How does the f-number of camera lenses affect image area?

[Gear300]

I was reading on the f-number of lenses for cameras...somewhere in the text, they stated that the area of the image is proportional to q², in which q is image distance. I figured the area to be proportional to q, but I don't really see it as q²...so how does this work out?

 

[Andy Resnick]

The f/# doesn't affect the field of view, so I'm not sure what you mean, exactly. The image *brightness* goes as the square of the f/#, but that's different.