I believe what Claus was trying to point out is that there is an error on line 5 of the equation in the last section of your article. The line directly above the big red arrow. The right side of the equation should show r sub 1, not r sub 2.

As you now know, the f-stop is a fraction.  As with all fractions, when the number in the denominator increases, the value of the fraction decreases.  For example, a ½ cup of sugar is a lot larger than ⅛ cup of sugar, even though the number 8 is greater than the number 2.

The f-stop takes both into consideration by normalizing the diameter of the opening to the focal length of the lens, which results in a relative aperture.  This way, the f-stop on one lens allows the same amount of light to hit the sensor as the same f-stop on a different lens.  As such, f-stops are relative, not absolute, values that represent a lens’ relative aperture.

Many lenses have an aperture range where each f-stop is one full stop of light different than the previous or the following f-stop.  Here is an example of an f-stop scale with full-stop increments:

This doubling or halving of the amount of light should sound familiar as it is leftover from the early days when metal stops were inserted into lenses to change exposure.

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i’m not a professional (good camera photographing ~ 6 years—canon). F-stop, aperture, diaphram??? really confused by the canon 100-400 4.5 -5.6 lens. you have the 4.5 – 5.6 and then there are all the F/#’s. how does this jive? geoff

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So, the f-stop scale appears as a wonky numerical list of numbers because they represent the doubling or halving the area of a circle, a change that is dependent on the radius (or diameter) changing by a factor of √2 between each f-stop.

Brenda, Wonderful explanation which I enjoyed reading. I will likely forward your link to my photography salon members for those that have interest in the mathematical explanation. One comment. Section on why Sq rt 2 is used as a factor, 4th line into equation solving, far right, did you mean sq2×Sq rt r1 Sq instead of r2 Sq? Pardon, I don’t have proper symbols to convey mathematically. I so do appreciate an algebraic solution. Much enjoyed your article

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If you want to know more about the math behind it, then keep reading.  If not, just remember that as the f-stop numbers increase, the aperture opening decreases.

Hopefully, this helped demystify the numbers behind the f-stop scale! If you want to continue learning about aperture and how to use it in outdoor photography, be sure to check out the related links below.

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focal length does not change the amount of light allowed through the lens. The aperture/f-stop determine the focal length. The bigger the hole (the smaller the number) the less in focus and more shallow the focal length.

Hi Geoff, I’m a little confused by your question. Are you asking what the 4.5-5.6 indicates? If so, that’s the range of the widest aperture of that lens. The widest aperture of a variable aperture lens, like your Canon, changes depending on the focal length. So at 100mm, the widest aperture you can get is f/4.5 and at 400mm the widest aperture you can get is f/5.6. I hope that helps.

This is an excellent explanation of f-stop. Thank you. But I do wish it included an explanation of why and how focal length changes the amount of light allowed through the lens.

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Why is aperture measured in f-stops?  Why is it called an f-stop? Why is the f-stop numerical scale so weird?  I had many of these same questions when I was first learning photography, so if this is you, you are not alone.

In the expresion directly above the read arrow in the equation it should have been the radius of circle 1 and not the radius of circle 2. Othervise the squate robot of the radius of circle 2 to the second would have to be equal to the squate robot of the radius of circle 2 to the second multiplied by the squate root to 2, which it it not. It could disturb.

A stop of light is a unit that represents relative exposure.  One stop of light is equivalent to one exposure value (EV).

The f stands for focal length and the number in the denominator is the quotient of the focal length divided by the diameter of the entrance pupil.

The intensity of light that travels through a lens and exposes the camera sensor is dependent on both the length of the lens and the diameter of the opening.

Aperture is one of three camera settings that control relative exposure.  The aperture is the opening in the lens diaphragm, which functions a lot like a human iris.  The aperture is like the pupil of an eye. It opens and closes to let more or less light into the lens.  Aperture is measured in f-stops.

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If you have additional f-stop options on your lenses, these likely represent ⅓ or ½ stop increments in addition to the full-stop increments.

I appreciate the effort that you went to. However, I suggest that you drop all reference to radius. It is just an additional confusion factor. Lens references are in diameter such as for filters,in drawings, etc. I know that accuracy would require (d/2)^2 in your equations, but you can explain omission; d^2 for clarity. Just something to consider.

