der Bogen
arch / curve
A curved line or shape; also an architectural element such as an archway or bridge arch.
Die alte Steinbrücke führt in einem weiten Bogen über den Fluss.
The old stone bridge crosses the river in a wide arch.

arch / sheet / bow
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arch / curve
A curved line or shape; also an architectural element such as an archway or bridge arch.
Die alte Steinbrücke führt in einem weiten Bogen über den Fluss.
The old stone bridge crosses the river in a wide arch.

'Bogen' is a base word in many compounds that all refer to a curved shape or a sheet of paper. Examples are 'Regenbogen' (rainbow), 'Ellenbogen' (elbow), 'Torbogen' (archway), or 'Fragebogen' (questionnaire).
Nach dem Regen erschien ein wunderschöner Regenbogen am Himmel.
After the rain, a beautiful rainbow appeared in the sky.
You can easily recognize the many meanings of 'Bogen' from context: 'Pfeil' (arrow) points to the weapon (m03), 'Geige' (violin) to the musical instrument bow (m05), 'ausfüllen' (to fill out) to the paper/form (m02), and 'Brücke' (bridge) to architecture (m01).
Er spannte den Bogen und zielte mit dem Pfeil.
He drew the bow and aimed with the arrow.
When 'Bogen' means a sheet of paper, it rarely stands alone. It is usually part of a compound that describes the purpose more precisely, e.g. 'Fragebogen' (questionnaire), 'Antwortbogen' (answer sheet), or 'Notenbogen' (sheet music).
Der Lehrer verteilte die Antwortbögen für den Test.
The teacher handed out the answer sheets for the test.
As a weapon, 'Bogen' almost always appears in the fixed combination 'Pfeil und Bogen' (bow and arrow). These two words belong together.
Im Mittelalter jagte man mit Pfeil und Bogen.
In the Middle Ages, people hunted with a bow and arrow.
The idiom 'einen Bogen um jemanden/etwas machen' means to deliberately avoid a person or thing. The object that one avoids is in the accusative.
Ich mache einen großen Bogen um das Thema Politik, wenn ich mit ihm spreche.
I steer well clear of the topic of politics when I talk to him.
The idiom 'den Bogen raushaben' is colloquial and means that you have understood how something works. It is often used after a learning phase.
Nach ein paar Versuchen hatte ich den Bogen raus und konnte endlich Fahrrad fahren.
After a few tries, I got the hang of it and could finally ride a bike.
paper
'das Papier' and 'der Bogen' can both have to do with a piece of paper. 'Papier' means the material in general. 'Bogen' means a single larger or specific sheet of paper, often in a fixed format, for drawing, printing, or as a form. Learners may confuse the words because a 'Bogen' is usually made of paper. Still, you do not automatically call every small sheet a 'Bogen'. 'Papier' fits when you mean the material; 'Bogen' fits when the individual format or form is important.

Bitte füllen Sie den Bogen aus und geben Sie ihn am Schalter ab.
Please fill out the form and hand it in at the counter.
Das Papier ist zu dünn für diese Farbe.
The paper is too thin for this paint.