## A Tedious Explanation of the f/stop

by Matthew Cole

Our Home Page | A History of Photography: The 2000s | A Tedious Explanation of Depth of Field

Photographers set their exposure using a combination of shutter speeds and f/stops to get the correct amount of light on the film/sensor. The shutter speed regulates how long the film is exposed to light coming through the lens. The f/stop regulates how much light is allowed through the lens by varying the area of the hole the light comes through. For any given film speed (ISO) and lighting combination there is one correct amount of light to properly expose the film. This amount of light can be achieved with many different combinations of f/stops and shutter speeds.

Despite being one of the exposure controls in photography, the f/stop remains a source of confusion and mystery to many photographers, even to some who use it all the time. I find it interesting that the local camera shop has pictures under glass on the counter showing a scene using a range of focal lengths (for a good example of this, see my friend Dave Dahms' Lens Focal Length Chart), photos showing the same scene printed at different sizes and photos showing an action scene shot at different shutter speeds. All that is assumed to be of interest and comprehension to the customers. What they don't have is a set of photos showing depth of field, or a scene shot at a range of exposure combinations where the f/stop's effects are shown. Maybe it just takes too much explanation. Well, too much explanation is what this page is all about, and I'll go over the f/stop and especially its initially-confusing numbering at some length.

#### Fill That Bucket!

My favorite analogy for exposure is filling a bucket with water. A bucket is of fixed size and needs a certain amount of water to fill it, just like film or a digital sensor, which is of a set sensitivity (the ISO) and needs a certain amount of light to optimally capture an image. To fill your bucket, you can pour a small stream of water for a long time or a fast stream of water for a short time. Either way, you end up with the same amount of water. In photography, the size of the stream of the water is analogous to the f/stop, the length of time you pour is analogous to the shutter speed, and the size of the bucket is analogous to the film speed. Broadly speaking, from the bucket's point of view, it doesn't matter which combination of stream size and length of time you choose as long as the right amount of water ends up coming in. Film is the same; within limits, it is indifferent to the combination of time and amount of light as long as the right amount of light eventually arrives.

#### Shutter Speeds

Shutter speeds are a bit easier to understand, so I'll start with those. Both exposure controls run through a sequence of settings which involve doubling and halving the amount of light reaching the film. Shutter speeds are measured in seconds and fractions of a second and so the doubling and halving is fairly self-evident. One quarter second is half as long as one-half second but is twice as long as one-eighth. One second is twice as long as half a second and half as long as 2 seconds. It's pretty easy, and this works through the whole sequence of shutter speeds. On my old Nikon FE, for instance, the shutter speed sequence is:

8 seconds 4 seconds 2 seconds 1 second 1/2 second 1/4 1/8 1/15 1/30 1/60 1/125 1/250 1/500 1/1000

Each of these settings is clearly half/double the length of time of its immediate neighbours (OK, I know, 1/15th isn't exactly half the time of 1/8th and 1/125th isn't exactly half the time of 1/60th, but it's close). This doubling/halving is thus pretty simple to comprehend for this exposure control.

F/Stops

f/stops are a bit more confusing because the numbers appear so arbitrary. This is the standard sequence of f/stops from f/1.4 to f/22. Although it may not seem intuitive at first, in this sequence the f/1.4 setting lets in the most light while the f/22 setting lets in the least. Also, each of these f/stops has precisely the same halving/doubling relationship as the shutter speed sequence.

1.4 2.0 2.8 4 5.6 8 11 16 22

On the face of it, going from f/4 to f/5.6 doesn't sound like halving the amount of light. What's more, 5.6 is a larger number and sounds like it ought to be more light, not less. Neither does f/4 to f/2.8 sound like doubling the amount of light. In fact, each of the numbers in this sequence is a halving/doubling of the amount of light from its immediate neighbours, just like the shutter speed settings are. Not only that, but it makes sense, as I shall show below.

The reason that both the halving and doubling and the smaller numbers mean more light things make sense is that the f/stop is a ratio. The ratio is between the diameter of the aperture in the lens and the focal length of the lens. The focal length is generally measured in millimeters, so we'll stick with those as our unit of measure. On a 50mm lens, f/2 is saying that the diameter of the aperture is 25mm. The ratio is: 50/25 = 2. That seems pretty straightforward. A good question might be, what is the area of that aperture? Well, the aperture is usually a set of five to fifteen blades which form a roughly circular hole, so we'll use the formula for the area of a circle, which as I'm sure you'll recall is π * radius2. For π I'll use 3.14159265. On our 50mm lens, the aperture at f/2 has a diameter of 25mm which is a radius of 12.5mm. The area of the aperture is thus π X 12.52, or 3.14159265 X 156.25, or 490.9 square millimetres.

