Units · Reference table

Named derived SI units

Coherent units expressed through SI base units. The radian and steradian have unit one, but retain named symbols for clarity.

Values: 22
Edition / reference: SI Brochure, 9th edition, current online version 2026

Reference values

Reviewed September 30, 2026

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Named derived SI units
Quantity	Unit	Symbol	In SI base units
plane angle	radian	rad	m·m⁻¹ = 1
solid angle	steradian	sr	m²·m⁻² = 1
frequency	hertz	Hz	s⁻¹
force	newton	N	kg·m·s⁻²
pressure	pascal	Pa	kg·m⁻¹·s⁻²
energy, work, heat	joule	J	kg·m²·s⁻²
power	watt	W	kg·m²·s⁻³
electric charge	coulomb	C	A·s
electric potential	volt	V	kg·m²·s⁻³·A⁻¹
capacitance	farad	F	kg⁻¹·m⁻²·s⁴·A²
electric resistance	ohm	Ω	kg·m²·s⁻³·A⁻²
electric conductance	siemens	S	kg⁻¹·m⁻²·s³·A²
magnetic flux	weber	Wb	kg·m²·s⁻²·A⁻¹
magnetic flux density	tesla	T	kg·s⁻²·A⁻¹
inductance	henry	H	kg·m²·s⁻²·A⁻²
Celsius temperature	degree Celsius	°C	t/°C = T/K − 273.15
luminous flux	lumen	lm	cd·sr
illuminance	lux	lx	cd·sr·m⁻²
activity of a radionuclide	becquerel	Bq	s⁻¹
absorbed dose	gray	Gy	m²·s⁻²
dose equivalent	sievert	Sv	m²·s⁻²
catalytic activity	katal	kat	mol·s⁻¹

Additional reference tables

More common derived quantities

Twenty lookup rows for compound units without additional special SI names. The official list of 22 named units remains separate. Notes help distinguish quantities sharing a dimension.

More common derived quantities
Quantity	Coherent SI unit	In base units	Interpretation
Area	m²	m²	Product of two lengths.
Volume	m³	m³	Product of three lengths.
Velocity, speed	m/s	m·s⁻¹	Length per time.
Acceleration	m/s²	m·s⁻²	Velocity change per time.
Wavenumber	m⁻¹	m⁻¹	State the convention: 1/λ or angular wavenumber 2π/λ.
Mass density	kg/m³	kg·m⁻³	Mass per volume.
Specific volume	m³/kg	m³·kg⁻¹	Volume per mass; reciprocal density for the same sample.
Electric current density	A/m²	A·m⁻²	Current per cross-sectional area; consider distribution and direction.
Magnetic field strength H	A/m	A·m⁻¹	Do not equate with magnetic flux density B in T.
Luminance	cd/m²	cd·m⁻²	Directional photometric quantity, not illuminance in lx.
Amount concentration	mol/m³	mol·m⁻³	Amount per volume; identify the species.
Molar mass	kg/mol	kg·mol⁻¹	Mass per amount of substance.
Dynamic viscosity	Pa·s	kg·m⁻¹·s⁻¹	Not kinematic viscosity in m²/s.
Torque	N·m	kg·m²·s⁻²	Notation preserves its distinction from energy in J.
Surface tension	N/m	kg·s⁻²	Force per length, not pressure per area.
Heat flux density / irradiance	W/m²	kg·s⁻³	Power per area; specify the transfer kind.
Heat capacity	J/K	kg·m²·s⁻²·K⁻¹	For the whole sample; entropy shares the same unit.
Specific heat capacity	J/(kg·K)	m²·s⁻²·K⁻¹	Per mass; state conditions such as constant pressure.
Thermal conductivity	W/(m·K)	kg·m·s⁻³·K⁻¹	Material coefficient, not heat flow alone.
Energy density	J/m³	kg·m⁻¹·s⁻²	Energy per volume; same base-unit expression as pressure.

References: NIST SP 811 §4.2 and Tables 2, 4: quantities and units · BIPM §2.3.4: coherent derivations and quantity distinctions

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Understand and use this reference

Derived SI units are products and powers of base units. The 22 special names form a complete official list, but do not exhaust all possible derived units. Quantity equations explain the units of pascal, watt and volt, how compound units convert, and where dimension checking alone reaches its limits.

