Traditionally the thermography community have used round infrared windows. This began in the old days where we primarily used fragile crystal optics. Square crystals were much more expensive to manufacture and much more fragile, so it was practical to make round windows. With todays modern polymer infrared transmissive materials it’s no longer necessary to stick to a round shape and compared to a round aperture, a square (or rectangular) polymer infrared window provides significantly more viewing area.
A modern thermal imager has a square or rectangular detector that produces an image with right angled corners. Subsequently a square or rectangular polymer infrared window allows the thermographer to use the full width and height of the window aperture without clipping the edges of the image. With a round window you are limited to the centre most part of the window. As you begin to move the camera toward the edge or start angling the camera through the window, the natural round of the window will begin to crop the image.
In real terms the viewing area goes well beyond the surface area calculations because you are fitting a “square image” through a “square hole”, not a “square image” through a round hole.