Depending on how the projected images fulfill the image sensor, fisheye lenses can be classified into three types: Circular Fisheye Lenses, Diagonal Fisheye Lenses (also known as Full-frame fisheye lenses), and Cropped Circle Fisheye Lenses.
The circular fisheye lenses provide 180°field of view over the width of the sensor, projecting the entire hemispherical space onto the image sensor, and the borders of the sensor will appear black.
The diagonal fisheye lenses or full-frame fisheye lenses provide 180°field of view over the diagonal of the camera sensor (this means the vertical and horizontal FOVs will be smaller than 180°). The image circle extends beyond the sensor frame.
The cropped circle fisheye Lenses (also known as the portrait fisheye lenses) are optimized for the width of the format cameras, producing images cropped at the top and bottom. The result is an image in the center and black shades at the corners ( the proportion of black shapes is much less than in the case of the circular fisheye lenses).
The figure below clearly illustrates the differences among the three types of fisheye lenses:
Figure 1. Three Types of Fisheye Lenses