**SI derived units** are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate power of exponentiation (see: Buckingham π theorem). Some are dimensionless, as when the units cancel out in ratios of like quantities.

- Special names
- Examples of derived quantities and units
- Other units used with SI
- Supplementary units
- See also
- References
- Bibliography
- External links

The SI has special names for 22 of these derived units (for example, hertz, the SI unit of measurement of frequency), but the rest merely reflect their derivation: for example, the square metre (m^{2}), the SI derived unit of area; and the kilogram per cubic metre (kg/m^{3} or kg⋅m^{−3}), the SI derived unit of density.

The names of SI derived units, when written in full, are always in lowercase. However, the symbols for units named after persons are written with an uppercase initial letter. For example, the symbol for hertz is "Hz", while the symbol for metre is "m".^{ [1] }

The International System of Units assigns special names to 22 derived units, which includes two dimensionless derived units, the radian (rad) and the steradian (sr).

Name | Symbol | Quantity | Equivalents | SI base unit Equivalents |
---|---|---|---|---|

hertz | Hz | frequency | 1/s | s^{−1} |

radian | rad | angle | m/m | 1 |

steradian | sr | solid angle | m^{2}/m^{2} | 1 |

newton | N | force, weight | kg⋅m/s^{2} | kg⋅m⋅s^{−2} |

pascal | Pa | pressure, stress | N/m^{2} | kg⋅m^{−1}⋅s^{−2} |

joule | J | energy, work, heat | m⋅N, C⋅V, W⋅s | kg⋅m^{2}⋅s^{−2} |

watt | W | power, radiant flux | J/s, V⋅A | kg⋅m^{2}⋅s^{−3} |

coulomb | C | electric charge or quantity of electricity | s⋅A, F⋅V | s⋅A |

volt | V | voltage, electrical potential difference, electromotive force | W/A, J/C | kg⋅m^{2}⋅s^{−3}⋅A^{−1} |

farad | F | electrical capacitance | C/V, s/Ω | kg^{−1}⋅m^{−2}⋅s^{4}⋅A^{2} |

ohm | Ω | electrical resistance, impedance, reactance | 1/S, V/A | kg⋅m^{2}⋅s^{−3}⋅A^{−2} |

siemens | S | electrical conductance | 1/Ω, A/V | kg^{−1}⋅m^{−2}⋅s^{3}⋅A^{2} |

weber | Wb | magnetic flux | J/A, T⋅m^{2},V⋅s | kg⋅m^{2}⋅s^{−2}⋅A^{−1} |

tesla | T | magnetic induction, magnetic flux density | V⋅s/m^{2}, Wb/m^{2}, N/(A⋅m) | kg⋅s^{−2}⋅A^{−1} |

henry | H | electrical inductance | V⋅s/A, Ω⋅s, Wb/A | kg⋅m^{2}⋅s^{−2}⋅A^{−2} |

degree Celsius | °C | temperature relative to 273.15 K | K | K |

lumen | lm | luminous flux | cd⋅sr | cd |

lux | lx | illuminance | lm/m^{2} | cd⋅m^{−2} |

becquerel | Bq | radioactivity (decays per unit time) | 1/s | s^{−1} |

gray | Gy | absorbed dose (of ionizing radiation) | J/kg | m^{2}⋅s^{−2} |

sievert | Sv | equivalent dose (of ionizing radiation) | J/kg | m^{2}⋅s^{−2} |

katal | kat | catalytic activity | mol/s | s^{−1}⋅mol. |

Name | Symbol | Quantity | Expression in terms of SI base units |
---|---|---|---|

metre per second | m/s | speed, velocity | m⋅s^{−1} |

metre per second squared | m/s^{2} | acceleration | m⋅s^{−2} |

metre per second cubed | m/s^{3} | jerk, jolt | m⋅s^{−3} |

metre per second to the fourth | m/s^{4} | snap, jounce | m⋅s^{−4} |

radian per second | rad/s | angular velocity | s^{−1} |

radian per second squared | rad/s^{2} | angular acceleration | s^{−2} |

hertz per second | Hz/s | frequency drift | s^{−2} |

cubic metre per second | m^{3}/s | volumetric flow | m^{3}⋅s^{−1} |

Name | Symbol | Quantity | Expression in terms of SI base units |
---|---|---|---|

