PetaPixel

A Mathematical Look at Focal Length and Crop Factor

focallengthmath

I’m hoping this post can explain a lot of the confusion beginning photographers have about focal length and crop factors. Some understanding of basic geometry is required for you to fully grasp this post. Also note that I’ll be rounding the math decently well. Just run through the calculations if you want exact answers.

The ‘Normal’ Focal Length

diagonal

There is a focal length on every system called the “normal”. This length is defined by the Pythagorean theorem. A2 + B2 = C2. The length of the two sides of the sensor are A and B. The “normal” is the length of the diagonal.

Nikon D800
24mm * 36mm sensor size
242=576, 362=1296
576+1296=1872
 1876= a normal of 43mm

Canon T2i
15mm * 22.5mm sensor size
152=225, 22.52=506
225+506=731
 731= a normal of 27mm

iPhone 5
4.54mm * 3.42 mm sensor size
4.54^2=20.6, 3.422=11.7
20.6+11.7=32.3
 32.3= a normal of 5.7mm

Calculating the Field of View

Now you have the normal length for each of this cameras. What do you do with it? You calculate their field of view.

a=2arctan(d/(2*FL))

In other words, angle of view (a) is 2 multiplied by the inverse tangent of your sensor size (either vertical, horizontal, or diagonal) divided by twice your given focal length (FL).

angleofviewill

We have a normal length for each camera, so let’s start there.

“BUT MISTER! What about my mount? I’m using full frame EF glass on my crop Canon T2i!” you might say, but hold on. The formula didn’t ask for that information yet.

Nikon D800 (horizontal) angle of view for a 43mm lens: 2*arctan(36/(43*2))
Canon T2i (horizontal) angle of view for a 27mm lens: 2*arctan(22.5/(27*2))
iPhone 5 (long side) angle of view for a 5.7mm lens: 2*arctan(4.54/(5.7*2))

For those without fancy calculators, just ignore the 2*arctan part. For those with fancy calculators, arctan is the same as tan-1. Run the calculations yourself and see what you get. No peeking because I’m not posting the answers.

The Concept of ‘Normal Lenses’

normallenses

Notice how all of the cameras produce nearly the exact same result? It’s as if they’re only a bit off because I rounded!

These cameras, with significantly different focal lengths, are producing an image with the exact same framing. I lied about not posting the answers: it’s approximately 45 degrees (0.4 for those just doing division). That’s crazy, right? It’s because the normal is a special focal length defined by the sensor size. Neither wide nor narrow, short nor long. These lengths just feel normal.

sigma3014Sigma’s 30mm f/1.4 was created because it’s close to the APS-C sensor’s normal focal length. Companies created 35mm and 50mm lenses for a long time because these lengths are a bit wider and narrower than 35mm film’s normal length. This is the reason the 50mm lens is so popular and inexpensive. I’ll cover these lenses later.

So now you understand the normal length. It’s that silly geometric thing that defines what looks normal for any different sensor size. (Homework assignment: What’s the normal length of a Hasselblad H4D-60: 40.2mm × 53.7mm? Comment with your answer.)

Practical Applications

Click

Let’s put the normal into practice. Let’s say we want to take a portrait of our friend Sally. You fill the frame with his head from ear to ear. Now Sally’s friend John wants to join, so you need a lens twice as wide to fit John’s big girly head in. Running back to your calculator, you punch in numbers!

The horizontal angle of the normal is 45.18 degrees. Twice as wide means twice as many degrees. 45.18*2=90.36, and now you run the calculation in reverse for your Nikon D800. 90.36 = 2*arctan(36/2X), so X=17.88.

A 17.88mm lens is exactly twice as wide as your 43mm lens. This works on all cameras; check for yourself.

Want something twice as narrow? Just divide your angle of view by 2. Three times as wide? Multiple that angle by 3. What matters is your angle of view. What doesn’t matter is your focal length. Your focal length is only one part of the formula that provides real images.

Let’s compare two cameras:

Canon T2i vs. Hasselblad H4D-60, 15*22.5 vs. 40.2*53.7, both using an 80mm lens:

faceoff

Canon T2i: a=2arctan(22.5/(2*80))=16 degrees
Hasselblad H4D-60: a=2arctan(53.7/(2*80))=37.1 degrees

The exact same 80mm is more than twice as wide on the Hasselblad than it is on the Canon. Keep in mind we haven’t talked about mounts yet. They’re not affecting any of our numbers.

Full Frame vs. Crop Sensor Lenses

efs

Time to explain why some lenses work on some cameras and not others (EF vs EF-S). Let’s talk circles of light. Your sensor is 24mm by 36mm if you shoot 135 format (I don’t like the term “full frame”).

