Toronto Photographer :: On f/stops and trigonometry
Fair warning: this is going to be a very geeky post, involving, as the title suggests, f/stops and a whole bunch of basic trigonometry. I assure you, it’s easy enough to follow (and, in my opinion, is fairly critical for any photographer to understand), but if these types of things don’t concern you, then by all means feel free to check out the rest of the site.
Now, on with the show.
Have you ever taken a look at the front of your lenses? If so, have you ever noticed that the physical size of the “hole” in the front (the aperture) varies wildly between different lenses?
For instance, take a look below at the front of two of my favourite lenses, the Canon EF 35 f/1.4L and the Sigma 50 f/1.4.
(Top: Sigma 50 f/1.4; Bottom: Canon EF 35 f/1.4L)
Both of these lenses feature a wide maximum aperture of f/1.4. What this means, if you don’t already know, is that the maximum aperture diameter for each lens is the focal length divided by 1.4 (incidentally, this is why it’s written as “f/1.4″ and not “F 1.4″, as you sometimes see it…that little forward slash means something).
For the 35mm lens, this means that the maximum physical diameter of the aperture is:
35mm/1.4 = 25mm
For the 50mm lens, the calculation is:
50/1.4 = 35.7mm
(And this is apparent in the photos above, where the physical size of the aperture is visibly larger on the 50mm than the 35mm)
Since the aperture is roughly a circle, and since the area of a circle is equal to pi times the radius squared, the area of the maximum physical aperture for these two lenses is:
35mm
Pi x (25/2)^2 = 490.87 mm^2
50mm
Pi x (35.7/2)^2 = 1000.98 mm^2
So the maximum physical aperture area of the 50mm lens is slightly more than 2 times larger than the 35mm lens (2.04, actually).
But this begs the question, why does setting each lens to f/1.4 give the same exposure if the 50 has an aperture that is twice as large (in area) than the 35? Doesn’t that mean that the 50 lets in twice as much light, and thus the exposure at f/1.4 would be one stop greater for the 50 than the 35? This question causes some degree of confusion.
As anyone with a light meter knows, f/1.4 (or any f/stop) gives the same exposure, irrespective of focal length. Here’s why:
A 35mm lens “sees” an angle of view of about 63 degrees, while a 50mm lens “sees” an angle of about 46 degrees.
Now, let’s assume that we are standing 10 metres away from a light source.
35mm
The angle of view is 63 degrees, the lens-to-subject distance is 10m, the radius of the circle that the lens “sees” is x. The hypotenuse of the triangle is y, as shown below.
Using basic trigonometry (which I can’t believe I actually remember), we can solve for x as follows:
Cos(63/2) = 10/y
y = 10/Cos(63/2) = 11.72m
Sin(63/2) = x/11.72
x = 6.12m
Thus, the area of the circle that the 35mm lens “sees” (let’s call this the “imaging circle”) is:
Pi x (6.12)^2 = 117.81 m^2
50mm
Again, we can easily solve for x by just changing the angles here from (63/2) for the 35mm to (46/2) for the 50mm lens, and we get an image circle at 10m of 56.48 m^2.
Now, let’s divide 117.81 (the area of the 35mm’s imaging circle @ 10m) by 56.48 (the area of the 50mm’s imaging circle) and we get 2.08.
Thus, the 35mm lens captures light from a circle that is two times larger (in area) than the 50mm lens.
But hold on a sec. We previously determined that at the same f/stop, the 35mm lens has an aperture that is about 2 times smaller than the 50mm lens.
Let’s assume that we put an even light source 10m from our camera. The 35mm lens will “see” about 2 times more of that light source than the 50mm lens, but it will also let in about 2 times fewer photons of light (at equivalent f/stops, of course) than the 50mm because of it’s smaller physical aperture size.
That is, the two cancel out perfectly. And this is why setting a lens to f/1.4 (or any aperture) gives the same exposure, regardless of the focal length.
(Just for fun, take a look at the massive size of the maximum aperture on the Canon 85 f/1.2L II. Despite the large physical aperture, I assure you that f/1.4 on this lens produces the same exposure as f/1.4 on any other lens, for precisely the reasons given above.)
I hope this helps to clear up any confusion regarding f/stops, focal length, and exposure.