Have you ever wondered why lenses are round, and photographs rectilinear? Obviously square lenses would not work, but why not round photographs? Well, lenses do indeed produce a circular image, however the quality of this image with respect to sharpness and brightness is not at all uniform. It is sharpest and brightest near the centre of the lens, becoming progressively less sharp and bright towards the outer edge of the circle. This deterioration is due to factors such as lens aberrations which become more pronounced towards the edges of the image. In terms of the photograph, only the inner, portion of the circular image should be used, hence why photographs are rectangular, or historically more square (before 35mm film).

Basically for lenses on a particular sensor, the diameter of the circle has to be larger than the diagonal of the frame. The example below shows a Full Frame 24mm×36mm sensor and its associated image circle with a diameter of 43.27mm.

This basically means that the image sensor only makes use of roughly 59% of the image circle (the sensor is 864mm², the image circle 1470mm²). Using a circular fisheye lens, or one that is smaller than the sensor, will result in a circular image. For example, using a small 16mm cinematographic lens on a full frame sensor.

In some cases, such in the case of the Leica D-LUX 6, the camera allows swapping between a bunch of aspect ratios: 16:9, 4:3, 3:2, and 1:1. This camera has a 1/1.7″ sensor (crop factor of 4.6). The actual sensor size is 3678 x 2745 pixels.