The camera lens transmits a circular image that completely covers the camera sensor. The area of a circle is calculated by taking the pure number Pi (approximately 3.14 and pronounced like cherry 'pie') times the radius squared. When we square a number, that just means to take the number and multiply it by itself. Four squared means 4 * 4, which equals 16. 10 squared means 10 * 10, which equals 100. The number Pi is a pure number (without any units) and never changes.
Area = (Pi) * (radius) * (radius)
The radius of a circle is the length from the center of a circle to the circle itself. The diameter of a circle is the length from one side of the circle to the other side, passing through the center. Obviously, the diameter of a circle equals exactly twice the radius.
So, let's calculate the area of a circle with a radius of 1 unit (millimeters, feet, miles, doesn't matter):
Area = (Pi) * (radius) * (radius)
Putting in our numbers, this gives us:
Area = (3.14) * (1 unit) * (1 unit) giving us 3.14 square units. This will be our starting point.
Now, if we want to double the area of a circle, what can we change? The number Pi never changes so the only thing we can change is the radius of the circle. Our first thought might be to simply double the length of the radius to double the area of the circle. Let's see how those numbers come out:
Area = (Pi) * (radius) * (radius)
Area = (3.14) * (2 units) * (2 units) - giving us 12.56 square units. Wow! That's 4 times the starting area, not twice. Now, we could try guessing a few times using a calculator to get close to a final area of 6.28 (twice our starting point area) or we can figure it out mathematically. Or I can simply tell you that the answer is the square root of 2. What's a square root? Earlier, we said that a number squared means to take a number and multiply it by itself. A square root is going the other direction. It means you start with a number and figure out what other number, when multiplied by itself gives you the starting number. So the square root of 100 would be 10 (because 10 * 10 = 100) and the square root of 25 would be 5 (because 5 * 5 = 25).
So what is the square root of 2? It is approximately 1.414. The number actually continues on forever past that (1.414213562...) but let's just use the 1.414 part.
Go ahead and get out your calculator. I know you're thinking you've got to check it yourself. That's ok, I'll wait.
Check out ok? Thought so.
Area = (3.14) * (1.414 units) * (1.414 units) gives us 6.28 square units. Now THIS is twice the area of our first circle. So what happened to the diameter in this exercise? In the starting circle, the radius was 1 unit which makes the diameter 2 units. In our second circle, the radius was 1.414 units so the diameter would be 2.828 units. The diameter increased by the same factor as the radius, ie, radius 1 to 1.414 and diameter 2 to 2.828. Increasing the diameter by the same square root of 2 value doubles the area of the resulting circle. The diameter of the opening through which the light is transmitted is the actual Aperture measurement.
So, what can we conclude? We can conclude that each time we want to double the area of a circle, we need to increase the radius (or diameter) of the circle by a factor of the square root of two (multiply by 1.414...).
What happens if the focal length of the lens and the aperture values are the same? Let's start with a 100mm focal length lens and give it a 100 mm aperture.
As we know, F/stop = (focal length) / (aperture) so the F/stop in this example would be:
F/stop = (100 mm) / (100 mm) - which gives us that magic value, F/stop = '1'.
The area of this aperture opening equals (Pi) * (radius) * (radius) or:
(3.14) * (50 mm) * (50 mm) = 7853.98 sq mm
Now let's start decreasing the diameter (aperture) by that square root of two value (to cut the area of the circle in half) with this lens and see what happens.
Remember, each step will divide the diameter by 1.414 to get the next diameter or Aperture value:
So, do the numbers in that last column look familiar yet? Yup, those are the F/stop values on your lens. Each change above represents cutting the area of the resulting circle in half, thereby cutting the amount of light allowed through the lens in half for the same shutter speed.
As you change the F/stop from a setting of 1 to 1.4 you are cutting the amount of light reaching the sensor in half. As you change the F/stop from a setting of 1.4 to 2 you cut the amount of light in half again. This means that the amount of light reaching the sensor at an F/stop setting of 1 is four times the amount of light reaching the sensor at an F/stop setting of 2.0. Remember, we cut the amount of light in half twice (1 to 1.4 and again from 1.4 to 2.0) and 2 * 2 equals 4.
The kit lens I bought with my Nikon D80 has a variable zoom range from 18mm to 135mm. This is a wonderful working range for focal lengths, covering the moderate wide angle to medium zoom range. It's also a variable aperture lens. This means that as I change the focal length, the maximum aperture value I can use (largest opening) also changes. Let's look at how the maximum aperture changes as I vary the focal length for this lens:
Once I get past 70mm focal length, the amount of light reaching my camera's sensor is half the amount it received when the focal length was between 24 and 30mm (F/5.6 vs F/4.0). This is an excellent "walkaround" lens. Yes, there are distortion and vignetting issues at specific settings, but once you understand how to work around these points the lens performs very well. It is very sharp throughout the F/stop range. Unfortunately, the variable F/stop range available with this lens means it cannot be used in poor light without using a flash or very slow shutter speeds.
I also have a Tamron 17-50mm constant F/2.8 and a Nikon 50mm F/1.8 lens. At 50mm, the Tamron can let in more than 3 times as much light as the variable aperture kit lens and the 50mm Nikon can let in more than 7 times as much light! This has several effects. First, I can shoot the same shot at much higher shutter speeds and have the same exposure. Second, I have a greater range of apertures to choose from so I can control the depth of field to a tighter range. Third, the viewfinder is much brighter, making it easier to see what you're shooting and fourth, the camera is able to focus better in low light.
How can the camera focus better in low light with a larger F/stop? When using your camera to set up a shot, the aperture remains wide open (lowest F/stop number, largest opening). It is only when you actually take the shot that the aperture closes down to the selected position. [Note: If your camera has a "Depth of Field Preview" button, you can push this to manually close down the aperture to the selected position and see how much of your picture will be in focus.] With your camera using the largest opening possible for the given lens, the autofocus circuitry has the maximum amount of light (and thereby contrast) available to control the focus. When lens A lets in 2 or 4 or 7 times as much light as lens B, the camera can control focus MUCH easier with lens A.
Now you see why that Nikon 70-200mm, constant F/2.8 VR lens costs four times as much as my 70-300mm, F/4.5-5.6 VR lens.