Derived units are measurement units built by combining the base units of a measurement system through multiplication, division, or both. In the International System of Units (SI), every quantity beyond the seven base quantities (length, mass, time, electric current, temperature, amount of substance, and luminous intensity) is expressed using a derived unit. Speed, for instance, is meters per second; force is kilograms times meters per second squared. Some of these combinations are used so frequently that they earn their own names, like the newton or the watt, but underneath, they are still just algebraic recipes of the base ingredients.
How Base Units Become Derived Units
The logic is straightforward. You start with the seven SI base units: the meter, kilogram, second, ampere, kelvin, mole, and candela. Any physical quantity you can measure that is not one of those seven gets its unit by combining them. Area is length times length, so its unit is the square meter. Velocity is length divided by time, so its unit is meters per second. Acceleration adds another division by time, giving meters per second squared. Force is mass times acceleration, giving kilogram-meters per second squared, which we call the newton.
This system is described as “coherent” because no extra numerical factors are needed when you combine units according to the equations of physics. If you multiply one newton by one meter, you get exactly one joule of energy, not 4.184 joules or some other conversion factor. That coherence is the central design principle of the SI. It means that when you write down a physics equation, the units on the left side and the right side automatically match, as long as you stick to SI throughout.
The distinction between base and derived is, in a sense, a matter of convention rather than physics. You could define the system differently, starting from a different set of seven. What matters practically is that the system is internally consistent, and that every measurable quantity has a well-defined unit traceable back to the same foundation.
Named Derived Units and Why They Exist
Writing out “kilogram-meters per second squared” every time you talk about force would be tedious and error-prone. So the SI assigns special names to 22 derived units that come up constantly in science, engineering, and daily life. Some of the most familiar include:
- Newton (N): the unit of force, equal to one kilogram-meter per second squared.
- Pascal (Pa): the unit of pressure, equal to one newton per square meter.
- Joule (J): the unit of energy, equal to one newton-meter.
- Watt (W): the unit of power, equal to one joule per second.
- Volt (V): the unit of electric potential, equal to one watt per ampere.
- Ohm (Ω): the unit of electrical resistance, equal to one volt per ampere.
- Hertz (Hz): the unit of frequency, equal to one cycle per second.
- Coulomb (C): the unit of electric charge, equal to one ampere-second.
Notice how derived units can themselves be combined into further derived units. The volt is defined in terms of the watt and the ampere, and the watt is defined in terms of the joule and the second, and the joule is defined in terms of the newton and the meter, and so on. It is a chain of definitions, but every link traces back to the same seven base units. The named units exist purely for human convenience; a circuit designer would go cross-eyed if every voltage had to be written out as “kilogram-meters squared per ampere per second cubed.”
Physiological Units Are a Special Case
Most derived units describe purely physical phenomena: force, energy, pressure, charge. A few, however, are built to account for how the human body responds to a physical stimulus. These have been described as “physiological units,” a special class of derived units defined with respect to specified reference stimuli for the purpose of calibrating the nominal human response to external physical effects.1Metrologia. Physiological Units in the SI
The candela is the most familiar example. It measures luminous intensity, which is not the same as total radiant power. A light source could emit enormous energy in the infrared, completely invisible to your eye, and have zero luminous intensity. The candela weights the light output according to how the average human eye perceives different wavelengths, peaking in the green-yellow region around 555 nanometers. This is why an infrared heater pumping out hundreds of watts registers no candela at all: the unit is tuned to human vision, not to raw physics.
The sievert is another physiological unit, used to measure the biological impact of ionizing radiation rather than its raw energy. One gray of alpha radiation does far more biological damage than one gray of gamma radiation, so the sievert applies weighting factors to account for how different types of radiation affect living tissue. Both the candela and the sievert sit within the SI as fully legitimate derived units, but they carry this extra layer of biological calibration that makes them fundamentally different from a pascal or a hertz.
The Katal and the Long Road Into the SI
Not every derived unit has been welcomed into the system quickly. The katal, the SI unit for catalytic activity, took decades of debate before it was officially adopted by the General Conference on Weights and Measures. It measures how fast an enzyme (or any catalyst) converts a substrate, defined as one mole of substrate transformed per second. Before the katal, clinical laboratories expressed enzyme activity in an older “enzyme unit” (U), which equaled one micromole per minute. The conversion works out to 1 U equaling about 16.667 nanonkatals.2Clinical Chemistry. The Tortuous Road to the Adoption of katal for the Expression of Catalytic Activity by the General Conference on Weights and Measures
The controversy was never about physics or math. It was about practical convenience. Clinical chemists had been reporting enzyme results in U/L (units per liter) for generations. Switching to katal per liter meant changing reference ranges, recalibrating instruments, and retraining clinicians to read lab reports. The numbers changed by roughly eight orders of magnitude in their raw form, since the katal is vastly larger than the old enzyme unit. This sort of friction is common when the SI expands: the formal system gets tidier, but the humans working with it have to rewire decades of habit.