This fact by itself isn't all that useful. It is useful in relation to the adjacent f/stops. What is the area of the aperture at f/2.8? Well, because the f/stop is a ratio of the focal length to diameter, our 50mm lens at f/2.8 would have a diameter of 50/2.8 = 17.86mm. Remember, we have to divide that by 2 to get the radius of 8.93mm, so the area of the circle thus formed would be π X 8.932, or 250.5 square mm. Rounding off a bit, that's about 250 sq. mm at f/2.8 and 500 sq. mm at f/2, a double/half relationship. Aha! So that's it! The area of the hole doubles and halves, it's just represented by a ratio on the lens! No wonder it's so darn confusing.

Here's a table of the aperture areas for the common f/stops for a 50mm lens:

f/stop | Diameter of aperture (mm) | Radius of aperture (mm) | Area of Aperture (sq. mm)
f/1.0 | 50.0 | 25.0 | 1,963
f/1.4 | 35.7 | 17.9 | 1,002
f/2.0 | 25.0 | 12.5 | 491
f/2.8 | 17.9 | 8.9 | 250
f/4 | 12.5 | 6.3 | 123
f/5.6 | 8.9 | 4.5 | 63
f/8 | 6.3 | 3.1 | 31
f/11 | 4.5 | 2.3 | 16
f/16 | 3.1 | 1.6 | 8
f/22 | 2.3 | 1.1 | 4
(As shown on lens) | (50mm divided by f/stop) | (1/2 the diameter) | (pi X the radius squared)

If you look down the column of figures on the right, you can see the (more or less) doubling/halving going on up and down the column. You can see also how the big numbers make for smaller areas since the f/stop number is being divided into the focal length, then halved, then squared, then multiplied by π. It's no wonder this seems obscure.

Why not just call for the area of the aperture directly? A couple of reasons. First of all, if you have a 50mm lens on and say "I shot this with my 50mm at 1/125th of a second and an aperture area of 63 square millimeters" you will impart correct and exact information that precisely zero people will understand. It's much easier to say "I shot this at 1/125th at f/5.6". Also, 63 square millimeters is f/5.6 only with a 50mm lens; if your lens is a 35mm, or an 85, or a 300, the ratio changes and the exposure is different. That 63 sq. mm is about f/4 on the 35mm, f/9.5 on the 85mm and f/32 on the 300. Knowing only the area of the aperture requires also knowing the length of the lens to be informative as to the amount of light coming through the lens. The f/stop figure incorporates both of these in one useful if initially confusing measure and the lens length is immaterial. It's shorthand, in effect. When you say f/8, you mean for this focal length (the f?), give me an aperture whose area is such that diameter of the resulting circle goes eight times into my focal length. Fortunately, the lens makers figure out all these things and just mark the f/stops on the lens for us. They're doing us a big favor.

#### So, does that mean f/8 is the same brightness of illumination on the film/sensor regardless of focal length of the lens?

Yes. That's why they're so handy! It's also why handheld light meters work--they have no idea what film size or camera body or lens you're using, but they can tell you that 1/125th at f/8 is the correct exposure (or at least one of a whole host of equivalent exposures, we'll get to that later). It doesn't matter if you set that f/8 on a 20mm or a 400mm, or if the camera is from 1954 or from last July.

#### Got it, I think. What about other f/stop terms?

#### When people talk about a fast lens, what does that mean?

Lenses are referred to by their maximum aperture (that's the biggest hole, the smaller number). Thus, Nikon made (at least) four 28mm lenses at one point, a 28 f/1.4, a 28 f/2.0, a 28 f/2.8 and a 28 f/3.5. All four of these lenses had f/4, f/5.6, and so on up to f/16; they were distinguished by the maximum amount of light they could let in. The 28mm f/3.5, one of which I own, when set to its maximum aperture of f/3.5, lets in one third less light that the 28 f/2.8. The 28 f/2.8, in turn, at its maximum aperture, lets in only half the light of the 28 f/2.0 at its maximum aperture. The 28 f/2.0 lets in only half the light of the 28 f/1.4 at its maximum aperture. Lenses which have wide maximum apertures and let in lots of light are called fast lenses. Lenses which let in comparatively less light at their maximum apertures are called slow lenses. The 28 f/1.4 is a very fast 28mm; 28 f/2.0 would be a fast lens; the 28 f/2.8 would be sort of regular, for which there isn't really a name; the 28 f/3.5 would be kind of slow. That 28 f/1.4 sold for enough new that Nikon didn't make many; once discontinued, their price in the used market went up! You can find them on eBay from time to time going for $2,500 or more. Nikon in 2010 introduced a 24 f/1.4 lens, just a bit wider, so you don't have to buy used but you still have to pay around $2,500.