square metre | m^{2} | area | m^{2} |

cubic metre | m^{3} | volume | m^{3} |

newton-second | N⋅s | momentum, impulse | m⋅kg⋅s^{−1} |

newton metre second | N⋅m⋅s | angular momentum | m^{2}⋅kg⋅s^{−1} |

newton-metre | N⋅m = J/rad | torque, moment of force | m^{2}⋅kg⋅s^{−2} |

newton per second | N/s | yank | m⋅kg⋅s^{−3} |

reciprocal metre | m^{−1} | wavenumber, optical power, curvature, spatial frequency | m^{−1} |

kilogram per square metre | kg/m^{2} | area density | m^{−2}⋅kg |

kilogram per cubic metre | kg/m^{3} | density, mass density | m^{−3}⋅kg |

cubic metre per kilogram | m^{3}/kg | specific volume | m^{3}⋅kg^{−1} |

joule-second | J⋅s | action | m^{2}⋅kg⋅s^{−1} |

joule per kilogram | J/kg | specific energy | m^{2}⋅s^{−2} |

joule per cubic metre | J/m^{3} | energy density | m^{−1}⋅kg⋅s^{−2} |

newton per metre | N/m = J/m^{2} | surface tension, stiffness | kg⋅s^{−2} |

watt per square metre | W/m^{2} | heat flux density, irradiance | kg⋅s^{−3} |

square metre per second | m^{2}/s | kinematic viscosity, thermal diffusivity, diffusion coefficient | m^{2}⋅s^{−1} |

pascal-second | Pa⋅s = N⋅s/m^{2} | dynamic viscosity | m^{−1}⋅kg⋅s^{−1} |

kilogram per metre | kg/m | linear mass density | m^{−1}⋅kg |

kilogram per second | kg/s | mass flow rate | kg⋅s^{−1} |

watt per steradian square metre | W/(sr⋅m^{2}) | radiance | kg⋅s^{−3} |

watt per steradian cubic metre | W/(sr⋅m^{3}) | radiance | m^{−1}⋅kg⋅s^{−3} |

watt per metre | W/m | spectral power | m⋅kg⋅s^{−3} |

gray per second | Gy/s | absorbed dose rate | m^{2}⋅s^{−3} |

metre per cubic metre | m/m^{3} | fuel efficiency | m^{−2} |

watt per cubic metre | W/m^{3} | spectral irradiance, power density | m^{−1}⋅kg⋅s^{−3} |

joule per square metre second | J/(m^{2}⋅s) | energy flux density | kg⋅s^{−3} |

reciprocal pascal | Pa^{−1} | compressibility | m⋅kg^{−1}⋅s^{2} |

joule per square metre | J/m^{2} | radiant exposure | kg⋅s^{−2} |

kilogram square metre | kg⋅m^{2} | moment of inertia | m^{2}⋅kg |

newton metre second per kilogram | N⋅m⋅s/kg | specific angular momentum | m^{2}⋅s^{−1} |

watt per steradian | W/sr | radiant intensity | m^{2}⋅kg⋅s^{−3} |

watt per steradian metre | W/(sr⋅m) | spectral intensity | m⋅kg⋅s^{−3} |

Name | Symbol | Quantity | Expression in terms of SI base units |
---|---|---|---|

mole per cubic metre | mol/m^{3} | molarity, amount of substance concentration | m^{−3}⋅mol |

cubic metre per mole | m^{3}/mol | molar volume | m^{3}⋅mol^{−1} |

joule per kelvin mole | J/(K⋅mol) | molar heat capacity, molar entropy | m^{2}⋅kg⋅s^{−2}⋅K^{−1}⋅mol^{−1} |

joule per mole | J/mol | molar energy | m^{2}⋅kg⋅s^{−2}⋅mol^{−1} |

siemens square metre per mole | S⋅m^{2}/mol | molar conductivity | kg^{−1}⋅s^{3}⋅A^{2}⋅mol^{−1} |

mole per kilogram | mol/kg | molality | kg^{−1}⋅mol |

kilogram per mole | kg/mol | molar mass | kg⋅mol^{−1} |

cubic metre per mole second | m^{3}/(mol⋅s) | catalytic efficiency | m^{3}⋅s^{−1}⋅mol^{−1} |

