der Magen
stomach
An internal organ in the upper abdomen that digests the food that has been eaten.
Nach dem schweren Essen tut mir der Magen weh.
After the heavy meal, my stomach hurts.

stomach
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stomach
An internal organ in the upper abdomen that digests the food that has been eaten.
Nach dem schweren Essen tut mir der Magen weh.
After the heavy meal, my stomach hurts.

When the stomach is causing pain, the person who has the pain is often expressed with a dative pronoun. The structure is then: 'Mir tut der Magen weh.' (My stomach hurts.)
Dem Patienten tat nach der Operation der Magen weh.
The patient's stomach hurt after the operation.
Common phrases related to eating and drinking are 'auf nüchternen Magen' (on an empty stomach/without having eaten beforehand), 'ein leerer Magen' (an empty stomach), or 'ein voller Magen' (a full stomach).
Man sollte Medikamente nicht auf nüchternen Magen einnehmen.
You shouldn't take medication on an empty stomach.
The stomach is often used metaphorically for feelings. Positive excitement is described with 'Schmetterlinge im Magen haben' (to have butterflies in one's stomach), discomfort or nervousness with 'ein flaues Gefühl im Magen haben' (to have an uneasy feeling in one's stomach).
Vor jeder Prüfung habe ich ein flaues Gefühl im Magen.
Before every exam, I have an uneasy feeling in my stomach.
In the expression 'jemandem schwer im Magen liegen' (to weigh heavily on someone), the problem is the subject of the sentence. The person who is worried stands in the dative.
Die Entscheidung, umzuziehen, liegt ihr immer noch schwer im Magen.
The decision to move is still weighing heavily on her.
belly / stomach
'der Bauch' and 'der Magen' are in the same area of the body, but they do not mean exactly the same thing. 'der Bauch' is the outer and general area at the front of the body between the chest and the pelvis. 'der Magen' is an internal organ that digests food. In everyday speech, people often say 'Bauchschmerzen' (stomachache / belly pain) when they do not know exactly which organ hurts. 'Magenschmerzen' fits more precisely when it really is about the stomach, for example after bad food or with nausea. The words are easy to confuse because both have to do with digestion and feeling unwell.

Mein Magen ist heute empfindlich, deshalb esse ich nur Suppe.
My stomach is sensitive today, so I'm only eating soup.
Nach dem Sport legte das Kind die Hand auf den Bauch, weil es Schmerzen hatte.
After exercising, the child put a hand on their belly because it hurt.