How does crop factor change focal length and aperture?

Err... It doesn't!

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Before digital photography became a thing no one ever really talked about "crop factor". The most popular film format by far was 35mm. Both larger and smaller formats existed but the vast majority of people used 35mm film in their SLRs, rangefinders, and point and shoot cameras.

When digital cameras started becoming popular the sensors were smaller than 35mm, much smaller. The term "full frame" became popular when 35mm digital sensors started becoming more common. However, 35mm is no longer the dominant format. Larger sensors are expensive to manufacture which obviously pushes up the price of the camera and so smaller formats have become the norm.

It's rather confusing for a lot of people that the same focal length lens produces a completely different field of view (FoV) on a camera with a smaller sensor. A 50mm lens on a full frame camera, for example, is a standard lens. If you put that same lens on a Micro 4/3 camera it is a short telephoto - it's confusing.

To remove some of this confusion it became convenient to think of focal length in terms of its 35mm (full frame) equivalent.

Let's clear up a couple of common misconceptions:

- Crop factor does not change the focal length of a lens
- Crop factor does not change the aperture of a lens

A 50mm lens is a 50mm lens, no matter what camera it is fitted to, the same with aperture. The difference is the field of view. A smaller sensor simply records less of the image than a larger sensor. How much less is determined by the crop factor.

To calculate the crop factor you just work out (or Google) the diagonal length of the sensor and compare that measurement to the diagonal length of a 35mm sensor. A standard (normal) lens on any given camera will be similar in focal length to the diagonal measurement of the sensor (or film).

The equation to calculate the diagonal length of a rectangle is: a² + b² = c² (thanks, Pythagoras).

A full frame sensor is close enough to 36x24mm, 36² + 24² = √1872 (43mm)

An APS-C sensor is close to 24x16mm, 24² + 16² = √832 (29mm)

43mm (Full frame diagonal) divided by 29mm (APS-C) = 1.48 (round it off to give a crop factor multiplier of 1.5)

So, a 50mm lens on full frame has an equivalent FoV to a 75mm lens on APS-C (50 x 1.5 = 75)

A Micro 4/3 sensor has a diagonal length of about 21mm... 43/21=2.04 (rounded to 2.0 crop factor).

A 50mm lens on full frame has an equivalent FoV to a 100mm lens on M4/3 (50 x 2 = 100).

So, focal length multiplied by crop factor multiplier = 35mm equivalent focal length.

The same principle applies if the format is larger than full frame, i.e. medium format, large format. The crop factor multiplier will just be a number smaller than one.

The following table shows some common formats, their crop factor multiplier, and an equivalent focal length for a standard lens:

Size mm	Format	Diagonal	Crop Factor	Standard Lens
100 x 120	Large format film (4x5)	156mm	0.3	165mm
60 x 70	Medium format film (6x7)	92mm	0.5	100mm
60 x 45	Medium format film (645)	75mm	0.6	80mm
44 x 33	Medium format digital	55mm	0.8	60mm
36 x 24	Full frame	43mm	1	50mm
24 x 16	APS-C	29mm	1.5	35mm
17 x 13	Micro 4/3	21mm	2.0	20mm

The above measurements are not exact, the part of the film that captures the image is always slightly smaller than the stated film size. I have only included a few film sizes as examples. Sensor sizes vary between manufacturers and there are several other sensor sizes in use apart from the ones I have included. The crop factor multiplier has been rounded to one decimal point but it doesn't need to be exact anyway... Near enough is good enough.

A standard lens is a lens that gives a "normal" field of view on any given format. The ideal standard focal length would be the same as the diagonal length of the film. The standard focal length is actually more a range than a specific number. For example, on a 4x5 camera any lens between 135mm and 180mm would be considered standard. On a full frame camera it would be between 35mm and 50mm.

Various common film and sensor sizes using the same focal length lens

The above image might make the idea of crop factor easier to visualise. As you can see, as the film/sensor size become smaller it has a kind of zooming effect, more accurately - a cropping effect.

Crop Factor and Aperture

Just as the focal length is a physical property of the lens, so too is the size of the aperture pupil. Neither focal length nor aperture changes by attaching a lens to a different format camera body.

What does change in regard to aperture is the apparent depth of field at the same F-number. Larger film/sensors produce a smaller depth of field than using the same F-number on a smaller format. This is why people go ape over the bokeh produced by larger formats, it's because of the smaller DoF.

You can use the same crop factor multiplier to find an equivalent F-number to compensate the depth of field. Don't forget though, if you change the F-number you are changing the exposure. You'll also need to compensate the shutter speed.

Examples:

A 645 camera has a crop factor of 0.5. The DoF of a full frame camera at f/4 would be the same as the 645 camera at f/8... 8 (f-stop) x 0.5 (crop factor) = 4 (equivalent f-stop).

Comparing a full frame camera's DoF to an M4/3 (crop factor 2.0) would give the same result. F/8 on the full frame would give the equivalent DoF as f/4 on the M4/3.

Vignetting, slightly exagerated

Using a lens designed for a cropped sensor camera on full frame.

As we all know, the image projected through a lens is circular in shape but the film/sensor onto which it is projected is four-sided. A lens designed for use on a full frame camera will produce an image circle that is slightly larger than the media capturing the image so, the entire frame is exposed.

A lens that is designed for use on a smaller format will be unlikely to produce an image circle large enough to cover the entire frame. This can lead to unpleasantly darkened corners and edges, an effect known as vignetting. A vignette can be a nice addition to enhance some images and draw the viewer's eye toward the centre of the frame - if added in the darkroom/post-production. A vignette caused by too small an image circle will be quite ugly.

Using a full frame lens on a cropped camera is not a problem as the image circle will be much larger than the film/sensor. My advice, if you own a crop camera and think that you may upgrade to full frame eventually, is to buy full frame lenses. They will be more expensive up front but you won't need to buy all new gear if you upgrade your camera body.

Don't get hooked up on the details

This whole concept of crop factor is nothing more than a way of comparing two things that really don't need to be compared in most circumstances. If you own an M4/3 camera do you really need to be able to calculate the equivalent focal length to a full frame system that you don't own or use? It could be of value if you were planning to change formats and you wanted to make sure that you were getting lenses that gave a similar FoV to what you are already using. I'm sure a lot of people have bought a 50mm lens for their APS-C camera thinking they were buying a standard lens only to find that it's too long. Never mind, use it for portraits.

As someone that shoots film in a variety of formats, and also occasionally shoots digital in puny little full frame, I can tell you that this is something I rarely think about. After a while it becomes instinctive, if you shoot different format cameras regularly you get to know which focal length you need for that camera in any given situation.

I often get asked questions about my larger film cameras and referencing 35mm equivalence makes it easy to explain my set up to someone that has only used 35mm or smaller.

People that share lenses between different format camera bodies, say D500 (APS-C) and D5 (full frame), might think about it more. It is interesting stuff to know but otherwise kind of pointless.