Is the calculation of the opening diameter correct?

It is assumed that a 400 mm lens f4.0 has a maximum aperture diameter of 10 cm. When looking at a lens of this type, they do not seem large enough to fit that type of aperture, including all the control mechanics within them. Do modern lenses have a way of focusing light on a smaller aperture to behave like a larger one or are they really always the size suggested by the formula (fstop = focal length / aperture diameter)?