cheek / buttock
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cheek
The soft, fleshy part of the face to the side of the nose and mouth. Usually used in the plural, 'Backen' (cheeks).
Die Großmutter kniff ihrem Enkel liebevoll in die Backe.
The grandmother lovingly pinched her grandson's cheek.
buttock
Colloquial term for one of the two halves of the buttocks.
Er rutschte auf dem Eis aus und landete direkt auf der rechten Backe.
He slipped on the ice and landed right on his right buttock.
Which meaning is intended becomes immediately clear from the context. Actions involving the face (kissing, smiling, eating) refer to the cheek. Actions like falling down or sitting refer to the buttocks.
Das Baby hat süße, runde Backen.
The baby has cute, round cheeks.
In connection with the face, 'Backe' (cheek) is often used in the plural. Typical expressions are 'rote Backen bekommen' (to get rosy cheeks), 'dicke Backen haben' (to have swollen cheeks), or 'einen Kuss auf die Backe geben' (to give a kiss on the cheek).
Als sie das Kompliment hörte, bekam sie rote Backen.
When she heard the compliment, her cheeks turned red.
The colloquial idiom 'eine dicke Backe machen' does not mean that the cheek is swollen. It means to be offended or to sulk.
Jetzt mach doch keine dicke Backe, es war doch nur ein Scherz.
Come on, don't sulk now; it was only a joke.
Using 'Backe' (buttock) to refer to the buttocks is purely colloquial. In neutral or formal situations, this usage is inappropriate.
Aua, ich habe mich auf die Backe gesetzt!
Ouch, I landed on my butt!
cheek
Here, Wange and Backe refer to the same soft, lateral part of the face. Wange is the neutral standard-language choice, while Backe in this sense usually sounds colloquial or regional. Wange is therefore more suitable in factual situations, while Backe can also be used in casual conversation.
Sein Freund sagt, dass seine Backe ganz rot ist.
His friend says that his cheek is completely red.
Die Ärztin untersucht die gerötete Wange des Patienten.
The doctor examines the patient's reddened cheek.