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Aperture and f-stops can be fairly frustrating to learn at first.  Many photographers learning about aperture for the first time are left with a lot of questions.

Hello and good day Thanks for your excellent subjects and notes. I’ve a question about the relationship between Fnumber and AOV. may I have your idea about it if there’s any relationship? Best regards Jalal

The figure below shows what this would look like numbers-wise using a 50mm lens as an example.  Note that the difference between each f-stop is a factor of √2 or 1/√2 and that the resulting area of the opening halves or doubles with each successive change in f-stop.

To better understand the f-stop ratio and why it is so important in knowing how to properly use aperture in photography, be sure to check out What is Aperture in Photography: Key Concepts Explained.

Side note: this is also where the phrase “stopping down your lens” comes from, which means to make the diameter of the entrance pupil smaller.

Each f-stop number on the f-stop scale differs from the previous and the following f-stop by a factor of √2 resulting in the doubling or halving of the area of the entrance pupil, which changes the relative exposure by one stop of light (one exposure value) in either direction.

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Back in the old days, the diaphragm of the lens was adjusted manually by inserting metal plates into the front of a lens (can you imagine?).  Each plate was called a “stop” because it stopped light from entering the lens by changing the area of the opening.

The significance of √2 comes from solving the following equation, where A1 is the area of Circle1 and A2 is the area of Circle2, which is double the area of Circle1.

So, there you have it! That’s why the f-stop scale is so odd, thanks to the formula for solving the area of a circle. That wasn’t too bad, was it?

Not being a brilliant mathematician, I worked out this relationship (halving/doubling of area) by hand so to speak, made a spreadsheet and saw the relationship. A very revealing exercise.

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An f-stop (or f-number) is the ratio of the lens focal length divided by the diameter of the entrance pupil of the aperture.  As such, an f-stop represents the relative aperture of a lens; it is basically a way to normalize the aperture setting across different lenses.  F-stop and f-number are terms used interchangeably to indicate the aperture setting on a lens.

Unless you geek out on math, you likely don’t readily know why the √2 is the factor used to double or halve the area of a circle, and you might find working with square roots a painful experience.

Similarly, as f-stop numbers increase, the relative aperture opening decreases.  Lower numerical f-stops let in more light than higher numerical f-stops.

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By convention, increasing the relative exposure value by one EV, or one stop of light, will double the intensity of light exposing the sensor.  Likewise, decreasing the relative exposure value by one EV will halve the intensity of light.

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I’m glad the info was helpful, Jalal. Can you clarify your question please? Are you asking whether there is a relationship between f-number and field of view (FOV)?

Now, I could stop here and tell you to just accept this mathematical reality, but I found it to be helpful to understand why √2 was the factor needed to have this doubling or halving effect on the area of the entrance pupil.

If that’s you, I totally get it!  It took me a while to figure it out too.  But once I worked it out, it made a lot more sense to my little brain, and the numbers of the f-stop scale seemed less confusing to me.

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The aperture is the physical opening of the lens diaphragm.  The amount of light that the aperture allows into the lens is functionally represented by the f-stop, which is a ratio of the lens focal length and the diameter of the entrance pupil.

The word “stop” has another meaning in photography as well.  As part of the exposure triangle, aperture, shutter speed, and ISO each use exposure values to increase or decrease relative exposure by equivalent stops of light.

Note – many modern camera lenses no longer have an aperture ring, so the aperture is controlled by the camera body and is viewed via the LCD display.  On some cameras, you can choose whether exposure is adjusted by full, ⅓ or ½ stops by choosing the increments of exposure control in a menu setting.

The f-stop represents the size of the aperture. Each f-stop number on the f-stop scale differs from the previous and the following one by a factor of √2 resulting in the doubling or halving of the area of the aperture, which changes the relative exposure by one stop of light (one exposure value) in either direction.

Each “stop” was designed to either double or halve the intensity of light that entered through the lens depending on whether it was removed or added.  The word just stuck and while it makes little sense to us today, it is the terminology used for better or worse.