phase change the temperature does not move, so Q = mcΔT does not apply.Closest calculator →
Steady conduction through a slab	`Q/t = k·A·ΔT / d`	k = thermal conductivity (W·m⁻¹·K⁻¹), A = area (m²), d = thickness (m), ΔT = temperature difference across it (K). Steady state, one dimension.
Carnot maximum efficiency	`η_max = 1 − T_cold / T_hot`	Absolute temperatures in kelvin. An upper bound no real heat engine reaches — it assumes a reversible cycle.
Celsius and kelvin	`T(K) = T(°C) + 273.15`	An exact offset. Every gas-law and radiation formula needs kelvin, never Celsius.Closest calculator →
Volumetric thermal expansion	`ΔV = β·V₀·ΔT, β ≈ 3α`	β = volumetric expansion coefficient (K⁻¹), α = linear coefficient. β ≈ 3α holds for isotropic solids over small temperature ranges.Closest calculator →

# Fluids

Quantity	Equation	Notes & calculator
Density	`ρ = m / V`	Solve it: Density calculator →
Hydrostatic pressure with depth	`p = ρ·g·h`	ρ = fluid density (kg/m³), h = depth below the surface (m). This is the gauge pressure; add atmospheric pressure for the absolute value.Closest calculator →
Buoyant force (Archimedes)	`F_b = ρ_fluid·g·V_displaced`	The upward force equals the weight of the fluid pushed aside. An object floats when it can displace its own weight.Closest calculator →
Continuity, incompressible flow	`A₁·v₁ = A₂·v₂`	A = cross-sectional area (m²), v = mean flow speed (m/s). Narrow the pipe and the fluid must speed up.
Bernoulli's equation	`p + ½·ρ·v² + ρ·g·h = constant`	Along a streamline, for steady, incompressible, inviscid flow. Viscosity and turbulence break it, so treat it as an idealisation.
Drag force	`F_d = ½·ρ·v²·C_d·A`	C_d = drag coefficient (dimensionless), A = frontal area (m²). The v² term is why doubling speed quadruples drag.Closest calculator →

# Optics

Quantity	Equation	Notes & calculator
Snell's Law	`n₁ · sin(θ₁) = n₂ · sin(θ₂)`	Solve it: Snell's Law calculator →
Refractive index	`n = c / v`	v = speed of light in the medium (m/s). n = 1 in vacuum, about 1.33 in water and 1.5 in common glass.Closest calculator →
Critical angle for total internal reflection	`sin θ꜀ = n₂ / n₁ (requires n₁ > n₂)`	Beyond θ꜀ no light escapes into the second medium. This is what makes optical fibre work.Closest calculator →
Magnification of a thin lens	`m = −dᵢ / dₒ = hᵢ / hₒ`	dₒ, dᵢ = object and image distance (m); h = heights. A negative m means the image is inverted.Closest calculator →
Lens power	`P = 1 / f`	f = focal length in metres, so P is in dioptres — the number on a spectacle prescription. Negative for a diverging lens.Closest calculator →

# Everyday Life

Quantity	Equation	Notes & calculator
Electricity Cost	`Cost = (W / 1000) × hours × price per kWh`	Solve it: Electricity Cost calculator →

How to read this page

Exact vs measured. Since the 2019 redefinition of the SI, seven constants have fixed numerical values by definition — the units are built from them. Those rows say “exact (by definition)” and carry no uncertainty. Everything else was measured, and the Uncertainty column gives the standard (one-sigma) uncertainty published alongside it. A value like 8.314 462 618… is exact even though the decimals continue for ever: the trailing dots are NIST’s own notation for a non-terminating exact value, not a rounding.

Units are SI throughout. Mixing units is the single most common source of a wrong answer, and it is the reason every calculator on this site prints the unit next to the result. If an equation on this page needs an absolute temperature it says kelvin, and 273.15 is the exact offset from Celsius.

The equations state their conditions. Bernoulli’s equation holds for steady, incompressible, inviscid flow; the pendulum period holds for small swings; the Carnot efficiency is an upper bound no real engine reaches. Those are not disclaimers, they are part of the formula — an equation applied outside its conditions returns a number that looks fine and is wrong.

Provenance. Every constant here comes from one place: Fundamental Physical Constants — Complete Listing (2022 CODATA adjustment) (NIST, retrieved 2026-07-26; also published as NIST SP 961 (May 2024); Mohr, Newell, Taylor & Tiesinga, Rev. Mod. Phys. 97, 025002 (2025)). If you are citing a constant in work of your own, cite that adjustment rather than this page — and if you spot a digit here that disagrees with it, the NIST listing is right and we are wrong.

Frequently asked questions

Where do these constant values come from?

Every value on this page is transcribed from a single source: NIST's "Fundamental Physical Constants — Complete Listing", the 2022 CODATA adjustment (also published as NIST SP 961, May 2024), retrieved on 2026-07-26. Nothing here is derived by us or recalled from memory. If you need to cite a constant, cite NIST directly — the link is at the top and bottom of this page.

What does the number in brackets after a value mean?

It is the standard (one-sigma) uncertainty in the final digits of the value. Written out in full in the Uncertainty column here, so 6.674 30 with an uncertainty of 0.000 15 means the true value is very probably between 6.674 15 and 6.674 45, in units of 10⁻¹¹ m³·kg⁻¹·s⁻².

Why are some constants marked exact?

Since the SI was redefined in 2019, seven constants — including the speed of light, the Planck constant, the elementary charge, the Boltzmann constant and the Avogadro constant — have fixed numerical values by definition. The units are defined from them, so they cannot be measured more precisely. Anything built purely out of those seven, such as the molar gas constant R or the Stefan–Boltzmann constant, is exact too, even though its decimal expansion never terminates.

Which value of g should I use?

The standard acceleration of gravity is exactly 9.806 65 m/s² — but that is an agreed convention, not a measurement of your location. Real surface gravity runs from about 9.78 m/s² at the equator to about 9.83 m/s² at the poles, and varies with altitude and local geology. The calculators on this site use 9.806 65 m/s² internally and round it to 9.81 m/s² in worked examples; for schoolwork either is normally accepted.

Can I use this page in a class, a handout or on my own site?

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UsePhysics. "Physics Constants and Formula Sheet." UsePhysics.com. https://usephysics.com/reference. Constant values from the 2022 CODATA adjustment (NIST). Accessed [date].