Once adopted, the katal made enzyme measurements fully traceable to the SI, which matters for international standardization of clinical lab results. Further derived quantities like katal per liter (catalytic concentration) and katal per kilogram (specific catalytic activity) followed naturally from the coherent framework.3Clinical Chemistry. The Tortuous Road to the Adoption of katal for the Expression of Catalytic Activity by the General Conference on Weights and Measures
What the 2019 Redefinition Changed
In May 2019, the SI underwent its most sweeping revision in over a century. The kilogram, ampere, kelvin, and mole were all redefined. Instead of being pinned to physical artifacts (like the platinum-iridium cylinder that served as the world’s kilogram for 130 years) or to measurement recipes, all seven base units are now defined in terms of seven fixed fundamental constants of nature: the speed of light, Planck’s constant, the elementary charge, the Boltzmann constant, the Avogadro constant, the cesium hyperfine frequency, and the luminous efficacy of a specific radiation frequency.
This might sound like an abstract philosophical shift, but it reshapes how derived units work in practice. Since the base units are now defined in terms of constants, all units in the SI are now defined in terms of those constants.4PubMed Central. How to Define the Units of the Revised SI Starting from Seven Constants with Fixed Numerical Values The distinction between “base” and “derived” becomes less fundamental. A volt or an ohm is no longer derived from the ampere through a chain of experimental procedures; it is realized directly from the constants, using quantum phenomena. In theory, anyone with the right equipment can now realize any SI unit anywhere in the world, without reference to a central artifact stored in a vault near Paris.
Quantum Effects and Electrical Derived Units
Electrical units provide the clearest example of how the 2019 redefinition actually plays out in a laboratory. Two quantum effects are central here. The Josephson effect produces a voltage that depends only on Planck’s constant and the elementary charge. The quantum Hall effect produces a resistance that also depends only on those two constants. Since both constants now have exact, fixed numerical values in the SI, these effects provide direct, universal ways to realize the volt and the ohm with extraordinary precision.
This approach was already in use informally before 2019. International electrical reference standards based on the Josephson and quantum Hall effects were introduced in 1990 to improve the international uniformity of electrical measurements.5Metrologia. New International Electrical Reference Standards Based on the Josephson and Quantum Hall Effects But under the old definitions, those quantum-based values carried small offsets from the “official” SI values, because the SI ampere was still defined through an older thought experiment involving infinite parallel wires. The 2019 revision removed that mismatch. The Josephson effect and the quantum Hall effect now provide realizations of the volt and the ohm without the uncertainties inherited from those older electromechanical definitions, to the benefit of anyone who needs highly accurate electrical measurements.6Comptes Rendus. Physique. The ampere and the electrical units in the quantum era
Researchers have pushed this further by combining both effects to realize a derived unit of current. By applying Ohm’s law to a Josephson voltage source and a quantum Hall resistance standard, a team recently demonstrated a primary quantum current standard with a measurement uncertainty at the level of about one part per billion.7Nature Communications. A primary quantum current standard based on the Josephson and the quantum Hall effects That is, they built the ampere from two derived units (volts and ohms) rather than the other way around. In the new SI, this is perfectly legitimate, because the hierarchy between base and derived has loosened. Any route that connects you to the defining constants with low uncertainty is valid.
The kW Versus kWh Problem
One of the most persistent confusions involving derived units shows up every time people talk about electricity costs, battery capacity, or solar installations: mixing up kilowatts and kilowatt-hours. A kilowatt is a unit of power, meaning the rate at which energy is used or produced at a given instant. A kilowatt-hour is a unit of energy, meaning the total amount accumulated over time. Confusing them is like confusing speed with distance.