What coherent derivation means

Apply a quantity equation’s operations to its units. v = l/t gives m/s and density ρ = m/V gives kg/m³. A coherent unit introduces no extra numerical factor. km/h is a useful velocity unit but is not coherent with the system based on m, kg and s: 1 km/h = (1 000/3 600) m/s. Coherence concerns the unit system, not measurement quality.[2]

[\rho]=\frac{\mathrm{kg}}{\mathrm{m^3}},\qquad [a]=\frac{\mathrm{m/s}}{\mathrm s}=\mathrm{m\,s^{-2}}

Negative exponents represent denominator units.

The table of 22 names includes radian and steradian. Previously classified as supplementary units, they are now derived units. Quantities such as momentum, heat capacity and dynamic viscosity have valid coherent units without another official special name. The additional lookup block keeps those quantities separate.[2]

From acceleration to newton, pascal and watt

In F = ma, mass multiplies acceleration: kg·m/s² is called newton. Pressure under uniform normal loading is force per area, with unit N/m² called pascal. Work by a constant parallel force is F·s, with unit N·m called joule. Power is work or energy transfer per time, giving J/s = W. This unit structure remains valid when an integral replaces a simple constant-force model.

\mathrm N=\mathrm{kg\,m\,s^{-2}},\quad \mathrm{Pa}=\mathrm{kg\,m^{-1}\,s^{-2}},\quad \mathrm W=\mathrm{kg\,m^2\,s^{-3}}

Special names shorten these products without changing their factor.[2]

The chain also supports rearrangement: A = F/p has N/Pa = m², an area. Dividing force by length rather than area gives N/m, which can describe another quantity such as surface tension, but not pressure. The physical quantity determines which equation needs checking.

The previous table example remains as a short check: a uniform normal force of 600 N on 0.03 m² gives p = 600/0.03 Pa = 20 000 Pa = 20 kPa. In reverse, (20 000 Pa)·0.03 m² = 600 N. The area is already in m²; an additional millimetre factor would be wrong.

Electrical units from charge and energy

Charge transfer Q = IΔt gives A·s, the coulomb. Voltage is energy per charge, with unit J/C = V. Resistance R = U/I gives V/A = Ω; conductance is its reciprocal, A/V = S. Capital S for siemens differs from s for second. Current direction and voltage signs follow the chosen references, not the unit.

\mathrm V=\frac{\mathrm J}{\mathrm C}=\mathrm{kg\,m^2\,s^{-3}\,A^{-1}},\qquad \mathrm\Omega=\mathrm{kg\,m^2\,s^{-3}\,A^{-2}}

Voltage contains A⁻¹ and resistance A⁻².[2]

Capacitance has C/V = F and inductance V·s/A = H. Symbols can play different roles in a formula: upright F denotes farad while italic F often denotes force. Begin a long calculation by identifying quantities and known relations. Correct units do not establish that a component remains ohmic at every current.

Equal dimension does not mean equal quantity

Energy and torque both reduce to kg·m²/s². Energy is normally expressed in J and torque in N·m to preserve quantity information. Torque uses the perpendicular lever arm; work uses displacement along force. A loaded stationary lever can have torque without mechanical work currently being performed. Equal dimension does not permit reinterpreting one quantity as another.[2]

Hertz and becquerel both reduce to s⁻¹ but identify periodic frequency and radionuclide activity respectively. Gray and sievert share J/kg but refer to different dose quantities. A shared base-unit expression does not make values interchangeable without their quantity definitions and evaluation conditions. This unit table alone cannot assess a medical dose.[2]

Keeping the meaning of angles and temperatures

A radian is arc length divided by radius. Its dimensional product is one, while rad preserves angular meaning. In uniform circular motion, f is the positive number of complete revolutions per time and |ωz| is angular-speed magnitude about the selected axis. One revolution is 2π rad, so |ωz| = 2πf. The signed axial component also depends on rotation direction. For nonuniform motion, 2π/T describes only the mean magnitude over a revolution without reversal. 5 Hz therefore corresponds to approximately 31.42 rad/s as magnitude for uniform rotation, not 5 rad/s.

|\omega_z|=2\pi f,\qquad \theta=\frac{l_{\mathrm{arc}}}{r}

The factor 2π comes from a revolution, not a new time or length unit.

Degree Celsius is a special name for kelvin when expressing Celsius temperatures. Their increments are equal but their origins differ by exactly 273.15 K. Temperature differences transfer directly; individual values require the offset. For T = 300 K, t = 26.85 °C, while an increase of 10 K is also an increase of 10 °C.[3]

Example: force on a small area

Pascal with a squared conversion

Given: A uniform normal force F = 180 N acts on A = 75.0 mm².
Find: Find mean pressure in Pa and MPa.