#### Why wouldn't you always use a fast lens?

Weight and expense. To get those larger diameter apertures means you need larger pieces of glass mounted in correspondingly larger lens barrels. They're harder to manufacture, the lens barrel keeps getting heavier to hold all that glass in alignment so it all gets weighty in a hurry, they're harder to autofocus and they're more challenging optical designs. In the rangefinder world there start to be concerns about the accuracy of the focusing since depth of field gets to be very limited at large apertures. Also, there have been very fast lenses made which have the reputation of being really nice wide open but kind of doggy stopped down. If you normally do not use the fast lens at its widest settings, if you are mostly at, say, f/8, then you are carrying around a heavy and expensive optic which may be underperforming its cheaper brethren stopped down.

The size penalty is really obvious in the long lenses. The weight balloons and the cost skyrockets. For instance, I used to own a Nikon 300mm f/4.5 ED-IF lens. The IF is internal focus, the ED had to do with the Extra-low Dispersion glass used. It was a sweet lens, 300mm in length, with silky smooth (manual) focusing and weighed in at 2 lbs. 2.9 oz. (989g). If I stepped up to the 300 f/2.8 lens the weight went to 5 lbs. 8 oz (2500g). Not fast enough? How about Nikon's 300 f/2? It weighed in at 15 lbs. 6.9 oz. (7000g). The 300 f/2 picks up 2 1/3 stops over the 300 f/4.5 I owned, but it takes an eminently hand-holdable telephoto that fits in the camera bag and makes it into an unwieldy unit needing a tripod, requiring its own suitcase and weighing seven times as much. The f/2s still sell used in the $20,000+ range.

You also see the size in the zoom lenses. Most modern consumer zooms are handy, light and slow. They are also have a variable maximum f/stop across their zoom range. The professional level lenses tend to be fixed f/stop (the Nikon 70-200 f/2.8, for instance, is f/2.8 across the whole range, is a standard pro lens, but it weighs a lot and costs $2,500). In everything there are tradeoffs.

Even on shorter lenses the difference is noticeable; my brother-in-law's Nikon 55 f/1.2 is much heavier than my 50 f/1.8. His viewfinder sure is bright and that last fraction of a stop can be handy sometimes, but the camera weighs a lot on the neckstrap and you start to question its value if you're shooting at f/11 anyway. If you do decide you want the fastest possible lenses, go buy yourself a Leica M6 (film) or M9 (digital), for which you can buy a 50mm f/1.0 lens and a 75mm f/1.4. And before you think that it's modern technology that allows these wonders, recall that Canon made a 50mm f/0.95 for their rangefinder cameras back in the 1950s, and still make an 85 f/1.2 for their SLRs.

#### My Zoom lens says it's f/4.5 to 5.6. What's that mean?

This relates to the size versus lens speed issue. I own a Nikon 24-85 f/2.8-4 zoom lens, as an example. It's not exactly small and light, but it tries. If you think about it, the size of the elements needed to get f/2.8 at 24mm is much smaller than the size needed to get f/2.8 at 85mm. To keep lens sizes and costs down, the lens manufacturer accepts that the lens wil be faster at the wide end of its focal length range and slower at the long end. As a result, you can open up to f/2.8 at the 24mm end of the range, but only to f/4 at the 85mm end. This is particularly noticeable in small lenses that do, say, 70-300mm where at the 300mm end the maximum f/stop is f/5.6 or worse. It also shows up in the compact point and shoot superzooms, which can have unspeakably slow long focal lengths. You'd better have a sunny day!

#### I hear stops referred to a lot. Are these always f/stops?

No. A source of confusion is that "stops", as in f/stops, has become a handy shorthand for other doubling/halving relationships when referring to exposure. Thus, when someone says they "stopped down", they probably did change the aperture from, say, f/8 to f/11. However, if someone says they wish they had a stop more light, they mean they wish they had twice as much. If they say they got some ISO 400 film which is two stops faster than their Sensia II, it means it is four times as sensitive and you can infer that the Sensia was ISO 100 (from 400, 200 would be one stop, one halving, and 100 would be the second stop, the second halving). Even experienced photographers get confused sometimes; I had one guy tell me he "pulled his film 6 stops, from ISO 100 to ISO 6". Well, that's not six stops, it's four. Here, count along: 100 to 50 is one, 50 to 25 is two, 25 to 12 is three, 12 to 6 is four.