Name | Symbol | Quantity | Expression in terms of SI base units |
---|---|---|---|

coulomb per square metre | C/m^{2} | electric displacement field, polarization density | m^{−2}⋅s⋅A |

coulomb per cubic metre | C/m^{3} | electric charge density | m^{−3}⋅s⋅A |

ampere per square metre | A/m^{2} | electric current density | m^{−2}⋅A |

siemens per metre | S/m | electrical conductivity | m^{−3}⋅kg^{−1}⋅s^{3}⋅A^{2} |

farad per metre | F/m | permittivity | m^{−3}⋅kg^{−1}⋅s^{4}⋅A^{2} |

henry per metre | H/m | magnetic permeability | m⋅kg⋅s^{−2}⋅A^{−2} |

volt per metre | V/m | electric field strength | m⋅kg⋅s^{−3}⋅A^{−1} |

ampere per metre | A/m | magnetization, magnetic field strength | m^{−1}⋅A |

coulomb per kilogram | C/kg | exposure (X and gamma rays) | kg^{−1}⋅s⋅A |

ohm metre | Ω⋅m | resistivity | m^{3}⋅kg⋅s^{−3}⋅A^{−2} |

coulomb per metre | C/m | linear charge density | m^{−1}⋅s⋅A |

joule per tesla | J/T | magnetic dipole moment | m^{2}⋅A |

square metre per volt second | m^{2}/(V⋅s) | electron mobility | kg^{−1}⋅s^{2}⋅A |

reciprocal henry | H^{−1} | magnetic reluctance | m^{−2}⋅kg^{−1}⋅s^{2}⋅A^{2} |

weber per metre | Wb/m | magnetic vector potential | m⋅kg⋅s^{−2}⋅A^{−1} |

weber metre | Wb⋅m | magnetic moment | m^{3}⋅kg⋅s^{−2}⋅A^{−1} |

tesla metre | T⋅m | magnetic rigidity | m⋅kg⋅s^{−2}⋅A^{−1} |

ampere radian | A⋅rad | magnetomotive force | A |

metre per henry | m/H | magnetic susceptibility | m^{−1}⋅kg^{−1}⋅s^{2}⋅A^{2} |

Name | Symbol | Quantity | Expression in terms of SI base units |
---|---|---|---|

lumen second | lm⋅s | luminous energy | s⋅cd |

lux second | lx⋅s | luminous exposure | m^{−2}⋅s⋅cd |

candela per square metre | cd/m^{2} | luminance | m^{−2}⋅cd |

lumen per watt | lm/W | luminous efficacy | m^{−2}⋅kg^{−1}⋅s^{3}⋅cd |

Name | Symbol | Quantity | Expression in terms of SI base units |
---|---|---|---|

joule per kelvin | J/K | heat capacity, entropy | m^{2}⋅kg⋅s^{−2}⋅K^{−1} |

joule per kilogram kelvin | J/(K⋅kg) | specific heat capacity, specific entropy | m^{2}⋅s^{−2}⋅K^{−1} |

watt per metre kelvin | W/(m⋅K) | thermal conductivity | m⋅kg⋅s^{−3}⋅K^{−1} |

kelvin per watt | K/W | thermal resistance | m^{−2}⋅kg^{−1}⋅s^{3}⋅K |

reciprocal kelvin | K^{−1} | thermal expansion coefficient | K^{−1} |

kelvin per metre | K/m | temperature gradient | m^{−1}⋅K |

Some other units such as the hour, litre, tonne, bar, and electronvolt are not SI units, but are widely used in conjunction with SI units.

Until 1995, the SI classified the radian and the steradian as *supplementary units*, but this designation was abandoned and the units were grouped as derived units.^{ [3] }

The **candela** is the unit of luminous intensity in the International System of Units (SI). It measures luminous power per unit solid angle emitted by a light source in a particular direction. Luminous intensity is analogous to radiant intensity, but instead of simply adding up the contributions of every wavelength of light in the source's spectrum, the contribution of each wavelength is weighted by the standard luminosity function. A common wax candle emits light with a luminous intensity of roughly one candela. If emission in some directions is blocked by an opaque barrier, the emission would still be approximately one candela in the directions that are not obscured.

The **kilogram** is the unit of mass in the International System of Units (SI), having the unit symbol **kg**. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a **kilo** colloquially. It means 'one thousand grams'.

The **litre** or **liter** is a metric unit of volume. It is equal to 1 cubic decimetre (dm^{3}), 1000 cubic centimetres (cm^{3}) or 0.001 cubic metre (m^{3}). A cubic decimetre occupies a volume of 10 cm × 10 cm × 10 cm and is thus equal to one-thousandth of a cubic metre.

**Measurement** is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. The scope and application of measurement are dependent on the context and discipline. In natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the *International vocabulary of metrology* published by the International Bureau of Weights and Measures. However, in other fields such as statistics as well as the social and behavioural sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales.

