
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.

amount / level
The extent, amount, or level of something, especially of numbers or money.
Die genaue Höhe des Schadens ist noch unbekannt.
The exact amount of the damage is still unknown.

to be on top form
To be in top form physically or mentally; or also: to be very well informed about a topic.
Er war letzte Nacht krank und ist heute noch nicht wieder ganz auf der Höhe.
He was sick last night and still isn't quite back on top form today.

height / rise
A geographically elevated place, such as a hill or rise.
Von dieser Höhe hat man einen wunderbaren Blick über das ganze Tal.
From this height, you have a wonderful view over the whole valley.

upward
An indication of direction that describes movement upward. Often used in connection with a rapid increase.
Wegen der gestiegenen Nachfrage schießen die Preise in die Höhe.
Because of the increased demand, prices are shooting upward.
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.
level
Höhe and Niveau can both describe the quantitative state of an economic variable. Höhe tends to refer to its specific extent or value. Niveau tends to refer to the level it has reached, especially when this level is compared with an earlier or different one.

Die Höhe der Arbeitslosigkeit beträgt derzeit sechs Prozent.
The unemployment rate is currently six percent.
Die Arbeitslosigkeit bleibt auf dem Niveau des Vorjahres.
Unemployment remains at the level it was at the previous year.