SI coherent derived units

An SI derived unit is defined by the product of powers of one or more SI base unit. If a derived unit includes no numerical factor other than one, it is termed a coherent derived unit. The base units and coherent derived units of the SI form a coherent set, designated the set of coherent SI units.

Each physical quantity has only one coherent SI unit, even if this unit can be expressed in different forms by use of special names and symbols. The inverse, however, is not true: in some cases the same SI unit can be used to express the values of several different quantities.

All SI coherent derived units can be defined directly from the SI defining constants.

Coherence

When coherent units are used, equations between the numerical values of quantities take exactly the same form as the equations between the quantities themselves. i.e. If only units from a coherent set are used, conversion factors between units are never required.

The expression for the coherent unit of a derived quantity may be obtained from the dimensional product of that quantity by replacing the symbol for each dimension by the symbol of the corresponding base unit.

Examples of SI coherent derived units

Name	Symbol	Quantity	Base units
square metre	m²	area	m²
$1 \mspace{4mu} \text{m}^2 \mspace{6mu} = \dfrac{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^2}{(299 \mspace{4mu} 792 \mspace{4mu} 458)^2} \mspace{6mu} \dfrac{c^2}{{\Delta \nu _{Cs}}^2}\\ \\ \\ 1 \mspace{4mu} \text{m}^2 \mspace{6mu} \approx 9.402 \mspace{4mu} 391 \mspace{4mu} 313 \mspace{4mu} 904 \mspace{4mu} 046 \mspace{4mu} 861 \times10^{2} \mspace{6mu} c^2 \mspace{4mu} {\Delta \nu _{Cs}}^{-2}$
cubic metre	m³	volume	m³
$1 \mspace{4mu} \text{m}^3 \mspace{6mu} = \dfrac{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^3}{(299 \mspace{4mu} 792 \mspace{4mu} 458)^3} \mspace{6mu} \dfrac{c^3}{{\Delta \nu _{Cs}}^3}\\ \\ \\ 1 \mspace{4mu} \text{m}^3 \mspace{6mu} \approx 2.883 \mspace{4mu} 085 \mspace{4mu} 241 \mspace{4mu} 129 \mspace{4mu} 260 \mspace{4mu} 961 \times10^{4} \mspace{6mu} c^3 \mspace{4mu} {\Delta \nu _{Cs}}^{-3}$
metre per second	m/s	speed, velocity	m s^-1
$1 \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-1} \mspace{6mu} = \dfrac{1}{299 \mspace{4mu} 792 \mspace{4mu} 458} \mspace{6mu} c\\ \\ \\ 1 \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-1} \mspace{6mu} \approx 3.335 \mspace{4mu} 640 \mspace{4mu} 951 \mspace{4mu} 981 \mspace{4mu} 520 \mspace{4mu} 496 \times10^{-9} \mspace{6mu} c$
metre per second squared	m s^-2	acceleration	m s^-2
$1 \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-2} \mspace{6mu} = \dfrac{1}{(299 \mspace{4mu} 792 \mspace{4mu} 458)(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)} \mspace{6mu} c \mspace{4mu} \Delta \nu _{Cs}\\ \\ \\ 1 \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-2} \mspace{6mu} \approx 3.628 \mspace{4mu} 602 \mspace{4mu} 815 \mspace{4mu} 210 \mspace{4mu} 469 \mspace{4mu} 913 \times10^{-19} \mspace{6mu} c \mspace{4mu} \Delta \nu _{Cs}$
