Die Höhe beträgt zwei Meter.

Because Höhe is a feminine noun in German. Its basic form is die Höhe.

German nouns have grammatical gender:

• der = masculine
• die = feminine
• das = neuter

So in the nominative singular, you say die Höhe.

Betragen means something like to amount to, to come to, or in measurement contexts, to be.

So Die Höhe beträgt zwei Meter literally feels like:

• The height amounts to two meters

In natural English, this is usually just:

• The height is two meters

This verb is common in formal or factual statements about measurements, totals, sums, and quantities.

German main clauses normally follow the verb-second rule.

Here the sentence is:

• Die Höhe = subject
• beträgt = verb
• zwei Meter = measurement

So the structure is:

• Subject + verb + rest

That is why beträgt comes right after Die Höhe.

After betragen, the measurement is normally in the accusative, and plural measurement nouns usually look the same in nominative and accusative.

So:

• ein Meter
• zwei Meter

You would see Metern in some dative contexts, but not here.

Examples:

• Die Strecke beträgt zehn Kilometer.
• Die Mauer ist aus zwei Metern Entfernung sichtbar. → here Metern appears because of a different grammatical context

In German, measurements after numbers usually appear without an article.

So you say:

• zwei Meter
• drei Liter
• fünf Euro

Adding an article would usually sound wrong here.

Yes, that is understandable, but Die Höhe beträgt zwei Meter is more standard and idiomatic for an exact measurement.

German often prefers:

• Die Höhe beträgt zwei Meter.
• Die Länge beträgt fünf Kilometer.
• Die Kosten betragen 20 Euro.

With sein, native speakers more often use an adjective phrase:

• Das Haus ist zwei Meter hoch.

So these are both natural:

• Die Höhe beträgt zwei Meter.
• Das Haus ist zwei Meter hoch.

But Die Höhe ist zwei Meter is less typical.

Höhe is a noun meaning height.

Hoch is an adjective meaning high or tall.

Compare:

• Die Höhe beträgt zwei Meter. = The height is two meters.
• Die Tür ist zwei Meter hoch. = The door is two meters high/tall.

So:

• Höhe = the measurement as a thing
• hoch = describing how high something is

No. Betragen is used for many kinds of amounts and totals.

Examples:

• Die Kosten betragen 50 Euro. = The cost is 50 euros.
• Die Entfernung beträgt 10 Kilometer. = The distance is 10 kilometers.
• Die Wartezeit beträgt 30 Minuten. = The waiting time is 30 minutes.

So it is a very useful verb for facts involving numbers.

Because all German nouns are capitalized.

So:

• die Höhe
• der Meter
• das Haus

This is one of the most noticeable spelling rules in German.

Höhe is pronounced roughly like HUR-huh, but with the German ö sound, which does not exist exactly in English.

A rough guide:

• Hö sounds a bit like the vowel in French peu or like saying ay with rounded lips
• he at the end is a weak uh sound

Also, the word has two syllables:

• Hö-he

Die Höhe is in the nominative case because it is the subject of the sentence.

You can tell it is the subject because it is the thing doing the grammatical action of the sentence:

• The height amounts to two meters

So:

• Die Höhe = nominative subject
• zwei Meter = accusative measurement after betragen

Yes, but the finite verb still has to stay in second position in a normal main clause.

For example:

• Die Höhe beträgt zwei Meter.
• Zwei Meter beträgt die Höhe.

The second version is grammatically possible, but it sounds more marked or stylistic. The first version is the normal, neutral order.

In zwei Meter, it is plural.

German measurement words often have:

• singular after one: ein Meter
• plural after numbers greater than one: zwei Meter, drei Meter

So in your sentence, Meter is plural, even though the form looks the same as the singular. That is normal for this noun.