Go find a roll of duct tape or painters tape. The inside diameter should be around 2-3 inches. Now go find a roll of toilet paper. The inside diameter should be 1-2″. Grab a flashlight. Hold the tape or toilet paper roll about an inch off the surface of your desk. Shine the light straight in. Notice how each object creates a circle of light.

circlelightThe tape is clearly a bigger circle than the toilet paper dowel. The size of these doesn’t change as you move the flashlight in and out (until you get too close to the front of your “lens”).

That’s what camera sensors use. Your lenses all create various sizes of light circles. They make these circles at a specific distance. This is known as the flange distance. If you hold your lenses at the proper distance, they will make an image circle just big enough to fit your sensor inside.

Canon EF 135mm f/2L? It’ll create a 36mm circle of light when held 44mm from a surface. A Hasselblad has a larger sensor to use, so it uses a larger lens. The tape roll instead of the toilet paper. Your Micro 4/3rds camera uses an even smaller lens because it requires only an 18mm circle.

So now you know why a M4/3 lens won’t work well on an APS-C sensor camera. Or why there exist “crop only” lenses such as the kit 18-55mm lenses. These lenses just won’t fit a larger camera. Because their image circle is smaller, the manufacturer can be more creative with the range. Sigma has created an f/1.8 zoom. They could do this because it only needed to cover an APS-C frame.

How Lens Mounts Factor In

canonmountWhat’s the other reason? Mounts. Your DSLR has a mirror. If your lens was closer than the flange distance it’d hit the mirror. Mirrorless lenses are made to be closer to the sensor. Same goes for EF-S mount. The EF-S mount was created to allow Canon to make lenses that fit closer to the sensor because they didn’t have to worry about hitting the mirror.

There is no such thing as “crop glass”. (Bubble has been burst.) There is such a thing as a lens that sits too close to the mirror and isn’t physically large enough to cover the entire sensor. Switch sensor and you switch angle of view, but the mount or crop factor matter ZERO to your angle of view assuming of course that your image circle will cover the sensor.

Let’s run the numbers just to be sure:

Canon 5D Mark III with EF 50mm lens. Angle of view = 2arctan(36/(50*2))
Canon 5D Mark III with EF-S 50mm lens. Angle of view = 2arctan(36/(50*2))

Canon T2i with EF 50mm lens. Angle of view = 2arctan(22.5/(50*2))
Canon T2i with EF-S 18-55mm lens set to 50mm. Angle of view = 2arctan(22.5/(50*2))

Yep. Sensor size matters but mount does not.

TL;DR

Forget mounts and crop factors. Just calculate the angles of view.


About the author: Carlton Bassett is a photographer in Raleigh, NC, who primarily shoots portraits and weddings. He enjoys wildlife photography in his spare time. This article originally appeared here.


Image credit: Determining Lens Focal Length by dvanzuijlekom, sensor photo by Filya1, angle of view illustration by Moxfyre, 32 Normal Lenses Test 1977 by Nesster, Click! by JoséMa Orsini, Circle by Sarah_Ackerman, Canon EF-S 18-55mm (F3.5-5.6). by MIKI Yoshihito (´・ω・)


 
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  • Jonathan Maniago

    I wish that camera and lens companies had just used degrees/field of view instead of focal length. I think it’s more intuitive and more relevant for photographers (like aperture often being expressed as a ratio rather than physical length).

  • http://www.edphoto.co.za/ ..

    Awesome article, it explains so much about a topic that I considered myself knoweledgable in, but obviously not!! I am definitely going to use these calculations in some or other way in my photography.

  • Nado

    Which angle of view should I calculate? the diagonal? the horizontal? or the vertical? I wish lens manufacturers could tell us the angle of view instead of the focal length

  • D.G. Brown

    My apologies, but that’s silly, especially given what you just read. The sensor/film size is part of the equation, which means that you simply cannot calculate the angle of view by only knowing the focal length of a given lens. An EF lens is going to have different fields of view depending on which sensor is attached. This isn’t a new problem either, as camera systems have a history of switching out how big the film or sensor can be.

    In addition, field of view is just one of a group of attributes you have to take into account. How aperture plays in, for instance, requires both knowing focal length and sensor size (for calculating exposure and depth of field).

    Keep in mind that angle of view is, in many ways, just a number. Yes, you can imagine a protractor and see how much an area you are seeing, but that’s a process of having learned such a thing (and hopefully we all learned that in geometry class). However, photographers have just taken a different approach, which is simply taking your favorite sensor/film size, learning which focal lengths create which fields of view (not as a number, but as a perspective in their head), and then learning a coefficient for other sizes if needed.

