die Summe
sum / amount
A specific amount of money.
Für den Hauskauf benötigen wir eine beträchtliche Summe.
We need a considerable sum to buy the house.

sum
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sum / amount
A specific amount of money.
Für den Hauskauf benötigen wir eine beträchtliche Summe.
We need a considerable sum to buy the house.

'Die Summe' (the sum) belongs to a word family. The related verb is 'summieren' (to add up) and the adjective is 'summarisch' (summary, summarizing).
Wenn man alle Kosten summiert, kommt man auf einen hohen Betrag.
If you add up all the costs, you arrive at a high amount.
When 'Summe' (sum) is used in the mathematical sense, it is often followed by the preposition 'aus' to name the numbers being added. The structure is: 'die Summe aus ... und ...' (the sum of ... and ...).
Was ist die Summe aus 7 und 8?
What is the sum of 7 and 8?
When it is about money, 'Summe' (sum/amount) is often combined with adjectives such as 'hoch' (high), 'klein' (small), 'beachtlich' (considerable), or 'ganz' (whole/entire). Typical verbs are 'zahlen' (to pay), 'überweisen' (to transfer), or 'investieren' (to invest).
Er hat eine hohe Summe im Lotto gewonnen.
He won a large sum in the lottery.
The phrase 'in der Summe' means something like 'overall' or 'all in all'. It is used to draw a conclusion from various individual parts.
Jeder Teil war für sich gut, aber in der Summe hat das Projekt nicht überzeugt.
Each part was good on its own, but overall the project was not convincing.
height / amount
'die Höhe' and 'die Summe' can both be used with numbers or money. 'die Summe' is the specific amount of money or the result when you add numbers together. 'die Höhe' describes more how large this amount or level is. That is why people often say 'die Höhe der Summe' (the size of the sum) when they want to emphasize whether it is large or small. They can be confused because both appear in contexts involving calculations and money. If you are naming the total amount itself, 'Summe' fits. If you are talking about the size of an amount, 'Höhe' fits.

Nach dem Einkauf hat sie die Summe noch einmal im Kopf nachgerechnet.
After shopping, she added up the sum again in her head.
Simon war überrascht von der Höhe der Rechnung im Krankenhaus.
Simon was surprised by the size of the hospital bill.