This is not just a trivial vocabulary issue. Research on the economics of electricity storage has found that many reports on the cost of energy storage confuse the cost of charge and discharge power (measured in kW) with the cost of energy storage potential (measured in kWh), and that these parameters vary greatly and create fundamentally different cost structures for different storage technologies.8Solar Energy. Microeconomics of electrical energy storage in a fully renewable electricity system A battery that can deliver a lot of power very quickly (high kW) but holds little total energy (low kWh) has a very different cost profile than one that trickles out a modest current for hours on end. Lumping the two together leads to misleading comparisons between, say, lithium-ion batteries and pumped hydroelectric storage.
If you are reading about home solar or electric vehicles, keep the distinction in mind. Your solar panels have a rated power in kilowatts, meaning the peak rate they can generate electricity under ideal conditions. Your electricity bill charges you in kilowatt-hours, which is the total energy you consumed over the billing period. A 5 kW solar array running at full capacity for 4 hours produces 20 kWh. The two numbers answer different questions: “how fast?” versus “how much total?”
Units in Software and Computing
Derived units create headaches for software developers because the algebraic combinations that are simple on paper become ambiguous in code. The core problem is notation. When a human writes “kg·m/s²” on a whiteboard, the meaning is clear. But software needs strict parsing rules, and derived units often involve multiple levels of multiplication and division that can be written in conflicting ways.
One well-known issue is the “double solidus” problem: what does “kg/m/s” mean? Is it kilograms per meter per second, or kilograms per meter-second? Different conventions give different answers. The Unified Code for Units of Measure (UCUM), a system widely used in health informatics and data exchange, allows the double solidus and parses it left to right.9Computer Standards & Interfaces. Quantities, units and computing Other systems reject double solidus notation entirely and require parentheses to disambiguate. This matters in practice because a misinterpreted unit expression in a medical dosage calculation or an engineering simulation can produce results that are off by orders of magnitude.
The broader issue is that the SI was designed for human communication, and its conventions rely on context that humans absorb effortlessly. We see “N·m” and know it is a torque, not a joule expressed differently, because we know the physical context. A parser sees only symbols and rules. Making derived units machine-readable without losing the physical meaning is an active area of work in standards bodies and scientific computing. Any time you export data from one system and import it into another, the derived unit expressions have to survive the translation intact, or the numbers become meaningless.
Derived Units Outside the SI
The concept of building new units from a set of base units is not unique to the SI. The older CGS system (centimeter-gram-second) had its own derived units, some of which still linger in specialized fields. Astronomers still talk about the erg (the CGS energy unit, equal to one ten-millionth of a joule). Physicists working in electromagnetism sometimes encounter the gauss for magnetic flux density, which is the CGS counterpart to the SI tesla.
Natural unit systems used in theoretical physics go even further. By setting certain fundamental constants equal to 1, physicists effectively collapse multiple dimensions into the same unit. In Planck units, for instance, length, time, mass, energy, and temperature can all be expressed as powers of a single quantity. This makes equations cleaner but makes the “derived” versus “base” distinction essentially meaningless, since everything reduces to dimensionless ratios of fundamental constants. These unit systems are rarely used outside the realm of theoretical research, but they highlight that the SI’s separation of base and derived units is a practical choice, not a law of nature.
When Derived Units Mislead
Derived units can occasionally obscure the physics they are supposed to clarify. The sievert, discussed earlier, folds in biological weighting factors that depend on the type of radiation and the tissue being irradiated. Two exposures measured in sieverts can represent very different physical situations but look identical on paper. Someone reading that they received “2 millisieverts” during a medical scan might assume that number is directly comparable to the “2 millisieverts” an astronaut receives from cosmic rays over a week, even though the radiation types, energies, and exposure patterns are completely different. The derived unit compresses all of that complexity into a single number, which is both its strength and its weakness.
A similar subtlety applies to the decibel, which is not technically an SI unit but is accepted for use alongside the SI. The decibel is a logarithmic ratio, not a linear measurement, so “twice as many decibels” does not mean “twice as loud.” And decibels used for sound pressure, electrical power, and signal-to-noise ratios are all calculated differently even though they share the same name. This kind of overloading, where the same unit label means slightly different things in different contexts, is a recurring hazard in the world of derived measurements.
Even the humble degree Celsius can trip people up. It is a derived unit in the SI (defined as the kelvin offset by 273.15), and it works perfectly for expressing temperatures. But it breaks down when you try to use it in calculations involving temperature differences versus absolute temperatures. A change of 10 degrees Celsius is the same as a change of 10 kelvins, but 10 °C is not the same temperature as 10 K. The derived unit behaves differently depending on whether you are talking about a point on a scale or an interval, and that ambiguity has caused real errors in engineering and scientific computation.