die Höhe
height
The vertical extent or distance from a lower point to an upper point.
Der Mount Everest hat eine Höhe von 8848 Metern.
Mount Everest has a height of 8,848 meters.

height / amount
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height
The vertical extent or distance from a lower point to an upper point.
Der Mount Everest hat eine Höhe von 8848 Metern.
Mount Everest has a height of 8,848 meters.

The noun 'die Höhe' (height) is derived directly from the adjective 'hoch' (high). It turns the quality of being high into a noun.
Der Turm ist sehr hoch.
The tower is very high.
Pay attention to the preposition and case: 'in der Höhe' (at a height / in the air) with the dative describes a place, while 'in die Höhe' (upward) with the accusative indicates a direction upward.
Das Flugzeug befindet sich in großer Höhe.
The airplane is at a great height.
To state a specific height, the construction 'in einer Höhe von' (at a height of) + number + unit is often used.
Das Flugzeug fliegt in einer Höhe von 10.000 Metern.
The airplane is flying at a height of 10,000 meters.
You can recognize this meaning by the fact that it is not about a physical distance, but about the amount or level of something abstract. Typical contexts are finances, costs, or damages.
Die Höhe der Rechnung hat mich überrascht.
The amount of the bill surprised me.
The idiom 'auf der Höhe sein' (to be on top form / to be up to speed) is idiomatic. It is very often used in the negative to express that someone is not fit or not well informed.
Ich fühle mich heute nicht ganz auf der Höhe.
I don't feel quite on top form today.
The directional phrase 'in die Höhe' (upward) is often combined with verbs that describe a rapid increase, such as 'schnellen' (to soar), 'schießen' (to shoot), or 'steigen' (to rise).
Die Mieten sind in den letzten Jahren stark in die Höhe gestiegen.
Rents have risen sharply upward in recent years.
status / stand
'der Stand' and 'die Höhe' can both be used with numbers, values, or a level. 'der Stand' usually describes the current state or current value at a particular point in time, for example with an account, a project, or a development. 'die Höhe' refers more to the amount or extent itself, especially with money, costs, or numbers. They can be confused because both can point to a value. If you are asking how far along or how up to date something is, 'Stand' fits. If you are asking how much something amounts to, 'Höhe' fits.

Die Höhe der Miete hängt von der Größe der Wohnung ab.
The amount of the rent depends on the size of the apartment.
Kannst du mir den aktuellen Stand des Projekts schicken?
Can you send me the current status of the project?