On Angle of View, Focal Length & Crop Factor

It is fair to say that, for many photographers, their first foray into the world of DSLR cameras will include a copy of the ubiquitous 18-55 mm kit lens. Manufacturers label it a “versatile lens,” which effectively means that its focal length is neither particularly long nor particularly short and, as a consequence, it is compact and easy to carry – a very rational choice for a first lens.

Canon EF-S 18-55mm f/3.5-5.6 IS II kit lens.

However, what almost always follows the acquisition of the kit lens is a photographer’s first step down the slippery slope of camera gear. Almost without exception, the very next question a would-be Ansel Adams will ask is, Which long lens should I buy? If you are at all like me, this will translate into hours spent analysing price vs focal length. Should I spend the extra money and get a 300 mm rather than a 200 mm? The thing that makes this particular line of enquiry so satisfying is that, unlike other photographic purchases, the difference from one long lens to another appears so tangible. 300 mm simply feels like so much more than 200 mm. There is a true sense that for twice the money you get twice the amount of ‘stuff’.

Whether or not the association of longer focal lengths with increased value is valid is a matter for another day, but one of the side effects from that thinking is that it often leads us to question the comparatively small numerical changes that exist at the wide-angle end of the lens spectrum. For example, adding a 10-18 mm wide-angle lens to your arsenal – a lens with a focal length range of just 8 mm – will cost almost the same as adding a 55-250 mm lens and those numbers can be somewhat difficult to digest.

I suggest that a lot of the difficulty that accompanies comprehending focal ranges comes as a result of an important lens attribute – the angle of view – being relegated to the photography jargon backbenches. Buy a lens today, and its barrel will almost certainly come marked with a focal length – 18-55 mm in the case of the kit lens. To establish what the angle of view is, though, one would have to either whip out that old scientific calculator for a little trigonometry or, alternatively, locate the manufacturer's detailed specifications.

There is a commonly held belief which indicates that for every equation you add to your writing, the number of readers halves and, thus, I am going to avoid including an equation of my own at this point. Instead, you will simply have to believe me when I say that focal length and angle of view are inextricably linked, and if one were to graph the product of the equation, the result would look something like this.

For those with a taste for trig, the details of the Angle of View equation used to produce this graph can be found here. 

The diagram clearly illustrates how the angle of view ramps up as the focal length approaches zero. A change in focal length from 10 mm to 20 mm translates into a 20° change in angle of view while a change in focal length from 110 mm to 120 mm has little impact on that angle at all. In fact, I would go as far as to say that when talking about wide-angle lenses, manufacturers would be better off stamping the angle of view rather than the focal length onto the barrel of the lens – the numbers just make more sense.

Taking into consideration everything that I have said thus far, it might be easy to dismiss the topic of the focal length and angle of view as merely a conceptual one. Whether a manufacturer says that their lens has a focal length of 10 mm or, alternatively, that it has a 130° angle of view, seems rather academic except that there is a second variable in the angle of view equation that elevates its status – the crop factor.

As I discussed last week, the crop factor – or focal length multiplier – has the effect of making a subject appear larger in an image. If we recognise that the physical focal length of a given lens is the same whether it is attached to a crop or a full-frame body, then we can prove mathematically that the only difference between the images produced by each body when shooting the same scene is the angle of view. For instance, a 50 mm lens on a crop sensor body will yield a 30° angle of view while on a full-frame sensor it will produce a 46° angle of view.

The point is better made when one plots the angle of view against focal length for an APS-C and full-frame camera and, importantly, tracks the difference in angle of view between the two. Doing so graphically illustrates how the difference – or delta – in angle of view increases as the focal length approaches zero, with the greatest delta occurring at a focal length of 17 mm.

The significance of this to the landscape photographer cannot be overstated. Wide-angle lenses are a discipline staple and it is at the wide end that the influence of the crop factor is at its biggest. A 16 mm lens – which among professional lenses constitutes a bit of a wide-angle standard – will on a full-frame body produce a 107° angle of view. By contrast, the same lens on a crop sensor body will only produce an 80° angle of view, a notable 27° less.

These results could, no doubt, lead someone to conclude that full-frame cameras are better for landscape photography, and while the evidence suggests that they are useful for getting wider angles, the question of whether that 27° determines the difference between a good photo and a great photo is a rather philosophical one.

If you enjoyed this article, then please give it a like or leave a comment below!