There are two types of fisheye camera lens to choose from, full-frame fisheye camera lens and circular fisheye camera lens. The lens glass is circular, so why does the image become a rectangle? Because all lenses form a circle, so your original idea is correct, the circular glass does indeed form a circular image. Then this circle is projected onto the camera sensor or film, like this. Knowing this, the image circle of a circular fisheye lens falls completely on the film or digital sensor. However, the image circle of a full-frame fisheye lens completely covers the film or digital sensor. Therefore, it effectively shows you the rectangular cropping of the image circle.
When using a full-frame fisheye lens, the effect is an image with a normal full-size rectangular shape. The image circle they produce is large enough to cover the entire image sensor. Take a photo and it's rectangular, just like any other photo, only it's ultra wide-angle. When measured horizontally, the frame emits a 180-degree field of view. Only the outer edges of the image are distorted. The focal length of a full-frame camera is around 15mm, and for an APS-C camera, it's around 10mm, to achieve this effect. They are called full-frame fisheye lenses because they cover the entire frame. The sensor or image area is completely covered and surrounded by the image circle.
Full-frame fisheye lenses were created in response to the increasing popularity of fisheye camera lens. Historically, circular lenses paved the way for introducing full-frame lenses. As a result, when measured from edge to edge, the field of view is 180 degrees. The horizontal and vertical field of view is smaller. For example, the equivalent spherical angle of a 15mm full-frame fisheye lens measures 147 degrees horizontally and 94 degrees vertically.
They produce an image with a relatively smaller radius, suitable for the boundaries of the image sensor. As a result, the final product is a circular image with a wider field of view, measuring up to 180 degrees horizontally and vertically. You take a photo and the entire image appears in a circular frame in the center of the frame, with the rest of the frame being black. Circular fisheye camera lens were the earliest lenses of this kind to be produced. For this type of fisheye lens, the image circle fits inside the image sensor.
In certain cases, there may be vignetting in some corners of the image frame. This leads to a tunnel vision effect where the entire scene appears to be contained within a circle. The circular fisheye lens effect can be achieved at an 8mm focal length. This is applicable when using a fisheye lens on a full-frame camera. On an APS-C camera, you would need a 13mm focal length. This is applicable because it is equivalent to 35mm. Therefore, using 8mm as the focal length will not give you the desired circular fisheye effect.