der Bauch
belly
The front part of the human or animal body between the chest and the pelvis, which contains the digestive organs.
Nach dem großen Essen hatte ich Bauchschmerzen.
After the big meal, I had a stomachache.

belly / stomach
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belly
The front part of the human or animal body between the chest and the pelvis, which contains the digestive organs.
Nach dem großen Essen hatte ich Bauchschmerzen.
After the big meal, I had a stomachache.

Pay attention to the case with prepositions. Movement is expressed with the accusative, while a position or state requires the dative.
Der Boxer schlägt seinem Gegner in den Bauch.
The boxer hits his opponent in the belly.
The idiom 'aus dem Bauch heraus' (from the gut) is very common and means that you make a decision intuitively. The expression functions in the sentence like an adverbial phrase.
Ich kann es nicht erklären, ich habe einfach aus dem Bauch heraus gehandelt.
I can't explain it; I simply acted from the gut.
In connection with the body part, 'Bauch' (belly/stomach) is often associated with pain or feelings. Typical phrases are 'Bauchschmerzen haben' (to have a stomachache), 'Schmetterlinge im Bauch haben' (to have butterflies in one's stomach), or 'sich den Bauch vollschlagen' (to stuff oneself).
Vor der Prüfung hatte ich immer ein flaues Gefühl im Bauch.
Before the exam, I always had a queasy feeling in my stomach.
When 'Bauch' (belly) is used in connection with objects such as bottles, vases, or ships, it refers to the rounded, bulging shape of the object. This is a figurative transfer.
Der Wein lagerte im Bauch des alten Holzfasses.
The wine was stored in the belly of the old wooden barrel.
head
'der Kopf' and 'der Bauch' are body parts located in different places: the head is at the top and contains the brain, while the belly is at the front in the middle of the body. They also form a contrast in a figurative sense. 'Kopf' is often associated with thinking, planning, and reason. 'Bauch' in the expression 'aus dem Bauch heraus' (from the gut) stands for feeling or intuition. They can be confused especially in such figurative expressions. A simple guide is: the head thinks, the gut feels.

Sie hat aus dem Bauch heraus zugesagt, obwohl sie noch nicht alle Informationen hatte.
She agreed based on gut feeling, even though she did not yet have all the information.
Bei wichtigen Entscheidungen höre ich meistens auf meinen Kopf.
When making important decisions, I usually listen to my head.