The Mathematics of F-stops

I am occasionally asked the question: what is an f-stop.  Here is the answer.

The aperture is the hole in the camera where the light gets in when you press the shutter.  The f number  which is used to measure aperture is formally expressed in dimensionless numbers from the ratio of the lens (F) to the diameter (D) of the ‘pupil’ of the camera.  The pupil of the camera is the ‘hole’ into which the lens fits and is designed to equal the width of the sensor or unexposed film from corner to corner.  So, for a given combination of lens of given focal length and camera the ratio of F/D is defined.

The diameter D can be found by rearranging the formula for the area of a circle (Area = Pi x Radius squared).  The radius is, you may remember from school days, half the diameter.  So the f number is given by the Focal length of the lens divided by the diameter as defined by the Pi and the area of the hole in the front of the camera the light goes through.  A swift bit of algebra shows that the f number is given as the reciprocal of 2 x square root of the fraction of the hole through which light passes to hit the sensor times (still with me?) a constant value determined by the camera lens combination.  If 100% of the hole in front of the sensor is open to light then the f number is 1/(2*sqrt(1)) = 0.5.  Halve the exposed area and the f number is 1/(2*sqrt(0.5)) = 0.707.  Halve it again and f = 1/(2*sqrt(0.25)) = 1 and then it goes 1.4, 2, 2.8, 4, 5.6 and so on as we reduce the area through which light passes by a half each time.  Well, if you are up to the maths you will notice that it's actually 2.828 and 5.657.  By convention the number  is rounded.

An f-stop is, therefore, half the area and hence half the amount of light passing through the next highest aperture.  A reduction of three stops would reduce the area by 1/2 x 1/2 x 1/2 = 1/8th of its previous area. So the take home messages are:

• the f number is the reciprocal of the increase or decrease in the area through which light passes into the camera body.
• The smaller the f number  the larger the 'hole' or aperture.
• The more expensive lenses have wider apertures for a given focal length.   A 50mm lens may open to 1.4,  a professional 300mm lens probably two stops lower at 2.8 and a 'prosumer' lens maybe to only 5.6.

A f-stop represents the change in the amount of light that hits the sensor.  As far as aperture is concerned it represents either a doubling or a halving of the area of the hole through which light passes on its way to the sensor.   The same logic applies with shutter speed and ISO - an f-stop is a half or a doubling of the previous number.  So, if I am shooting at 1/500th of a second, I reduce the light by a stop if I increase my shutter speed to 1/1000th, I increase the light by a stop if I slow the shutter speed to 1/250th and so on.  With ISO 100 to 200 is a stop, 200 to 400 is a stop but this time what we are changing is the sensitivity of the sensor to light - in each case doubling it or reducing it.

I know all this looks horribly technical but, in terms of technicality, that's all there is to it.   From now on, the trick is to understand this and use it intuitively to estimate the impact of a change of aperture, shutter speed or ISO on the job of creating an image.