The size of the front lens element is not specifically related to the lenses angle of view, although that is a factor. That is more a function of the curvature of the front lens element, as a greater curve is what increases the angle of incident light that can be captured and bent by the lens. The diameter of the front lens element is more indicative of the "speed" of a lens, its maximum aperture. As lenses are rated in relative aperture these days, such as f/4, that means the longer the focal length (f), the wider the lens tube must be to accommodate a similar aperture as a wide angle lens.
If we take two lenses, one wide and one long, that have an f/4 aperture. One lens is 16mm, the other is 400mm. The apparent maximum aperture, or
entrance pupil diameter, is the size of the aperture as viewed through the front lens element. The pupil diameter must be 4mm for the wide angle lens, but 100mm for the long lens, at an f# (N) of 4. Assuming we had an f/2.8 lens, the wide angle would have a pupil diameter of 5.71mm, while the long lens would have to have a pupil diameter of 143mm.
When it comes to telephoto lenses, the picture gets a bit muddier. If you have ever used an actual 400mm prime, or 100-400mm/80-400mm zoom telephoto lens (which tend to have an f/5.6 or 71mm maximum aperture), you should know that the front lens elements are not actually 71mm in diameter. In the contrary, they are probably closer to 60-70mm in diameter, meaning it is impossible for the entrance pupil to appear as large as it should according to the f-number in any case. (The aperture will usually appear quite a bit smaller than the front lens element when viewed through the front of the lens.) The construction of telephoto lenses is rather complex, as they use back-focusing groups to allow the lens to be a different length than the focal length. Generally speaking though, the reason a telephoto lens has larger front lens elements is pretty much the same...a larger aperture requires a larger lens tube diameter, therefor requiring large front lens elements.
(NOTE: The last paragraph is based on actual observations with the Canon EF 100-400mm f/4.5-5.6 L series lens at 400mm. This lens has a 77mm filter thread, and the front lens element has an actual [roughly measured] diameter of about 62-63mm. At f/5.6, the 400mm focal length would dictate a 71mm entrance pupil diameter. Given that the front lens element is at most 63mm, and the entrance pupil diameter as viewed through the front of the lens is smaller than that (as you peer down a tube, due to perspective, the diameter shrinks the farther down the tube you look), it is impossible for the apparent aperture size at f/5.6 and 400mm to be as large as it "should be" if you simply calculate the number as
focal length \ f-number. Its difficult to get any kind of accurate measurement, but it appears to be about 22-25mm as viewed through the front of the lens.
In contrast, it should also be noted that in the case of the Canon 16-35mm f/2.8 L series lens, at 16mm, when the maximum aperture is viewed through the front of the lens and measured with a ruler, does appear to be just shy of 6mm, which is exactly how it should appear given an f/2.8 aperture, which should be exactly 5.71mm. This observed difference between the apparent size of the aperture of a wide-angle lens vs. a telephoto lens gives rise to my opinion that the specialized construction of a telephoto lens has an impact on the apparent aperture size, allowing it to be smaller than it "should be" according to the f-number.)