If you care about cameras you probably like learning about interesting new (and old) lenses. They’re described by two numbers: How wide they open (also: aperture, brightness, speed), and how long they are (also: focal length). That first number is generally comparable across lenses: Lower is better. The aperture unit, “F-stop”, is hardly intuitive, but whatever. Focal lengths are hard to compare, because how much the lens sees depends on how big the sensor behind it is, and there are lots of different sensor sizes.

Generally, focal lengths are expressed as “35mm equivalent”, meaning “If we stuck this lens in front of a sensor the size of traditional photographic film, which is the same size used in modern expensive digital cameras, here’s what the focal length would be.”

Which is dumb and irritating, because that’s not the number printed on the side of the lens, and anyhow, the number itself doesn’t really mean anything. So I was thinking of some uniform ways to express what focal length means.

Focal angle · What you really care about is how much of the scene in front of you the lens can take in. There’s a nice smooth trade-off: The less it takes in, the more magnification you get, and the further away you can see. The terminology is broadly “wide-angle”, “normal”, and “telephoto”.

The math isn’t hard. If your sensor is W mm wide and your lens is L mm long, your focal angle is:

Angle = 2 × arctan(W / 2×L))

Of course, that’s in radians, which most of us don’t think in. So the most obvious way to present them to humans would be as degrees, with extreme wide-angle lenses pushing toward 180° and huge telephotos getting down into the single digits.

I suppose that’s OK, but if we assume that the maximum practical field of view is 180°, why not just give the answer as a percentage of that? It turns out that if you think of 75% as 0.75, that is also a direct measurement of the angle, if you think the natural unit to describe angles is “π radians” (well, d’oh, of course it is).

Without further ado, here’s how that would work.

Mobile
iPhone 7 Plus (4.8mm sensor, 3.99mm) 62.1° 34.5%
Nexus 6P (6.17mm sensor, 4.67mm) 66.9° 37.2%
Micro Four Thirds (17.3mm sensor)
Olympus OM-D E-M1, Olympus M.Zuiko 8mm F1.8 94.5° 52.5%
Olympus OM-D E-M1, Leica Summilux 25mm F1.4 38.2° 21.2%
Olympus OM-D E-M1, M.Zuiko 300mm F1.8 3.3° 1.8%
APS-C (23.6mm sensor)
Fujifilm X-T1, XF 10-24 F4 at 10 99.4° 55.2%
Fujifilm X-T1, XF 35mm F1.4 37.3° 20.7%
Fujifilm X-T1, XF 100-400 F4.5-5.6 at 100 13.5° 7.5%
Fujifilm X-T1, XF 100-400 F4.5-5.6 at 400 3.4° 1.9%
Full frame (36mm sensor)
Sony a7 II, Rokinon 8mm F3.5 131.8° 73.2%
Leica M10, Summilux 50mm F1.4 39.6° 22.0%
Canon 1DX, EF 200-400mm F4 at 200 10.3° 5.7%
Canon 1DX, EF 400mm F2.8 5.2° 2.9%
Medium format (645) (43.8mm sensor)
Pentax 645z, FA645 35mm F3.5 64.1° 35.6%
Pentax 645z, FA645 120mm F4 20.7° 11.5%

Problem? · It’s a little weird that as lenses get longer, the numbers get smaller. I think marketers like thumping their chests about shipping a big-ass 300mm lens, so they might be unhappy.

I actually considered using inverse angles or percentages to counteract that, but it looked lame.

Notes on the cameras · Finding the specs on the phone sensors was a little tough; in each case, the focal length is what they report in the EXIF data.

The Micro Four Thirds combined with the Olympus 300mm lens, is actually the biggest telephoto you can get. But both Oly and Fuji ship tele-extenders if you want to go completely insane.

The Leica is sort of the canonical example of a “normal” lens. The Sony/Rokinon combo is the widest off-the-shelf combo that exists, as far as I know. The Canon combos are those big light-grey lenses you see clustered around the edges of the soccer pitch or football field on TV.

There actually are larger sensor formats than 645, but very rare in the field.



Contributions

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From: Bram (Sep 17 2016, at 13:18)

I love long lenses, I never owned one, but if I did, I would replicate this:

https://vimeo.com/56298775

Also I would like to note that if you are willing to sacrifice resolution, you can always fake a long lens with cropping!

If you crop the centre 400x300 pixels from a 20Mpixel sensor, you get a very narrow angle of view. You also sacrifice some sharpness of course, but mainly resolution.

And let's face it: most images end up on websites, where 20Mpixels are wasted anyway, on 1920x1080 screens.

Cropping... the poor man's zoom!