Note that stops always refer to exposure things. You would not say a 100mm lens is a "stop longer" than a 50mm because it was twice as long! You would say it was twice as long, or just that it's a 100mm.

#### What is stopping down?

I've had a number of emails asking about this. When you stop down a lens, you are going to a larger number/smaller aperture and therefore less light. Going from f/8 to f/11 is stopping down. The opposite is opening up; going from f/11 to f/8 is moving towards the smaller number/larger aperture and therefore more light.

#### How am I supposed to remember that f/stop sequence?

Easy, just get yourself a wristband! You can get it either with just white numbers, or, for us old Nikon hands, with colored numbers like those Nikon used to use to indicate depth of field!

#### What About my weird f/stops?

The f/stop sequence I listed is the full stops. Most things in photography work in 1/3 or 1/2 stop increments, and you will find lenses with maximum apertures at other-than-full f/stops. In fact, among the lenses I own or have owned, there are maximum apertures are f/1.4, f/2, f/2.8, f/4 and f/8, all right on the full stops, and others in between at f/1.8, f/2.5, f/3.2, f/3.5, f/3.8 and f/4.5. You see these much more in modern digital cameras which tend to show half or third-stop increments (or your choice!: my Nikon D700 allows me to select 1/2 or 1/3 stop increments) in their LCD displays, fine gradations you never saw on physical f/stop rings.

#### You Say Most things Double and Halve?

Yep. Shutter speeds do the 1/15 1/30 1/60 1/125 thing referred to earlier. The f/stops we have referred to extensively in their f/2.8 f/4 f/5.6 etc. sequence. Film speeds do the same thing. The doubling goes like this in the common range of film speeds:
25 50 100 200 400 800

Each step here is a doubling/halving of the film's sensitivity to light. Thus, an ISO 100 film requires twice as much light to be correctly exposed as an ISO 200 film but only half as much as an ISO 50 film. You would say it was a stop slower than the 200, a stop faster than the 50. On digital cameras we get to change the ISO from shot to shot, and as you change this you'll see the half or third-stop increments go by as you choose.

There are third-stop intervals in ISOs as well. Here are the third stop increments of film speed with the full-stops in bold.

25 32 40 50 64 80 100 125 160 200 250 320 400

There were lots of still films made some intermediate speeds, like Kodachrome 64 slide film, Plus-X Pan Professional black and white at ISO 125, and Fuji NPS and some Kodak Portra color negative film at ISO 160.

#### How do you refer to exposures between full f/stops?

Generally, I just say f/5.6 and a third, or halfway between f/5.6 and f/8, or something. I have a Sekonic L-308B handheld light meter that reads full f/stops plus a fraction in between expressed in tenths. If I took a reading that said 1/125th of a second at f/5.6 plus four of these ten segments, I could go through the machinations to figure out exactly what f/stop that is (f/6.25) but that's not all that handy, to tell you the truth. No lenses are incremented in tenths of stops and tenth-stops are a needless amount of precision anyway given all the sources of slop in photography. Half and third stops are about as fine a distinction as matters. I've had many inquiries about what the intermediate stops are, so I put together a Printable Sheet of Third-Stop Increments which you can look at if you are interested.

#### I took my lens apart. The aperture is nowhere near as big as the calculation shows. What's up?

You're right. I had an email from a guy who had taken apart a Rokkor (Minolta) 300mm f/4.5 (for other reasons, not to check my measurements) and he said the diameter of the f/stop blades was way smaller than the calculation would indicate. Yep. My calculations above would be accurate if the aperture blades were mounted right in front of the front element. In fact, they're buried in the lens somewhere and, on my Nikkor 300mm f/4.5 were actually located behind all the lens elements. They still have the same relationship but the manufacturer can make the aperture blades much smaller in the light path partway back. However, the relationship is the same between each of the adjacent stops.

#### Why are they called f/stops?

I have no idea. I've never read an authoritative description of where the name came from. I have a vague memory that the defunct magazine Modern Photography did an article about it in about 1974 but my vague memory also seems to recall that it might have been the April issue. I get a trickle of emails from various people offering a range of opinions of why the name, and one day perhaps I'll compile them into a page of their own.