cubic metre per second	m³ s^-1	volumetric flow rate	m³ s⁻¹
$1 \mspace{4mu} \text{m}^3 \mspace{4mu} \text{s}^{-1} \mspace{6mu} = \dfrac{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^2}{(299 \mspace{4mu} 792 \mspace{4mu} 458)^3} \mspace{6mu} \dfrac{c^3}{{\Delta \nu _{Cs}}^2}\\ \\ \\ 1 \mspace{4mu} \text{m}^3 \mspace{4mu} \text{s}^{-1} \mspace{6mu} \approx 3.136 \mspace{4mu} 300 \mspace{4mu} 151 \mspace{4mu} 321 \mspace{4mu} 367 \mspace{4mu} 418 \times10^{-6} \mspace{6mu} c^3 \mspace{4mu} {\Delta \nu _{Cs}}^{-2}$
reciprocal metre	m^-1	wavenumber	m⁻¹
$1 \mspace{4mu} \text{m}^{-1} \mspace{6mu} = \dfrac{299 \mspace{4mu} 792 \mspace{4mu} 458}{9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770} \mspace{6mu} \dfrac{\Delta \nu _{Cs}}{c}\\ \\ \\ 1 \mspace{4mu} \text{m}^{-1} \mspace{6mu} \approx 3.261 \mspace{4mu} 225 \mspace{4mu} 571 \mspace{4mu} 749 \mspace{4mu} 405 \mspace{4mu} 557 \times10^{-2} \mspace{6mu} c^{-1} \mspace{4mu} \Delta \nu _{Cs}$
kilogram per cubic metre	kg m⁻³	density, mass concentration	kg m⁻³
$1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m}^{-3} \mspace{6mu} = \dfrac{(299 \mspace{4mu} 792 \mspace{4mu} 458)^5}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^4} \mspace{6mu} \dfrac{h \mspace{4mu} {\Delta \nu _{Cs}}^4}{c^5}\\ \\ \\ 1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m}^{-3} \mspace{6mu} \approx 5.117 \mspace{4mu} 855 \mspace{4mu} 617 \mspace{4mu} 606 \mspace{4mu} 822 \mspace{4mu} 686 \times10^{35} \mspace{6mu} h \mspace{4mu} c^{-5} \mspace{4mu} {\Delta \nu _{Cs}}^4$
kilogram per square metre	kg m⁻²	surface density	kg m⁻²
$1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m}^{-2} \mspace{6mu} = \dfrac{(299 \mspace{4mu} 792 \mspace{4mu} 458)^4}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^3} \mspace{6mu} \dfrac{h \mspace{4mu} {\Delta \nu _{Cs}}^3}{c^4}\\ \\ \\ 1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m}^{-2} \mspace{6mu} \approx 1.569 \mspace{4mu} 304 \mspace{4mu} 393 \mspace{4mu} 397 \mspace{4mu} 563 \mspace{4mu} 377 \times10^{37} \mspace{6mu} h \mspace{4mu} c^{-4} \mspace{4mu} {\Delta \nu _{Cs}}^3$
cubic metre per kilogram	kg⁻¹ m³	specific volume	kg⁻¹ m³
$1 \mspace{4mu} \text{kg}^{-1} \mspace{4mu} \text{m}^3 \mspace{6mu} = \dfrac{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^4}{(299 \mspace{4mu} 792 \mspace{4mu} 458)^5} \mspace{6mu} \dfrac{c^5}{h \mspace{4mu} {\Delta \nu _{Cs}}^4}\\ \\ \\ 1 \mspace{4mu} \text{kg}^{-1} \mspace{4mu} \text{m}^3 \mspace{6mu} \approx 1.953 \mspace{4mu} 943 \mspace{4mu} 359 \mspace{4mu} 714 \mspace{4mu} 421 \mspace{4mu} 354 \times10^{-36} \mspace{6mu} h^{-1} \mspace{4mu} c^5 \mspace{4mu} {\Delta \nu _{Cs}}^{-4}$
kilogram metre per second	kg m s⁻¹	momentum	kg m s⁻¹
$1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-1} \mspace{6mu} = \dfrac{299 \mspace{4mu} 792 \mspace{4mu} 458}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)} \mspace{6mu} \dfrac{h \mspace{4mu} \Delta \nu _{Cs}}{c}\\ \\ \\ 1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-1} \mspace{6mu} \approx 4.921 \mspace{4mu} 809 \mspace{4mu} 606 \mspace{4mu} 482 \mspace{4mu} 064 \mspace{4mu} 722 \times10^{31} \mspace{6mu} h \mspace{4mu} c^{-1} \mspace{4mu} \Delta \nu _{Cs}$
