## A Mathematical Look at Focal Length and Crop Factor

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

There is a focal length on every system called the “normal”. This length is defined by the Pythagorean theorem. A^{2} + B^{2} = C^{2}. 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

24^{2}=576, 36^{2}=1296

576+1296=1872

√ 1876= a normal of **43mm**

**Canon T2i**

15mm * 22.5mm sensor size

15^{2}=225, 22.5^{2}=506

225+506=731

√ 731= a normal of **27mm**

**iPhone 5**

4.54mm * 3.42 mm sensor size

4.54^^{2}=20.6, 3.42^{2}=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).

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’

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.

Sigma’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

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:

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

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.

The 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

What’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･)