Volymenheter

$$\begin{gathered} 1m^3=1000L \\ 1m^3={\left(10dm\right)}^3=1000d{m}^3\iff 1d{m}^3=\frac{1}{1000}{m}^3 \\ Men vi vet att 1m^3=1000L \\ Alltså 1d{m}^3=\frac{1}{1000}·1000L=1L \end{gathered}$$

Sen kan man ju då ställa sig frågan, varför är $1m^3=1000L$

Det kan jag dessvärre inte ge dig någon matematisk förklaring till, men för att citera wikipedia:

Litres are most commonly used for items (such as fluids and solids that can be poured) which are measured by the capacity or size of their container, whereas cubic metres (and derived units) are most commonly used for items measured either by their dimensions or their displacements. The litre is often also used in some calculated measurements, such as density (kg/L), allowing an easy comparison with the density of water.

One litre of water has a mass of almost exactly one kilogram when measured at its maximal density, which occurs at about 4 °C. It follows, therefore, that 1000th of a litre, known as one millilitre (1 mL), of water has a mass of about 1 g; 1000 litres of water has a mass of about 1000 kg (1 tonne or megagram). This relationship holds because the gram was originally defined as the mass of 1 mL of water; however, this definition was abandoned in 1799 because the density of water changes with temperature and, very slightly, with pressure.