kilogram metre squared per second	kg m² s⁻¹	angular momentum	kg m² s⁻¹
$1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m}^2 \mspace{4mu} \text{s}^{-1} \mspace{6mu} = \dfrac{1}{6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34}} \mspace{6mu} h\\ \\ \\ 1 \mspace{4mu} \text{kg} \mspace{4mu} \text{m}^2 \mspace{4mu} \text{s}^{-1} \mspace{6mu} \approx 1.509 \mspace{4mu} 190 \mspace{4mu} 179 \mspace{4mu} 642 \mspace{4mu} 151 \mspace{4mu} 842 \times10^{33} \mspace{6mu} h$
ampere per square metre	m⁻² A	current density	m⁻² A
$1 \mspace{4mu} \text{m}^{-2} \mspace{4mu} \text{A} \mspace{6mu} = \dfrac{(299 \mspace{4mu} 792 \mspace{4mu} 458)^2}{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^3(1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19})} \mspace{6mu} \dfrac{{\Delta \nu _{Cs}}^3 \mspace{4mu} e}{c^2}\\ \\ \\ 1 \mspace{4mu} \text{m}^{-2} \mspace{4mu} \text{A} \mspace{6mu} \approx 7.221 \mspace{4mu} 234 \mspace{4mu} 035 \mspace{4mu} 654 \mspace{4mu} 436 \mspace{4mu} 424 \times10^{5} \mspace{6mu} c^{-2} \mspace{4mu} {\Delta \nu _{Cs}}^3 \mspace{4mu} e$
ampere per metre	m⁻¹ A	magnetic field strength	m⁻¹ A
$1 \mspace{4mu} \text{m}^{-1} \mspace{4mu} \text{A} \mspace{6mu} = \dfrac{299 \mspace{4mu} 792 \mspace{4mu} 458}{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^2(1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19})} \mspace{6mu} \dfrac{{\Delta \nu _{Cs}}^2 \mspace{4mu} e}{c}\\ \\ \\ 1 \mspace{4mu} \text{m}^{-1} \mspace{4mu} \text{A} \mspace{6mu} \approx 2.214 \mspace{4mu} 270 \mspace{4mu} 027 \mspace{4mu} 258 \mspace{4mu} 733 \mspace{4mu} 941 \times10^{7} \mspace{6mu} c^{-1} \mspace{4mu} {\Delta \nu _{Cs}}^2 \mspace{4mu} e$
mole per cubic metre	m⁻³ mol	amount concentration, concentration	m⁻³ mol
$1 \mspace{4mu} \text{m}^{-3} \mspace{4mu} \text{mol} \mspace{6mu} = \dfrac{(299 \mspace{4mu} 792 \mspace{4mu} 458)^3(6.022 \mspace{4mu} 140 \mspace{4mu} 76 \times 10^{23})}{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^3} \mspace{6mu} \dfrac{{\Delta \nu _{Cs}}^3}{c^3 \mspace{4mu} N_A}\\ \\ \\ 1 \mspace{4mu} \text{m}^{-3} \mspace{4mu} \text{mol} \mspace{6mu} \approx 2.088 \mspace{4mu} 783 \mspace{4mu} 458 \mspace{4mu} 112 \mspace{4mu} 816 \mspace{4mu} 111 \times10^{19} \mspace{6mu} c^{-3} \mspace{4mu} {\Delta \nu _{Cs}}^3 \mspace{4mu} {N_A}^{-1}$
candela per square metre	m⁻² cd	luminance	m⁻² cd
$1 \mspace{4mu} \text{m}^{-2} \mspace{4mu} \text{cd} \mspace{6mu} = \dfrac{(299 \mspace{4mu} 792 \mspace{4mu} 458)^2}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^4(683)} \mspace{6mu} \dfrac{h \mspace{4mu} {\Delta \nu _{Cs}}^4 \mspace{4mu} K_{cd}}{c^2}\\ \\ \\ 1 \mspace{4mu} \text{m}^{-2} \mspace{4mu} \text{cd} \mspace{6mu} \approx 2.781 \mspace{4mu} 027 \mspace{4mu} 075 \mspace{4mu} 972 \mspace{4mu} 537 \mspace{4mu} 548 \times10^{7} \mspace{6mu} h \mspace{4mu} c^{-2} \mspace{4mu} {\Delta \nu _{Cs}}^4 \mspace{4mu} K_{cd}$