Convert the area with the full squared factor.
$A=75.0(10^{-3}\,\mathrm m)^2=7.50\times10^{-5}\,\mathrm{m^2}$
Divide normal force by area.
$p=\frac{180\,\mathrm N}{7.50\times10^{-5}\,\mathrm{m^2}}=2.40\times10^6\,\mathrm{Pa}$
Apply mega as the factor 10⁶.
$p=2.40\,\mathrm{MPa}$

Result: Mean pressure is 2.40 MPa. The uniform-distribution assumption excludes local peaks.

Check: pA = (2.40 × 10⁶)(7.50 × 10⁻⁵) N = 180 N. Alternatively, 1 N/mm² = 10⁶ Pa, so 180/75 = 2.40 N/mm² gives the same result.

Example: a device’s energy and charge

Checking watt, joule and coulomb together

Given: Idealised DC device: constant U = 12.0 V and I = 0.250 A over Δt = 180 s.
Find: Find power, transferred energy and charge.

For the stated constant DC values use P = UI.
$P=12.0\,\mathrm V\cdot0.250\,\mathrm A=3.00\,\mathrm W$
Multiply power by time.
$E=P\Delta t=3.00\,\mathrm W\cdot180\,\mathrm s=540\,\mathrm J$
Multiply current by the same interval.
$Q=I\Delta t=0.250\,\mathrm A\cdot180\,\mathrm s=45.0\,\mathrm C$

Result: P = 3.00 W, E = 540 J and Q = 45.0 C; this is electrical energy transfer, not necessarily entirely useful output.

Check: UQ = 12.0 V·45.0 C = 540 J confirms the energy. Under constant ohmic conditions R = 48.0 Ω and I²R = 3.00 W. Conversion to 540 J = 0.150 Wh uses 1 h = 3 600 s.

Checking units and assumptions separately

For a unit check, substitute base units and combine exponents. Both sides of an equation must have the same dimension, as must terms being added. Then check quantity kind, sign and model conditions. P = UI is directly usable for the constant DC values above. For varying quantities instantaneous power is u(t)i(t); multiplying RMS values does not generally give average real power without further conditions.

Include prefixes as factors before cancelling units. In mPa·s the first m is milli, while in Pa·m it is metre. Parentheses in W/(m·K) place both quantities in the denominator. Looking up a row does not justify an equation: thermal conductivity has W/(m·K), but calculating heat flow also requires geometry, temperature behaviour and suitable assumptions.[4]

Three solved exercises

Derive the unit of momentum p = mv.

Show hint

Multiply mass by velocity, then expand newton in reverse.

Show solution and calculation

kg·m/s = N·s.
Both forms are coherent; N alone denotes force, not momentum.

[p]=\mathrm{kg\,m\,s^{-1}}=\mathrm{N\,s}

A 40 N force has a 0.30 m perpendicular lever arm. Find torque and its appropriate unit.

Show hint

Torque magnitude is force times perpendicular lever arm.

Show solution and calculation

M = 12 N·m.
The base-unit expression equals that of joule, but use N·m because this is torque.
It does not imply 12 J of work without displacement.

M=40\,\mathrm N\cdot0.30\,\mathrm m=12\,\mathrm{N\,m}

A rotor turns uniformly at 600 revolutions per minute. Find f and angular-speed magnitude.

Show hint

One minute is 60 s; each revolution is 2π rad.

Show solution and calculation

f = 10 Hz and |ωz| = 20π rad/s ≈ 62.83 rad/s.

Check: ω/(2π) = 10 s⁻¹. A result of 600 Hz would omit the minute conversion.

f=10\,\mathrm{Hz},\quad|\omega_z|=20\pi\,\mathrm{rad/s}

Sources for this explanation

Quanta · Updated October 1, 2026

Editorial responsibility: Quanta Study editorial team

Review scope

This explanation, 2 worked examples and 3 exercises are available in German and English. Structure, source references and mathematical notation undergo automated checks. Calculations, model assumptions and both language versions receive a separate AI review.

An internal human subject review is documented for an earlier release of this page. That confirmation does not apply automatically to the revised explanation and new exercises.

An independent, named human subject review has not yet been documented. Our exercise numbers are worked examples, rather than measurements published by the cited sources.

Editorial responsibility, sources and methods

Cite this reference page

Quanta. Named derived SI units. Updated October 1, 2026. https://quanta-study.de/en/formula-sheet/tables/si-abgeleitete-einheiten

When citing a specific scientific statement, also cite the specialist source linked to that statement.

