An explanation of the mathematics behind the F-stop
When making the jump to manual photography, one of the most confusing topics can be that of the f-stop. Even after learning that the aperture controls your depth of field, you can still be very confused by why the numbers change the way they do. It's no surprise, as circle geometry isn't something you use in your daily life.
However, understanding the mathematics can give you an excellent grip on the f stop scale, especially if you're the left-brain type. Let's assume that you are already familiar with the full f-stop scale (1.4 - 2 - 2.8 - 4 - 5.6 - 8 - 11 - 16 etc.). Why is it that only increasing .6 from 1.4 to 2 is the same time of adjustment as moving 5 from 11 to 16?
The reason is that the f/stop number is actually a ratio between the diameter and focal length of the lens. The inverse relation of light stems from the diameter becoming smaller as the f/stop number increases. For instance, an 85mm lens at f/2 will yield a diameter of 42.5mm (85 / 2), If you stop down (increase the f/stop and reducing the light 1 stop) to f/2.8, the diameter is now 30.3 (85 / 2.8). Now I know what you are thinking, 30 isn't half of 43, so how did we halve the light if the diameter didn't get cut in half?
This brings us back to circle geometry. We need to look at the area of light that passes through the opening. The area of a circle is found by π x radius^2. The radius is half of the diameter, and pi is a constant that represents the circumference divided by the diameter. So, lets do the area math for the example above with the 85mm lens.
At F/2 we have a diameter of 42.5, and F/2.8 is 30.3. This gives us a radius of 21.25 and 15.15 respectively.
So for f/2 we have π x 21.25^2 = 3.14 x 451.5 = 1418 square mm (rounded)
For f/2.8, its π x 15.15^2 = 3.14 x 229.5 = 720 square mm (rounded)
As you can see (while looking past some rounding), we have reduced the area of the opening by half. This is why when you increase the F/stop number by one stop and reduce the size of the opening, you are reducing the amount of light that can get through by half. The ratio also explains while the numbers start to have bigger intervals as you move up the scale.
Hope this helps and please feel free to keep asking questions so I can provide you with more educational content!
Robert Hall is a professional photographer in Southeast Michigan. His work primarily consists of weddings, commercial and editorial. He is constantly improving his skills through discussion of techniques and critique with fellow photographers. Robert is always looking for new connections on social networks!