Красный сектор проходит через кольцо утроения.

Because красный has to agree with сектор.

• сектор is masculine
• singular
• nominative case here, because it is the subject

So the adjective also takes the masculine singular nominative form: красный.

Compare:

• красный сектор — red sector
• красная зона — red zone
• красное кольцо — red ring

Сектор is the subject of the sentence, so it is in the nominative case.

The basic dictionary form is сектор, and that is exactly the form used here.

In this sentence:

• Красный сектор = the red sector
• проходит = passes / goes / runs
• через кольцо утроения = through the triple ring

So сектор is simply the thing doing the action.

Проходит is the 3rd person singular present tense of проходить.

In different contexts, проходить can mean:

• to pass
• to go through
• to run through
• to cross

In this sentence, it does not necessarily mean literal movement. It can describe the path or position of something, like:

• Линия проходит через центр. — The line passes through the center.
• Дорога проходит через лес. — The road goes through the forest.

So here проходит can be understood as passes through or runs through.

A few different grammar points are involved here:

1. The infinitive is проходить, not проходить.
   Russian infinitives normally end in -ть, so the correct dictionary form is проходить.

2. Проходит is the finite verb form used with he/she/it or any singular noun.
   Since сектор is singular, the verb must also be singular:

• сектор проходит — the sector passes
• секторы проходят — the sectors pass

3. The ending -ит here marks 3rd person singular in the present tense.

The verb проходить is imperfective. That is natural here because the sentence describes a general fact, layout, or ongoing characteristic, not one completed action.

Russian often uses the imperfective when stating how something is arranged or where it goes:

• Граница проходит по реке. — The border runs along the river.
• Линия проходит через точку. — The line passes through the point.

If you used a perfective form like пройдёт, it would usually sound more like will pass through as a single completed event in the future.

Через usually means through or across.

It is the natural preposition when something passes from one side to another, or goes through an area/object:

• идти через парк — to go through the park
• смотреть через окно — to look through the window
• проходить через кольцо — to pass through the ring

So через is the right choice for through the ring.

It actually is in the accusative — it just happens to look the same as the nominative.

Через takes the accusative case, so we need кольцо in the accusative. But:

• nominative singular: кольцо
• accusative singular: кольцо

These forms are identical because кольцо is a neuter inanimate noun.

This is very common in Russian:

• окно → через окно
• море → через море
• кольцо → через кольцо

Утроения is the genitive singular form of утроение.

Here it depends on кольцо:

• кольцо чего? — ring of what?
• кольцо утроения — ring of trebling / triple ring

This is a very common Russian pattern: one noun is followed by another noun in the genitive to specify the type or function.

Examples:

• стакан воды — a glass of water
• центр города — the center of the city
• кольцо утроения — the trebling ring / triple ring

So утроения is not the subject or object by itself; it is modifying кольцо.

Yes. Russian word order is more flexible than English word order.

The neutral order here is:

• Красный сектор проходит через кольцо утроения.

But other orders are possible for emphasis, for example:

• Через кольцо утроения проходит красный сектор.
• Кольцо утроения красный сектор проходит...
  This last one would sound much more marked or context-dependent.

The original version is the most straightforward and natural if you are simply stating the fact.

Russian has no articles.

So Russian does not have separate words for:

• a
• an
• the

Whether something is understood as a red sector or the red sector depends on context.

That is why Красный сектор can mean either:

• a red sector
• the red sector

depending on the situation.

The usual stress pattern is:

• кра́сный
• се́ктор
• прохо́дит
• че́рез
• кольцо́
• утрое́ния

So the full sentence is pronounced:

Кра́сный се́ктор прохо́дит че́рез кольцо́ утрое́ния.

Stress is important in Russian because it is not always predictable and can affect pronunciation quite a lot.