SI Derived Units in Base Units: Full Table and Derivations
How the newton, joule, watt, coulomb, volt, ohm and the other named SI units break down into the seven base units, with worked derivations.
Last reviewed on October 2, 2026
1 N = kg·m·s⁻² · 1 J = kg·m²·s⁻² · 1 W = kg·m²·s⁻³ · 1 C = A·s · 1 V = kg·m²·s⁻³·A⁻¹ · 1 Ω = kg·m²·s⁻³·A⁻² · 1 Pa = kg·m⁻¹·s⁻²
There are 22 SI derived units with special names. Each is a product of powers of the seven base units (kg, m, s, A, K, mol, cd) with no extra numerical factor.
What a Derived Unit Is
The SI has seven base units: the kilogram (kg), metre (m), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). Every other SI unit is a derived unit — a product of powers of those seven, with no numerical factor other than 1. A newton, for example, is exactly one kilogram metre per second squared.
Twenty-two of these combinations are used so often that they have their own names and symbols. Writing 5 kW is easier than 5 × 10³ kg·m²·s⁻³, but the two mean exactly the same thing. Being able to unpack a named unit into base units is the most reliable way to check an equation’s dimensions. The base units themselves are explained in the SI base units and prefixes guide.
All 22 Named SI Derived Units in Base Units
| Unit | Symbol | Quantity | In other SI units | In SI base units |
|---|---|---|---|---|
| radian | rad | Plane angle | m/m | m·m⁻¹ = 1 |
| steradian | sr | Solid angle | m²/m² | m²·m⁻² = 1 |
| hertz | Hz | Frequency | — | s⁻¹ |
| newton | N | Force, weight | — | kg·m·s⁻² |
| pascal | Pa | Pressure, stress | N/m² | kg·m⁻¹·s⁻² |
| joule | J | Energy, work, heat | N·m | kg·m²·s⁻² |
| watt | W | Power, radiant flux | J/s | kg·m²·s⁻³ |
| coulomb | C | Electric charge | — | A·s |
| volt | V | Electric potential difference | W/A | kg·m²·s⁻³·A⁻¹ |
| farad | F | Capacitance | C/V | kg⁻¹·m⁻²·s⁴·A² |
| ohm | Ω | Electrical resistance | V/A | kg·m²·s⁻³·A⁻² |
| siemens | S | Electrical conductance | A/V | kg⁻¹·m⁻²·s³·A² |
| weber | Wb | Magnetic flux | V·s | kg·m²·s⁻²·A⁻¹ |
| tesla | T | Magnetic flux density | Wb/m² | kg·s⁻²·A⁻¹ |
| henry | H | Inductance | Wb/A | kg·m²·s⁻²·A⁻² |
| degree Celsius | °C | Celsius temperature | — | K |
| lumen | lm | Luminous flux | cd·sr | cd |
| lux | lx | Illuminance | lm/m² | cd·m⁻² |
| becquerel | Bq | Activity of a radionuclide | — | s⁻¹ |
| gray | Gy | Absorbed dose | J/kg | m²·s⁻² |
| sievert | Sv | Dose equivalent | J/kg | m²·s⁻² |
| katal | kat | Catalytic activity | — | mol·s⁻¹ |
A negative exponent means “per”: s⁻² is “per second squared”. The radian and steradian reduce to the number 1 but keep their names so angles stay distinguishable. The becquerel and hertz share the dimension s⁻¹, and the gray and sievert share m²·s⁻², but each is reserved for its own quantity.
Step-by-Step Derivations
Each named unit comes from a defining equation. Substitute the base units of every quantity in the equation and simplify the exponents.
Newton (force): N = kg·m·s⁻²
Force = mass × acceleration (F = ma). Mass is in kg; acceleration is metres per second per second, m·s⁻². So 1 N = 1 kg × 1 m·s⁻² = 1 kg·m·s⁻². One newton accelerates one kilogram by one metre per second, every second.
Pascal (pressure): Pa = kg·m⁻¹·s⁻²
Pressure = force ÷ area. 1 Pa = 1 N/m² = kg·m·s⁻² ÷ m² = kg·m⁻¹·s⁻².
Joule (energy): J = kg·m²·s⁻²
Work = force × distance. 1 J = 1 N·m = kg·m·s⁻² × m = kg·m²·s⁻². The same result comes from kinetic energy, ½mv²: kg × (m·s⁻¹)² = kg·m²·s⁻².
