Mit dem Maßband messen wir, wie breit das Regal sein darf.

Because mit always takes the dative case in German.

• das Maßband = nominative/accusative
• dem Maßband = dative

Since Maßband is a neuter noun, its dative singular article is dem.

So:

• das Maßband = the tape measure
• mit dem Maßband = with the tape measure

German main clauses normally follow the verb-second rule. That means the conjugated verb comes in the second position.

Here, the sentence starts with Mit dem Maßband, which takes the first position. So the verb must come next:

• Mit dem Maßband | messen | wir

If you started with wir, then you would say:

• Wir messen mit dem Maßband, wie breit das Regal sein darf.

Both are correct. Starting with Mit dem Maßband just puts more emphasis on the tool being used.

Maßband means tape measure or measuring tape.

It is a compound noun:

• Maß = measure
• Band = band / strip / tape

So literally it is something like measure-band.

Because wie breit das Regal sein darf is a subordinate clause, more specifically an indirect question.

German uses commas before subordinate clauses much more consistently than English does.

Here the structure is:

• main clause: Mit dem Maßband messen wir
• subordinate clause / indirect question: wie breit das Regal sein darf

So the comma is required.

It is an indirect question.

The direct question would be:

• Wie breit darf das Regal sein? = How wide may the shelf be?

When this question is embedded inside a larger sentence, the word order changes:

• ..., wie breit das Regal sein darf.

In indirect questions, the verb goes to the end of the clause.

So this part means something like how wide the shelf is allowed to be.

In the clause wie breit das Regal sein darf, das Regal is the subject, not the object.

Why?

Because the verb is really sein darf:

• das Regal = the thing that is
• breit = the adjective describing it

So this clause works like:

• The shelf may be wide
• How wide may the shelf be?

That means das Regal is in the nominative case.

Even though the whole sentence contains messen, the embedded clause has its own internal grammar.

Because breit is a predicate adjective here.

Adjectives in German only take endings when they come before a noun:

• das breite Regal = the wide shelf

But after verbs like sein, werden, or bleiben, the adjective usually has no ending:

• Das Regal ist breit.
• Das Regal darf breit sein.

So in this sentence, breit stays plain.

Because this is a subordinate clause with a modal verb.

In a normal main clause, you would say:

• Das Regal darf breit sein.

Here:

• darf is the conjugated modal verb
• sein is the infinitive

But in a subordinate clause, German moves the verbal elements to the end:

• ..., wie breit das Regal sein darf.

So the end position sein darf is normal for this kind of clause.

Yes, but it is often a bit broader than personal permission.

dürfen usually means:

• to be allowed to
• may

In this sentence, it suggests a limit or allowed size:

• how wide the shelf is allowed to be
• how wide the shelf may be

So this is not necessarily someone verbally giving permission at that moment. It can also mean that there is some rule, requirement, or practical limit.

Yes, absolutely.

That version is grammatically correct and probably feels a bit more neutral to an English speaker because it starts with the subject:

• Wir messen mit dem Maßband, wie breit das Regal sein darf.

The original sentence:

• Mit dem Maßband messen wir, wie breit das Regal sein darf.

is also correct, but it gives a little more focus to Mit dem Maßband.

German is more flexible than English about what can come first in the sentence, as long as the conjugated verb still stays in second position.