#### Is the f/stop all I need to know about the light transmission through the lens?

Probably. It's good enough for virtually all amateurs and nearly all professionals. There is a concept called t/stop, for transmission stop, which is a measure of the actual light transmission of the lens rather than the simple ratio of the aperture to the focal length. The t/stop can vary from the f/stop because you have a lot of lens elements (big zoom lenses might have these) or at one time you might have had one lens coated and another not coated. About the only people who need this level of precision are professional cinemaphotographers who use the t/stop to set exposure. Their lenses sometimes have both f/stop and t/stop scales marked. Even when they know the t/stops of the lens, the f/stops remain important because depth of field is driven by the f/stop regardless of the light-passing ability of the glass. I have never seen a still photography lens marked in t/stops, I have never used t/stops personally and I doubt you'd ever find a professional still photographer who has, but the concept is out there so I thought I'd mention it.

Personally, the only time I have found the marked f/stop to be undependable was with a Vivitar 600mm f/8 Series One lens I had. This was a catadioptric (mirror) lens billed as a Solid Cat because rather than mirrors and airspace, it had mirrors with solid glass in between. This puppy weighed a lot! Anyway, the lens was nominally f/8 but my own experience was that if you used a separate meter you'd better think about it as an f/8 and a half or f/11 lens.

#### So What's Important in all this?

You need to know the doubling/halving relationship and how it works with shutter speeds in exposure. You need to understand that for a given amount of light and ISO, there are many combinations of shutter speeds and f/stops that give the same amount of light on the film/sensor (see next paragraph). This is key since the shutter speeds and f/stops you choose have implications in how your final photograph will look in ways other than purely the amount of light on the film. You need to know that as you stop down you get more depth of field. You do not need to go around calculating aperture areas for your lenses and f/stops. If you're like me, it's worth doing it once to see that it works, then forgetting about.

### How a Range of Settings Gives the Same Amount of Light

Now, to bring this all together, we know that the shutter speeds and f/stops both double and halve. Thus, we know that we can open up an f/stop (letting in twice the light) and move the shutter speed one step faster (cutting the time in half) and have the same amount of light on the film. It's like that bucket of water; run the water twice as fast for half the time and the bucket is still full. As an example, if we meter a scene and it tells us that 1/125th at f/8 is the correct exposure, any of the following combinations would work:

Shutter Speed | 1/4 second | 1/8 | 1/15 | 1/30 | 1/60 | 1/125 | 1/250 | 1/500 | 1/1000 | 1/2000 | 1/4000
f/stop | f/45 | f/32 | f/22 | f/16 | f/11 | f/8 | f/5.6 | f/4 | f/2.8 | f/2 | f/1.4

Practically speaking, you aren't going to have one lens which takes you from f/1.4 to f/45 and your camera body may not have the higher shutter speeds. Also, if you are without a tripod, there are limits to how slow your shutter speed can be before your body movements blur the photo, so there are some constraints. But the point remains, all these combinations yield the same amount of light on the film and an identical picture in terms of brightness. What does vary is the ability of the camera to stop action and the depth of field, or how much is in focus in front of and behind the subject (I've written a page about Depth of Field which goes into the factors which affect it, the f/stop being one of them). This gives you some control over how your photographs will turn out. You should understand it and use it.

It is also worth noting that in flash photography, the shutter speed pretty much controls the ambient light portion of the exposure and the f/stop controls the flash portion of the exposure. When you are ready to move into advanced flash techniques, I'd highly recommend David Hobby's Strobist blog (work your way through Lighting 101) but you're going to want to have this f/stop thing down cold first.

PS: After an exchange with someone who finally understood this after reading this page (which is very flattering I must say) it occurred to me that part of the reason this whole sequence of identical exposure combinations is absolutely second-nature to me yet so confusing to newer photographers is that I started out life metering everything with a handheld light meter. Initially this was some long-forgotten Sekonic reflected-only selenium meter my Dad had, later (1976) the Sekonic L28C2 meter, usually in incident mode. The common feature of these two meters was that they displayed the readings on a dial that showed every possible combination of shutter speed and f/stop for that particular light level and film speed. By the time I moved to modern cameras and a digital readout meter (my Sekonic L308B) the whole sequence thing was completely ingrained. Nowadays light meters like my little Sekonic and many cameras read out with needless and often confusing precision and it appears that there is one correct exposure, not a whole sequence that will give the same exposure result. Now, not everyone gets to use old light meters for a few years to nail down these relativities, so I did a sheet to help you see all your exposure combinations given a reading from your camera or light meter. Take a look at this sheet and see if it helps.

PPS: Thanks to Mike Sheehan of Aurora Colorado who emailed me with how to show pi in html; &#960; which will show as π.

Have fun with all this!

Support this site!
Please make a small donation to help support this website. Thanks!

All material Copyright Matthew Cole 2003/2011 Last edited 14 May 2011 |

hit counter