die Ausdehnung
expansion
The increase in the space that something takes up, for example through heat or growth.
Die Ausdehnung von Wasser zu Eis kann Rohre sprengen.
The expansion of water into ice can burst pipes.

expansion / extent
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expansion
The increase in the space that something takes up, for example through heat or growth.
Die Ausdehnung von Wasser zu Eis kann Rohre sprengen.
The expansion of water into ice can burst pipes.

'Ausdehnung' (expansion/extent) is a noun derived from the separable verb 'sich ausdehnen' (to expand). The ending '-ung' turns the verb into a feminine noun.
Das Universum dehnt sich ständig aus, und diese Ausdehnung wird von Astronomen gemessen.
The universe is constantly expanding, and this expansion is measured by astronomers.
The meaning depends strongly on the context. If it is about a process or a physical change (e.g., through heat), it refers to expansion. If it is about geography or a static description, it refers to spatial extent.
Die Ausdehnung des Metalls bei Hitze ist ein physikalisches Phänomen.
The expansion of metal in heat is a physical phenomenon.
In this meaning, 'Ausdehnung' (expansion) is often found in scientific contexts. Typical combinations are 'thermische Ausdehnung' (thermal expansion), 'die Ausdehnung von Gasen' (the expansion of gases), or 'die Ausdehnung eines Materials' (the expansion of a material).
Beim Bau von Brücken muss die thermische Ausdehnung des Stahls berücksichtigt werden.
When building bridges, the thermal expansion of the steel must be taken into account.
When referring to spatial extent, the word is often combined with geographical or abstract terms. For example: 'die räumliche Ausdehnung' (spatial extent), 'die geografische Ausdehnung' (geographical extent), or 'die Ausdehnung eines Reiches' (the extent of an empire).
Die volle Ausdehnung des alten Römischen Reiches war gewaltig.
The full extent of the ancient Roman Empire was enormous.
length
'die Länge' and 'die Ausdehnung' both describe how large something is in space. 'die Länge' usually means only one direction: from one end to the other. 'die Ausdehnung' is more general and means the spatial extent of an area or the expansion of something. They can be confused when talking about size. For a table, a cable, or a path, 'Länge' is more natural; for an area, a surface, or growth, 'Ausdehnung' fits better.

Auf der Karte sehen wir die Ausdehnung des Waldes.
On the map, we can see the extent of the forest.
Die Länge des Kabels reicht nicht bis zur Steckdose.
The length of the cable is not enough to reach the outlet.