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Making Sense of the Odd Progression of the F-Stop Scale?

We all know that aperture of a lens is marked with f/stop numbers that work in stop values where numbers in the progression represent doubles and halves. If you look at a standard f/stop series the values are:

f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 and f/32.

It is clear that the numbers themselves are not doubles and halves then on what basis are these values used.


Why such a weird progression of numbers?

Area of a Circle
Area of a Circle

The f/stop sequence represents the varying hole size in a lens (the aperture) through which light passes. Each aperture allows twice as much or half as much light to pass as the preceding or following aperture. The aperture is in the shape of a circle and to make each aperture double or halve the amount of light the area of the circle should be doubled or halved. F-stop numbers are actually fractions and thinking of them as fractions is the easiest way to understand why the larger numbers represents smaller lens openings and vice versa ( ½ is a lot larger than 1/22). Thus f/2 represents a fairly large hole that admits a lot of light while f/22 is quite a small opening admitting only a small amount of light.

If you look closely at the aperture scale you will find that each number is 1.4 times the preceding.

Since apertures are holes in the lens which is circular in shape we could use the following formula to calculate the area of a circle.

Area of a Circle (A) = πr^2  (kindly read "r^2" as r squared)

Where A = Area, r = radius, π = 3.14

So 2 πr^2 = πx^2. Solve for x and you get the square root of 2 (which is 1.414) times the radius.

For example

Lens Aperture
Lens Aperture

Let us assume that at a given aperture the size of the lens opening (read area of the circle) is 3.14.

i.e πr^2= 3.14 that translates to 3.14 x r^2= 3.14 so r^2= 3.14/3.14 = 1. Therefore r = √1 = 1.

so we have a circle with an area of 3.14(mm or cm or any other measure) and a radius of 1 (again any measure).

Now let us see what happens when we double the area of the circle.

New area = 3.14 x 2 = 6.28 now let’s find out the radius of our new circle

6.28 = 3.14 x r^2 i.e. r^2 = 6.28/3.14 = 2 therefore r = √2 = 1.4

Thus it is clear that the diameter varies as the squared root of the area. When we double the area the diameter will be 1.4 times (which is √2 )  larger and that is how the aperture scale is built using the multiplication function.

f/1 x 1.414 = f/1.4
f/1.4 x 1.414 = f/2
f/2 x 1.404 = f/2.8
f/2.8 x 1.414 = f/4 and so on….

Wow the f stop sequence of numbers.
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