
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.

sum
The result you get when you add numbers together.
Die Summe von 15 und 27 ist 42.
The sum of 15 and 27 is 42.

sum / total
The entirety of something; the whole that consists of several parts.
Die Summe seiner Erfahrungen machte ihn zu einem weisen Ratgeber.
The sum of his experiences made him a wise advisor.
'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
Höhe and Summe can refer to the same total value on an invoice. Höhe emphasizes how large the invoice amount is. Summe refers to the whole resulting from adding up the individual items.

Die Summe der einzelnen Posten auf der Rechnung beträgt 300 Euro.
The sum of the individual items on the invoice is 300 euros.
Die Höhe der Rechnung von 300 Euro überrascht mich.
The amount on the invoice, 300 euros, surprises me.