Watt (power): W = kg·m²·s⁻³
Power = energy ÷ time. 1 W = 1 J/s = kg·m²·s⁻² ÷ s = kg·m²·s⁻³.
Coulomb (charge): C = A·s
Charge = current × time (Q = It). 1 C = 1 A·s: the charge carried by one ampere flowing for one second. Since the ampere is a base unit, this is already in base units.
Volt (potential difference): V = kg·m²·s⁻³·A⁻¹
Voltage = power ÷ current (P = IV, so V = P/I). 1 V = 1 W/A = kg·m²·s⁻³ ÷ A = kg·m²·s⁻³·A⁻¹. Equivalently, 1 V = 1 J/C.
Ohm (resistance): Ω = kg·m²·s⁻³·A⁻²
Resistance = voltage ÷ current (Ohm’s law, R = V/I). 1 Ω = 1 V/A = kg·m²·s⁻³·A⁻¹ ÷ A = kg·m²·s⁻³·A⁻².
How to Convert Any Unit to SI Base Units
- Find a defining equation for the quantity, such as F = ma, P = E/t or V = IR.
- Replace each quantity with its unit, expanding any named units you already know (N → kg·m·s⁻²).
- Collect like units and add exponents: m × m² = m³, s⁻² ÷ s = s⁻³.
- Strip prefixes into powers of ten: 1 kN = 10³ kg·m·s⁻². Remember the base unit of mass is the kilogram, so a gram is 10⁻³ kg.
- Check dimensions: both sides of a valid equation must reduce to the same base units.
Example: kinetic energy of a 2 kg ball moving at 3 m/s is ½ × 2 kg × (3 m/s)² = 9 kg·m²·s⁻² = 9 J. If an equation produced kg·m·s⁻² instead, you would know it gave a force, not an energy — something went wrong.
Common Derived Units Without Special Names
Many derived units have no special name and are simply written as combinations:
| Quantity | Unit | In SI base units |
|---|---|---|
| Area | square metre | m² |
| Volume | cubic metre | m³ |
| Speed, velocity | metre per second | m·s⁻¹ |
| Acceleration | metre per second squared | m·s⁻² |
| Density | kilogram per cubic metre | kg·m⁻³ |
| Momentum | kilogram metre per second | kg·m·s⁻¹ |
| Torque | newton metre | kg·m²·s⁻² |
| Electric field strength | volt per metre | kg·m·s⁻³·A⁻¹ |
| Molar concentration | mole per cubic metre | mol·m⁻³ |
| Specific heat capacity | joule per kilogram kelvin | m²·s⁻²·K⁻¹ |
Torque and energy share the base units kg·m²·s⁻², but torque is always written N·m and never J, to keep the two quantities distinct.
Related Converters and Guides
- SI base units & prefixesThe 7 base units and all 24 prefixes.
- Joules to caloriesEnergy conversions.
- Horsepower to kWPower conversions.
- psi to barPressure conversions.
Frequently Asked Questions
What is 1 N in SI base units?
1 newton = 1 kg·m·s⁻² (one kilogram metre per second squared). It comes from Newton’s second law, force = mass × acceleration.
What combination of SI base units is force?
Force is measured in newtons, and 1 N = kg·m·s⁻². That is mass (kg) multiplied by length (m) divided by time squared (s²).
What are the SI base units of power (the watt)?
1 W = 1 J/s = kg·m²·s⁻³. A watt is one joule of energy per second, and a joule is kg·m²·s⁻².
What is the coulomb in base units?
1 C = 1 A·s, one ampere flowing for one second. Electric charge is the only common electrical quantity that needs just two base units. Its inverse, C⁻¹, is A⁻¹·s⁻¹.
What is the joule in SI base units?
1 J = 1 N·m = kg·m²·s⁻². It is the work done when a force of one newton moves its point of application one metre.
How many SI derived units have special names?
There are 22: radian, steradian, hertz, newton, pascal, joule, watt, coulomb, volt, farad, ohm, siemens, weber, tesla, henry, degree Celsius, lumen, lux, becquerel, gray, sievert and katal.
Is the degree Celsius a derived unit?
Yes. The degree Celsius is one of the 22 SI derived units with a special name. It is the same size as the kelvin; a Celsius temperature is the kelvin temperature minus 273.15.
How do I convert a unit to SI base units?
Write a defining equation for the quantity, replace each term with its unit, expand any named units, then combine exponents. For example, pressure = force ÷ area gives N/m² = kg·m·s⁻² ÷ m² = kg·m⁻¹·s⁻².