Suggest a correction with a reference

Source trail

Edition and sources

Each link opens the cited source. Any source note clarifies whether it supports the table values or only the curriculum context.

BIPM: SI Brochure

Quanta Study editorial team

Quantity

Unit

Symbol

In SI base units

plane angle

radian

rad

m·m⁻¹ = 1

solid angle

steradian

m²·m⁻² = 1

frequency

hertz

s⁻¹

force

newton

kg·m·s⁻²

pressure

pascal

kg·m⁻¹·s⁻²

energy, work, heat

joule

kg·m²·s⁻²

power

watt

kg·m²·s⁻³

electric charge

coulomb

A·s

electric potential

volt

kg·m²·s⁻³·A⁻¹

capacitance

farad

kg⁻¹·m⁻²·s⁴·A²

electric resistance

ohm

kg·m²·s⁻³·A⁻²

electric conductance

siemens

kg⁻¹·m⁻²·s³·A²

magnetic flux

weber

kg·m²·s⁻²·A⁻¹

magnetic flux density

tesla

kg·s⁻²·A⁻¹

inductance

henry

kg·m²·s⁻²·A⁻²

Celsius temperature

degree Celsius

°C

t/°C = T/K − 273.15

luminous flux

lumen

cd·sr

illuminance

lux

cd·sr·m⁻²

activity of a radionuclide

becquerel

s⁻¹

absorbed dose

gray

m²·s⁻²

dose equivalent

sievert

m²·s⁻²

catalytic activity

katal

kat

mol·s⁻¹

Additional reference tables

More common derived quantities

Twenty lookup rows for compound units without additional special SI names. The official list of 22 named units remains separate. Notes help distinguish quantities sharing a dimension.

More common derived quantities
Quantity	Coherent SI unit	In base units	Interpretation
Area	m²	m²	Product of two lengths.
Volume	m³	m³	Product of three lengths.
Velocity, speed	m/s	m·s⁻¹	Length per time.
Acceleration	m/s²	m·s⁻²	Velocity change per time.
Wavenumber	m⁻¹	m⁻¹	State the convention: 1/λ or angular wavenumber 2π/λ.
Mass density	kg/m³	kg·m⁻³	Mass per volume.
Specific volume	m³/kg	m³·kg⁻¹	Volume per mass; reciprocal density for the same sample.
Electric current density	A/m²	A·m⁻²	Current per cross-sectional area; consider distribution and direction.
Magnetic field strength H	A/m	A·m⁻¹	Do not equate with magnetic flux density B in T.
Luminance	cd/m²	cd·m⁻²	Directional photometric quantity, not illuminance in lx.
Amount concentration	mol/m³	mol·m⁻³	Amount per volume; identify the species.
Molar mass	kg/mol	kg·mol⁻¹	Mass per amount of substance.
Dynamic viscosity	Pa·s	kg·m⁻¹·s⁻¹	Not kinematic viscosity in m²/s.
Torque	N·m	kg·m²·s⁻²	Notation preserves its distinction from energy in J.
Surface tension	N/m	kg·s⁻²	Force per length, not pressure per area.
Heat flux density / irradiance	W/m²	kg·s⁻³	Power per area; specify the transfer kind.
Heat capacity	J/K	kg·m²·s⁻²·K⁻¹	For the whole sample; entropy shares the same unit.
Specific heat capacity	J/(kg·K)	m²·s⁻²·K⁻¹	Per mass; state conditions such as constant pressure.
Thermal conductivity	W/(m·K)	kg·m·s⁻³·K⁻¹	Material coefficient, not heat flow alone.
Energy density	J/m³	kg·m⁻¹·s⁻²	Energy per volume; same base-unit expression as pressure.

References: NIST SP 811 §4.2 and Tables 2, 4: quantities and units · BIPM §2.3.4: coherent derivations and quantity distinctions

Understand and use this reference

What coherent derivation means

[\rho]=\frac{\mathrm{kg}}{\mathrm{m^3}},\qquad [a]=\frac{\mathrm{m/s}}{\mathrm s}=\mathrm{m\,s^{-2}}

Negative exponents represent denominator units.

