The Magic Numbers of Photography - Calculating the F number

F numbers or F stops are an important concept to grasp in Photography. The F stop that you select will have important implications for your depth of field and the amount of light that enters the camera. But where does this value come from?

F numbers express the relationship between the focal length of a lens and the diameter of the opening through which light can pass (sometimes called a pupil or iris). The F number for any given lens is obtained by dividing the focal length by the diameter of the opening.

If they are equal the F number is 1.

If the focal length is smaller than the opening the F number is less than 1.

If the focal length is greater than the opening (the most common scenario) then the F number is greater than 1.

Now the truly interesting part about F numbers is the way that the relate to the amount of light that enters the lens. While doubling the shutter speed halves the amount of light that enters the camera doubling the f number more than halves the amount of light entering the camera. This is because the F number is related to the diameter of the opening while the amount of light that enters is determined by the area of the opening.

To illustrate this point we will compare the area and diameter of the opening for 3 F stop values in a 50mm lens: 1, 1.4, and 2.

F = 1

Diameter = 50/1 = 50mm

Area = 25 x 25 pi = 1964.29mm

F = 1.4

Diameter =  50/1.4 = 35.71...mm

Area = 17.86... x 17.86... x pi = 1002.19mm

F = 2

Diameter = 50/2 = 25mm

Area = 12.5 x 12.5 x pi = 491.07mm

NB:// This relationship holds for any focal length as F numbers take the focal length into account. 50mm simply used as a concrete example.

As it turns out in order to halve the amount of light that enters the camera you need to increase the f number by the sqaure root of 2 (1.4-ish). This is why the standard F stop scale goes:

1 - 1.4 - 2 - 2.8 - 4 - 5.6 - 8 - 11 - 16 - 22 - 32

Each value is an approximation to the preceding value multiplied by the square root of 2 (1.4-ish)

F = 1

Diameter = 50/1 = 50mm

Area = 25 x 25 pi = 1964.29mm

F= Square root of 2

Diameter = 35.53...

Area = 982.14

Now if you're super inquisitive you may be wondering why the square root of 2 is the magical number. If you're also super observant you may have noticed earlier that the change from an F number of 1 to an F number of 2 reduced the area by a factor of 4. If you extend that out to an F number of 4 the area is 16 times smaller than when you have an F number of 1.

F = 1

Diameter = 50/1 = 50mm

Area = 25 x 25 pi = 1964.29mm

F = 2

Diameter = 50/2 = 25mm

Area = 12.5 x 12.5 x pi = 491.07mm

F = 4

Diameter =  50/4 = 12.5.mm

Area = 6.25 x 6.25 x pi = 122.76mm

What's happening?

Every time you double the F number, by going from 1 to 2 or 2 to 4, the amount of light entering the camera reduces by 4 times (or 2 squared). 

If you quadruple the F number, by going from 1 to 4, the amount of light entering the camera reduces by 16 times (or 4 squared). Turning this around if you wanted to decrease the amount of light entering the lens by 16 times you could quadruple your f number (4 is the square root of 16). If you wanted to decrease the amount of light entering your camera by 4 times you could double your F number (2 is the square root of 4). Finally if you want to reduce the amount of light entering your camera by two times (i.e. if you wanted to halve the amount of light entering your camera) you could increase your F number by ~1.4 (~1.4 is the square root of 2).

In fact this concept pops up a lot in photography and is known as the inverse square law.