A **derived unit** is a unit of measurement in the International System of Units (SI) that is derived from one or more of the seven base units. Derived units are either dimensionless or else are the product of base units.

### Derived Unit Names and Symbols

The names of the derived units are all written using lowercase letters. Most of the names are just combinations of base units, but there are 22 derived units with special names. The symbols for units named for persons begin with an uppercase letter.

### Examples of Derived Units

For example, the watt, hertz, and coulomb are derived units named for people. Their symbols are W, Hz, and C, respectively. Other examples of derived units include meters per second (m/s), cubic meters (m^{3}), and joule per kelvin (J/K).

### How Many Derived Units Are There?

There are 22 derived units with special names, including the dimensionless derived units the radian (rad) and steradian (sr). However, there are over 100 other derived units that are expressed in terms of their base units.

### Derived Unit List

Here are the 22 derived units with names:

Name | Symbol | Quantity | SI Base Units |
---|---|---|---|

hertz | Hz | frequency | s^{-1} |

radian | rad | angle | 1 |

steradian | sr | solid angle | 1 |

newton | N | force | kg⋅m⋅s^{−2} |

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

joule | J | energy | kg⋅m^{2}⋅s^{−2} |

watt | W | power | kg⋅m^{2}⋅s^{−3} |

coulomb | C | electric charge | s⋅A |

volt | V | voltage or potential difference | kg⋅m^{2}⋅s^{−3}⋅A^{−1} |

farad | F | electrical capacitance | kg^{−1}⋅m^{−2}⋅s^{4}⋅A^{2} |

ohm | Ω | electrical resistance | kg⋅m^{2}⋅s^{−3}⋅A^{−2} |

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

weber | Wb | magnetic flux | kg⋅m^{2}⋅s^{−2}⋅A^{−1} |

tesla | T | magnetic flux density | kg⋅s^{−2}⋅A^{−1} |

henry | H | electrical inductance | kg⋅m^{2}⋅s^{−2}⋅A^{−2} |

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

lumen | lm | luminous flux | cd |

lux | lx | illuminance | cd⋅m^{−2} |

becquerel | Bq | radioactive decays per unit time | s^{−1} |

gray | Gy | absorbed dose of ionizing radiation | m^{2}⋅s^{−2} |

sievert | Sv | equivalent dose of ionizing radiation | s^{−1}⋅mol |

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

### Derived Units and Dimensional Analysis

Many other derived units are mathematical combinations of SI base units, just without special names. Here are some examples:

Quantity | Symbol | Unit | Abbreviation | Derivation |
---|---|---|---|---|

area | A | square meter | m^{2} | length x width |

volume | V | cubic meter | m^{3} | length x width x height |

density | ρ | kilograms per cubic meter | kg/m3 | mass / volume |

concentration (molarity) | c or M | moles per liter | mol/L | amount / volume |

speed | v | meters per second | m/s | length / time |

acceleration | a | meters per second per second | m/s^{2} | speed / time |

angular velocity | rad/s | radians per second | rad/s | angle / time |

current density | A/m^{2} | ampere per square meter | A/m^{2} | current / time |

wave number | m^{-1} | reciprocal meter | m^{-1} | 1 / length |

specific volume | ν | cubic meter per kilogram | m^{3}/kg | volume / mass |

Use dimensional analysis when deriving units and converting them to other units. Squaring or cubing units, as in area and volume, means you also square conversion factors.

For example, converting cubic meters to cubic centimeters:

(100 cm / 1 m)^{3} = 100 cm^{3} / 1 m^{3} = 1000000 cm^{3} / 1 m^{3}

Avoid the common mistake of thinking there are 100 cm^{3} per 1 m^{3} just because there are 100 cm in 1 m!

### Who Can Make a Derived Unit?

There are so many derived units that you might think anyone could make one up, providing they use the base units as a starting point. But, a unit only comes into being if it is published in The International System of Units (SI).

The General Conference on Weights and Measures (CGPM) manages the International System of Units (SI or metric system) and makes recommendations to the International Committee for Weights and Measures (CIPM). The International Bureau of Weights and Measures (BIPM) periodically reports updates to the unit list and definitions.

### Non-SI Units

The metric system also includes several units which are neither base units nor derived units. These units exist within the metric system either because they are multiples or fractions of SI units or else they are practical.

The permitted non-SI units are:

Name | Symbol | Quantity |
---|---|---|

minute | min | time |

hour | h | time |

day | d | time |

astronomical unit | au | length |

degree | ° | plane angle |

minute | ′ | plane angle |

second | ″ | plane angle |

hectare | ha | area |

liter or litre | l | volume |

tonne | t | mass |

dalton | Da | mass |

electronvolt | eV | energy |

neper | Np | logarithmic ratio |

bel, decibel | B, dB | logarithmic ratio |

Several permitted units in the 8th SI brochure did not make it into the 9th brochure. Among these are the bar (pressure), mmHg (pressure), ångström (length), and gauss (magnetic flux density).

### References

- Atkins, Tony; Escudier, Marcel (2019).
*A Dictionary of Mechanical Engineering*. Oxford University Press. ISBN 9780199587438. - International Bureau of Weights and Measures (2006).
*The International System of Units (SI)*(8th ed.). ISBN 92-822-2213-6. - Bureau International des Poids et Mesures (2019).
*The International System of Units (SI)*(9th ed.). - Mills, I.; Cvitas, Tomislav; Homann, Klaus; Kallay, Nikola (1993).
*Quantities, Units and Symbols in Physical Chemistry*(2nd ed.). Blackwell Science Inc.