    As a note, if you shoot Canon and have to go back and forth between crop and FF a lot, you’ll have an easier time driving in Canada ;-)

  • tekmonkey

    I have a Canon T2i. I would like to purchase another prime lens, my current one is a 50mm “nifty fifty”. It works great but I have to move away from my subjects most of the time. Since the T2i is not a full frame camera, what would be a good lens to buy where I don’t have to move away from my subjects so much, meaning the lens takes pictures close to what my eyes see?

  • D.G. Brown

    Canon’s crop factor is 1.6x, so if you want the feel of a 50mm, you’d want a something close to 31mm, which is right between the 28mm and 35mm lenses. I had the 28mm f/1.8 and it’s pretty nice, though you’d be able to get the 35mm f/2 for cheaper. There’s also the super-cheap pancake 40mm, though that’ll still be pretty narrow.

  • Nigel Donald

    I dont agree with the above! If you replace “focal length” with “field of view” I’ll agree with it. Focal length is all in the lens & sensor size makes no difference otherwise with a tiny sensor like an iphone behind a lens you would be able to read the name on the side of a container ship on the horizon! Crop factor is not zoom.

  • Jonathan Maniago

    Yes, I’m well aware of the mathematics involved and I have no issues
    with juggling with numbers. My point is that angle of view is still a
    more intuitive measurement that even beginners could understand,
    regardless of whether they know the sensor size or not.

    Of course, the
    focal length would still be given by manufacturers, and those using
    adapters and/or those switching between different lens lines (ex. EF and
    EF-S) could utilize that for their calculations.

  • D.G. Brown

    You’re missing the point, however. Even if angle of view is intuitive, it’s not always possible to know it for a given lens.

    Take for instance, any EF lens. The same lens on a 7D (1.6x crop), a 1D mark V (1.3x crop), and a 5D mark III (FF) has very different angles of view. For camera systems with varied sensor sizes, it’s not possible to have one value.

  • Jonathan Maniago

    A table of values then. For example, the manual that comes with my Samyang fisheye provides the focal length -AND- a table for APS-C (1.5 and 1.6 crop) and Four Thirds. At the very least they should make those numbers more available for the common camera lines compatible with the lenses they manufacture because focal length on its own is a meaningless value from the user’s point of view.

    Besides, there have been lens comparisons on the Net which have noted that field of view may have slight variations even for lenses with supposedly identical focal lengths. Our calculations using focal length could be decent ballpark figures at best.

  • Graf Almassy

    Usually you can read the field of view parameter (in degrees) on your lens specs page. However this is useless, because the FOV calculated via the reference sensor or film size (35mm film or full frame sensor). This is the same method as the equivalent focal lenght calculating.

  • http://csabaszilvasi.com/ Csaba

    I know it is nowhere near scientific, but in Nikon’s lens simulator you can check the angle of view. You can select sensor sizes (DX, FX) and lenses from 10-600mm.

    http://www.europe-nikon.com/en_GB/learn_explore/simulator.page

  • ehemax

    Crop factor IS zoom factor limited to the resolution of the lens. A lens x mounted on a FF and a APS-C system will give different results (APS-C “zoomed-in”). If the lens x has a high enough resolution in the center area (means it can resolve opticaly 18+ MP in the APS-C crop of the lens) then you get a zoomed in image, also if the optical effect is just an “Illusion”.

  • Dale Dale

    This has been one of the best reads of the week.

  • Greg

    Dan, I agree with you. Coming from many years in shooting film I would choose a lens many times to create the depth of field I was after. Of course we all know that focal length, f-stop and how close you are to the subject will effect the overall bokeh you wish to create. Correct me if I am wrong, but crop factor is what I did in the darkroom with an enlarger. I might shoot a portrait with an 85 m.m. lens and then crop it tighter for the actual print, however it did not change the depth of field to a 135 m.m. lens because of the cropping done in the enlarger. I believe young photographers would be better off learning how to use their lenses and lighting to create effects rather than be worried about crop factor!

  • AGS

    Great article. I’ve got a question tough. Do the lenses made for dx cameras “simulate” the look of what that focal length would be on an fx lens with an fx camera? In other words, if I have a 50mm dx lens mounted on a dx camera… is it equivalent to an fx 50 mm lens mounted on an fx camera? Or do I need to calculate the crop factor? Which would give, for that same dx configuration, a real focal length of around 80 mm. I hope I made myself clear enough. Thanks!

  • Oj0

    I also wondered this when I swapped to digital. Turns out you still need to calculate the crop factor.