Derived SI Units

We can derive many units from the seven SI base units, building our toolbox of standard units.


Volume is the measure of the amount of space occupied by an object. The standard SI unit of volume is defined by the base unit of length (the meter). The standard volume is a cubic meter (m3), a cube with an edge length of exactly one meter. To dispense a cubic meter of water, we could build a cubic box with edge lengths of exactly one meter. This box would hold a cubic meter of water or any other substance.

A more commonly used unit of volume is derived from the decimeter (0.1 m, or 10 cm). A cube with edge lengths of exactly one decimeter contains a volume of one cubic decimeter (dm3). A liter (L) is the more common name for the cubic decimeter.

A cubic centimeter (cm3) is the volume of a cube with an edge length of exactly one centimeter. The abbreviation cc (for cubic centimeter) is often used by health professionals. A cubic centimeter is equivalent to a milliliter (mL) and is 1/1000 of a liter.

(a) The relative volumes are shown for cubes of 1 m3, 1 dm3 (1 L), and 1 cm3 (1 mL) (not to scale). (b) The diameter of a dime is compared relative to the edge length of a 1-cm3 (1-mL) cube.


We use the mass and volume of a substance to determine its density. Thus, the units of density are defined by the base units of mass and length.

$$density = {mass \over volume}$$

The density of a substance is the ratio of the mass of a sample of the substance to its volume. The SI unit for density is the kilogram per cubic meter (kg/m3). For many situations, however, this is an inconvenient unit, and we often use grams per milliliter (g/mL) or gram per cubic centimeter (g/cm3) for the densities of liquids and solids respectively, and grams per liter (g/L) for gases. Although there are exceptions, most liquids and solids have densities that range from about 0.7 g/mL (the density of gasoline) to 19 g/cm3 (the density of gold, remember 1 mL=1 cm3). The density of air is about 1.2 g/L. The table below shows the densities of some common substances.

Solids Liquids Gases (at 25 °C and 1 atm)
ice (at 0 °C) 0.92 g/cm3 water 1.0 g/cm3 dry air 1.20 g/L
oak (wood) 0.60–0.90 g/cm3 ethanol 0.79 g/cm3 oxygen 1.31 g/L
iron 7.9 g/cm3 acetone 0.79 g/cm3 nitrogen 1.14 g/L
copper 9.0 g/cm3 glycerin 1.26 g/cm3 carbon dioxide 1.80 g/L
lead 11.3 g/cm3 olive oil 0.92 g/cm3 helium 0.16 g/L
silver 10.5 g/cm3 gasoline 0.70–0.77 g/cm3 neon 0.83 g/L
gold 19.3 g/cm3 mercury 13.6 g/cm3 radon 9.1 g/L
Densities of Common Substances

Example: Calculation of Density

Gold — in bricks, bars, and coins — has been a form of currency for centuries. In order to swindle people into paying for a brick of gold without actually investing in a brick of gold, people have considered filling the centers of hollow gold bricks with lead to fool buyers into thinking that the entire brick is gold. It does not work: Lead is a dense substance, but its density is not as great as that of gold, 19.3 g/cm3.

What is the density of lead if a cube of lead has an edge length of 2.00 cm and a mass of 90.7 g?


The density of a substance can be calculated by dividing its mass by its volume. The volume of a cube is calculated by cubing the edge length. $$volume\,of\,lead\,cube\,=\,2.00\,cm\,×\,2.00\,cm\,×\,2.00\,cm\,=\,8.00\,cm^3$$ $$density\,=\frac{mass}{volume}=\frac{90.7\,g}{8.00\,cm^3}=\, 11.3 \frac{g}{cm^3}$$


The derived SI unit for pressure is the pascal (Pa), defined as:
$$\text{Pa} = \frac{\text{kg}}{\text{m s}^2} = \frac{\text{N}}{\text{m}^2}$$.

Since a Pascal is a relatively small force, many forces are reported in kilopascals (kPa). Watch your units in calculations to make sure you haven’t lost a factor of 1000.

The bar is another commonly used unit, though it is not strictly part of the SI. $$1\; \text{bar} = 100 000\; \text{Pa}$$. The bar is part of the modern definition of standard atmospheric pressure. (Standard pressure is 1 bar).