A **physical quantity** is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a *value*, which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For example, the physical quantity of mass can be quantified as '32.3 kg ', where '32.3' is the numerical value and 'kg' is the Unit.

The **radian**, denoted by the symbol **rad**, is the unit of angle in the International System of Units (SI), and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit. The radian is defined in the SI as being a dimensionless unit with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing.

The **International System of Units,** known by the international abbreviation **SI** in all languages and sometimes pleonastically as the **SI system**, is the modern form of the metric system and based on the metre as the unit of length and either the kilogram as the unit of mass or the kilogram-force as the unit of force.</ref> and the world's most widely used system of measurement. Established and maintained by the General Conference on Weights and Measures (CGPM), it is the only system of measurement with an official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.

A **metric prefix** is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The prefix *kilo-*, for example, may be added to *gram* to indicate *multiplication* by one thousand: one kilogram is equal to one thousand grams. The prefix *milli-*, likewise, may be added to *metre* to indicate *division* by one thousand; one millimetre is equal to one thousandth of a metre.

The **steradian** or **square radian** is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a *length* on the circumference, a solid angle in steradians, projected onto a sphere, gives an *area* on the surface. The name is derived from the Greek στερεός *stereos* 'solid' + radian.

The **caesium standard** is a primary frequency standard in which the photon absorption by transitions between the two hyperfine ground states of caesium-133 atoms is used to control the output frequency. The first caesium clock was built by Louis Essen in 1955 at the National Physical Laboratory in the UK. and promoted worldwide by Gernot M. R. Winkler of the United States Naval Observatory.

The **metric system** is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the International System of Units (SI) in the mid-20th century, under the oversight of an international standards body. Adopting the metric system is known as *metrication*.

A **dimensionless quantity** is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of **one**, which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time. **Dimensionless units** are dimensionless values that serve as units of measurement for expressing other quantities, such as radians (rad) or steradians (sr) for plane angles and solid angles, respectively. For example, optical extent is defined as having units of metres multiplied by steradians.

In science and engineering, the **parts-per notation** is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are **parts-per-million**, **parts-per-billion**, **parts-per-trillion** and **parts-per-quadrillion**. This notation is not part of the International System of Units (SI) system and its meaning is ambiguous.

A **geometrized unit system**, **geometric unit system** or **geometrodynamic unit system** is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, *c*, and the gravitational constant, *G*, are set equal to unity.

The **International System of Quantities** (**ISQ**) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities. It is inherently incomplete because the number of quantities is potentially infinite.

The **Unified Code for Units of Measure** (**UCUM**) is a system of codes for unambiguously representing measurement units. Its primary purpose is machine-to-machine communication rather than communication between humans.

A **unit of measurement** is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement.

**Quantity calculus** is the formal method for describing the mathematical relations between *abstract* physical quantities. Its roots can be traced to Fourier's concept of dimensional analysis (1822). The basic axiom of quantity calculus is Maxwell's description of a physical quantity as the product of a "numerical value" and a "reference quantity". De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full axiomatization has yet to be completed. Such axiomatization needs to begin from a definition of *quantity* in terms of *physical dimension*(see dimensional analysis) which is indeed a more fundamental concept than of *unit* or *unit quantity* or *unit of measurement*.

In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram. Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are now defined by setting exact numerical values, when expressed in SI units, for the Planck constant, the elementary electric charge, the Boltzmann constant, and the Avogadro constant, respectively. The second, metre, and candela had previously been redefined using physical constants. The four new definitions aimed to improve the SI without changing the value of any units, ensuring continuity with existing measurements. In November 2018, the 26th General Conference on Weights and Measures (CGPM) unanimously approved these changes, which the International Committee for Weights and Measures (CIPM) had proposed earlier that year after determining that previously agreed conditions for the change had been met. These conditions were satisfied by a series of experiments that measured the constants to high accuracy relative to the old SI definitions, and were the culmination of decades of research.

A **coherent system of units** is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities.

- ↑ Suplee, Curt (2 July 2009). "Special Publication 811".
- ↑ International Bureau of Weights and Measures (2006),
*The International System of Units (SI)*(PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 4 June 2021, retrieved 16 December 2021 - ↑ "Resolution 8 of the CGPM at its 20th Meeting (1995)". Bureau International des Poids et Mesures . Retrieved 23 September 2014.

- Media related to SI derived units at Wikimedia Commons

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