Změříme šířku a hloubku.

Změříme comes from the infinitive změřit.

Grammatically, it looks like a present-tense form, but because změřit is a perfective verb, its present form usually refers to the future.

So změříme means:

• we will measure
• or in natural English, often we’ll measure

This is very common in Czech:

• měřit = to measure, be measuring (imperfective)
• změřit = to measure completely / to take a measurement (perfective)

So in this sentence, the idea is not just the process of measuring, but the completed action.

Because Czech often leaves out subject pronouns when they are already clear from the verb ending.

The ending -me in změříme tells you the subject is we.

So:

• Změříme šířku a hloubku. = We will measure the width and depth.
• My změříme šířku a hloubku. = We will measure the width and depth.

Adding my is possible, but it usually gives extra emphasis or contrast, for example:

• Oni udělají výšku, my změříme šířku a hloubku.
  They’ll do the height, we’ll measure the width and depth.

Because they are direct objects, so they are in the accusative case.

Their dictionary forms are:

• šířka = width
• hloubka = depth

Both are feminine nouns of the common -a pattern. In the singular accusative, that -a changes to -u:

• šířka → šířku
• hloubka → hloubku

So the sentence literally has:

• measure what? → šířku a hloubku

That is why the accusative is used.

The dictionary forms are:

• šířka = width
• hloubka = depth

In vocabulary lists, you would usually learn them in that nominative singular form.

You may also see similar measurement words:

• délka = length
• výška = height
• šířka = width
• hloubka = depth

These often behave the same way in the accusative:

• délku
• výšku
• šířku
• hloubku

This is about aspect, which is very important in Czech.

• měřit is imperfective
  It focuses on the process, duration, or repeated action.
• změřit is perfective
  It focuses on the action as completed, with a result.

So:

• Měříme šířku a hloubku. can mean We are measuring the width and depth / We measure the width and depth
• Změříme šířku a hloubku. means We will measure the width and depth / We’ll take the measurements

Since this sentence is about carrying out the measurement as a complete task, změřit is the natural choice.

Sometimes it can sound a bit like a suggestion in context, but its basic meaning is still We will measure the width and depth.

If you want a true let’s form, Czech usually uses the 1st person plural imperative:

• Změřme šířku a hloubku. = Let’s measure the width and depth.

That said, in real conversation Změříme... can be used in a practical, decision-like way, similar to:

• We’ll measure the width and depth.

Depending on tone and context, English might translate that as a suggestion, but grammatically it is not the main imperative form.

No, Czech word order is fairly flexible because the case endings show the grammatical roles.

The neutral order here is:

• Změříme šířku a hloubku.

But you could also say:

• Šířku a hloubku změříme.

This puts more emphasis on šířku a hloubku, as if contrasting them with something else.

For example:

• Délku zatím ne, ale šířku a hloubku změříme.
  Not the length yet, but we’ll measure the width and depth.

So the default order is natural and neutral, but other orders are possible for emphasis.

Because Czech does not have articles like English a or the.

So šířku a hloubku can mean:

• width and depth
• the width and depth

Which one sounds best in English depends on the context.

Czech expresses definiteness mostly through context, not with articles. That means learners often need to get used to the fact that a Czech noun can correspond to several English possibilities.

A rough pronunciation guide is:

• Změříme ≈ z-mnye-rzhee-me
• šířku ≈ sheerzh-koo
• a = and, pronounced like a short ah
• hloubku ≈ hlohp-koo

A few important points:

• mě is pronounced something like mně, with a softened m
• ř is the famous Czech sound that does not exist in English; it is something like a mix of r and zh
• In hloubku, the b is pronounced more like p because it comes before k; this is normal Czech sound assimilation

If you want to say it in syllables:

• Změ-ří-me šíř-ku a hloub-ku

A simply means and.

So:

• šířku a hloubku = the width and depth

It connects the two objects of the verb.

This is a very common and straightforward conjunction in Czech:

• délku a šířku = length and width
• výšku a hloubku = height and depth

So in this sentence, a works exactly as you would expect.