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From: Anthony Coates (Sep 17 2016, at 13:38)

I looked at a 75-300mm zoom once for my Olympus OM-D EM-5, but that's equivalent to a 600mm on a 35mm camera, and it was so impossible to hold steady at full zoom that I decided it wasn't for me - I was rarely going to have the camera bolted to a tripod to allow me to make use of it.

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From: EricH (Sep 17 2016, at 15:57)

I too have long thought that lenses should be described by their angle of view, instead of their focal length.

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From: Janne (Sep 17 2016, at 16:30)

Problem: the angle of view is not easy to relate to what you actually get. It's really no more intuitive than length, unless you happen to be a surveyor or something. I could not show what 10 degrees cover; I wouldn't even get close. It's really no less arbitrary than focal length.

Problem: the same 300mm lens is sold for both aps and full frame pentax cameras. How should they describe the reach in their materials? Two numbers? Then a zoom will need four. And it doesn't take into account using it on their small-sensor camera with the official adapter.

Problem: sensors aren't all the same proportions. You really need to give two angles to describe the view, or people will continue to mentally adjust the actual coverage for their particular camera.

I'm not saying giving angles is worse than focal length. It's not. But I am saying it is not significantly better; certainly not enough that the industry would switch.

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From: Rick Levine (Sep 17 2016, at 17:44)

I'm trying to warp my (admittedly ancient) sense of lenses into the math, and it mostly aligns. However, I don't really use the "35mm equivalent" in that way. It's mostly a check on how it compares to a working lens - 50mm is about normal vision, standing in front of a scene, up around 105mm is a nice portrait/candid lens, and when it's heading towards 30mm, it's noticeably wide. If you can look through/at a viewfinder at a scene, and see about the same thing when you look through your other eye or lift your head to look at the scene irl, then that's about a "normal" lens equivalent. Viewfinder magnification might also contribute some size dissonance, as there's a range of magnifications in cameras.

These days, with a phone camera, I no longer think about lens choices - I shoot what I have. Zooming or center crop isn't always a useful choice, as I use still images for pan & scan in video, and I want the dots. Getting closer to the subject works. :)

Most of the "normal" lens numbers relate back to sensor, uh, film stock diagonal size, which still seems to be a useful way to look at lenses. I did a quick list of the formats I've worked, and the lenses :

35mm (24 x 36mm)

1.70in/43.27mm diagonal

lenses: 50mm normal, 105mm portrait. Less than 50mm always seems to be slightly wide. Viewfinder mag?

2 1/4 x 2 1/4in (6x6)

3.18in/80.82mm diagonal

lenses: 80mm normal. The 120/Rollei/Hasselblad medium format "standard" lenses always seemed slightly wide to me, as well, and I opted for longer lenses when I could, if the camera took them.

2 1/4 x 3 1/4in (6x9)

3.95in/100.40mm diagonal

lenses: 100mm normal. More lens options on mamiya/horseman/alpa cameras

3 1/4 x 4 1/4in

5.35in/135.90mm diagonal

lenses: 127mm normal. A 127mm Ektar was a cheap standard lens; the weight and added expense of covering the format with a longer lens made it a marketing choice, not a practical one.

4 x 5in

6.40in/162.64mm diagonal

lenses: ~160mm/6 1/2" is the diagonal but shooting with a 210mm/8" always felt more "normal." Heading more towards a candid/portrait lens, I guess.

5 x 7in

8.60in/218.50mm diagonal

lenses: longer here, too - 240mm/9.5-10" is more comfortable. Longer lenses in view cameras were always a cost issue. Having enough coverage to use swing/tilt was a nice luxury.

8 x 10in

12.81in/325.28mm diagonal

lenses: 12-14" is comfortable, but I tended towards slightly longer than the diagonal, 325mm. (And yeah, I think about 5x7 and 8x10 lenses in inches, 4x5 sometimes in metric, and smaller down in metric. Hmpf.)

Rick

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From: Michael Zajac (Sep 18 2016, at 08:07)

It’s usual to talk about the diagonal fov, because it represents the extent of a lens’s projection circle, and the maximum fov captured by any sensor, regardless of its proportions (square, HD, multi-aspect sensor, whatever).

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From: foresmac (Sep 20 2016, at 08:47)

Yeah, I was just about to say you need the diagonal for W in your formula. That’s how angle of view has traditonally be calculated, since 35mm, medium format, and large format have traditionally had different proportions. The for digital cameras with selectable proportions, how do you describe the FOV?

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From: CraigM (Sep 21 2016, at 08:55)

In case you hadn't found them, Apple do provide some FOV, etc details form their lenses/sensors in recent models. You can find them in the tables (3.3 and above) on the Apple Developer site at https://developer.apple.com/library/content/documentation/DeviceInformation/Reference/iOSDeviceCompatibility/Cameras/Cameras.html#//apple_ref/doc/uid/TP40013599-CH107-SW1

HTH

Craig

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