From acceleration to newton, pascal and watt

\mathrm N=\mathrm{kg\,m\,s^{-2}},\quad \mathrm{Pa}=\mathrm{kg\,m^{-1}\,s^{-2}},\quad \mathrm W=\mathrm{kg\,m^2\,s^{-3}}

Special names shorten these products without changing their factor.[2]

Electrical units from charge and energy

\mathrm V=\frac{\mathrm J}{\mathrm C}=\mathrm{kg\,m^2\,s^{-3}\,A^{-1}},\qquad \mathrm\Omega=\mathrm{kg\,m^2\,s^{-3}\,A^{-2}}

Voltage contains A⁻¹ and resistance A⁻².[2]

Equal dimension does not mean equal quantity

Keeping the meaning of angles and temperatures

|\omega_z|=2\pi f,\qquad \theta=\frac{l_{\mathrm{arc}}}{r}

The factor 2π comes from a revolution, not a new time or length unit.

Example: force on a small area

Pascal with a squared conversion

Given: A uniform normal force F = 180 N acts on A = 75.0 mm².
Find: Find mean pressure in Pa and MPa.

Convert the area with the full squared factor.
$A=75.0(10^{-3}\,\mathrm m)^2=7.50\times10^{-5}\,\mathrm{m^2}$
Divide normal force by area.
$p=\frac{180\,\mathrm N}{7.50\times10^{-5}\,\mathrm{m^2}}=2.40\times10^6\,\mathrm{Pa}$
Apply mega as the factor 10⁶.
$p=2.40\,\mathrm{MPa}$

Result: Mean pressure is 2.40 MPa. The uniform-distribution assumption excludes local peaks.

Check: pA = (2.40 × 10⁶)(7.50 × 10⁻⁵) N = 180 N. Alternatively, 1 N/mm² = 10⁶ Pa, so 180/75 = 2.40 N/mm² gives the same result.

Example: a device’s energy and charge

Checking watt, joule and coulomb together

Given: Idealised DC device: constant U = 12.0 V and I = 0.250 A over Δt = 180 s.
Find: Find power, transferred energy and charge.

For the stated constant DC values use P = UI.
$P=12.0\,\mathrm V\cdot0.250\,\mathrm A=3.00\,\mathrm W$
Multiply power by time.
$E=P\Delta t=3.00\,\mathrm W\cdot180\,\mathrm s=540\,\mathrm J$
Multiply current by the same interval.
$Q=I\Delta t=0.250\,\mathrm A\cdot180\,\mathrm s=45.0\,\mathrm C$

Result: P = 3.00 W, E = 540 J and Q = 45.0 C; this is electrical energy transfer, not necessarily entirely useful output.

Check: UQ = 12.0 V·45.0 C = 540 J confirms the energy. Under constant ohmic conditions R = 48.0 Ω and I²R = 3.00 W. Conversion to 540 J = 0.150 Wh uses 1 h = 3 600 s.

Checking units and assumptions separately

Three solved exercises

Derive the unit of momentum p = mv.

Show hint

Multiply mass by velocity, then expand newton in reverse.

Show solution and calculation

kg·m/s = N·s.
Both forms are coherent; N alone denotes force, not momentum.

[p]=\mathrm{kg\,m\,s^{-1}}=\mathrm{N\,s}

A 40 N force has a 0.30 m perpendicular lever arm. Find torque and its appropriate unit.

Show hint

Torque magnitude is force times perpendicular lever arm.

Show solution and calculation

M = 12 N·m.
The base-unit expression equals that of joule, but use N·m because this is torque.
It does not imply 12 J of work without displacement.

M=40\,\mathrm N\cdot0.30\,\mathrm m=12\,\mathrm{N\,m}

A rotor turns uniformly at 600 revolutions per minute. Find f and angular-speed magnitude.

Show hint

One minute is 60 s; each revolution is 2π rad.

Show solution and calculation

f = 10 Hz and |ωz| = 20π rad/s ≈ 62.83 rad/s.

Check: ω/(2π) = 10 s⁻¹. A result of 600 Hz would omit the minute conversion.

f=10\,\mathrm{Hz},\quad|\omega_z|=20\pi\,\mathrm{rad/s}

Sources for this explanation

Quanta · Updated October 1, 2026

Editorial responsibility: Quanta Study editorial team

Review scope

An internal human subject review is documented for an earlier release of this page. That confirmation does not apply automatically to the revised explanation and new exercises.

An independent, named human subject review has not yet been documented. Our exercise numbers are worked examples, rather than measurements published by the cited sources.

Editorial responsibility, sources and methods

Cite this reference page

Quanta. Named derived SI units. Updated October 1, 2026. https://quanta-study.de/en/formula-sheet/tables/si-abgeleitete-einheiten

When citing a specific scientific statement, also cite the specialist source linked to that statement.

Suggest a correction with a reference