Another factor to consider is the equation which includes the focal length of the lens: lens focal length/diameter of the aperture = f-stop. The f-stop number is a convenient way of describing the exposure settings rather than noting the area of the aperture. And, a given f-stop produces the same exposure regardless of the lens focal length (for a given shutter speed). For example, 1/500th of a second at f/5.6 on a 50mm lens gives the same exposure as those settings on a 300mm lens.
This becomes obvious if you think about an exposure meter - it only gives shutter speed and aperture regardless of the lens or camera type.