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What is meant when someone says that a 1/2.3 sensor has a crop factor of 6.22? How many times the area of a 1/2.3 sensor is a full-frame sensor? Please include plenty of details.
Another question on this site, What is crop factor and how does it relate to focal length? addresses the angle of view, but doesn't say anything about how the crop factor relates to the surface area of a sensor compared to a full-frame sensor.
Crop factor is expressed as a ratio of the linear measurements of a sensor compared to a 36x24mm 35mm film frame or a full frame sensor. This is because a sensor exactly half as large as another will also provide exactly half the angle of view as the other with a lens of the same focal length. Or conversely, a sensor half as large requires a lens of half the focal length to provide the same angle of view as the larger sensor. Since not all sensors have the same aspect ratio, the comparison is almost always made to the measured diagonals of each format. This is due to the fact that, regardless of the aspect ratio of the lens, in order to allow the entire sensor to be covered by the virtual image cast by the lens the usable image circle must be at least as wide in diameter as the length of the sensor's diagonal measurement.
Most sensors referred to as 1/2.3" format have a diagonal of 7.66mm (a crop factor of 5.64X) and an area of 28.50mm². The Sony Exmoor IMX220, also considered a 1/2.3" camera, has a slightly larger sensor with a 7.87mm diagonal (a crop factor of 5.49) and an area of 29.73mm². An FF sensor has an area of 864mm². Thus a 1/2.3" sensor has an area about 1/30 of an FF sensor.
Since crop factor is a linear ratio, to get the ratio of two sensors' areas one would need to square the crop factor. Thus your 1/2.3" sensor with crop factor of 6.22 (I know of no such sensor), which means the diagonal of your sensor is 1/6.22 the length of a full frame diagonal, would have an area roughly 1/38.688 of a full frame sensor. Differences in aspect ratios could alter the relative areas of two sensors with identical diagonal lengths.
The Wikepedia article on Image Sensor Formats includes a table with measurements, areas, and calculated crop factor for many sensors.
Michael's answer covers well how the area and dimensions of the two sensor sizes relate, but does not conceptually explain how or why that appears to affects focal length. I'll try to explain that.
A lens designed for full frame 35mm (FF35) is designed to project a circular image large enough to cover a 36x24mm sensor / film frame. So what we have to imagine is: what would happen if we used a smaller sensor that only recorded a portion of that circle?
Get a camera with a 36x24mm sensor. Make Image A with a 200mm lens, and then Image B with a 300mm lens. Image A has a 12.3 degree diagonal field of view, Image B 8.2 degree.
Now, open Image A in Photoshop and crop it down to 2/3rds (1/1.5) its original size. It will now have the same 8.2 degree field of view as Image B. You've cropped your image so that it has a field of view equivalent to a lens with a focal length 1.5 times longer. Well, that's exactly what happens when you put your FF35-format 200mm lens on a camera with an APS-C (24x16, 2/3rd the size of FF35) sensor! So APS-C can be said to have a "1.5x crop factor" compared to FF35.
FF sensor area (fxA)= 36x24mm= 864mm squared (^2), Any area of a given sensor can be represented by MP's as well as square mm Therefore, APS-C effective area (efA)= 24x16mm= 384mm^2 or possibly XMP (only as an example)
FF crop factor for angle of view (cf) of lens is inversely related to the ratio of diagonal lengths of the rectangle made by the sensor (dr)= Xd/fxd or Xd/43.3mm. In the case of APS-C, the ratio is 2/3 or 28.87/43.3mm. So, the crop factor for the angle of view (cf) for a FF 35mm lens on an APS-C sensor would be the inverse of the dr 1/dr, or 3/2= 1.5. cf can be multiplied.
efA is the square of the ratio of the diagonal lengths (dr)^2 multiplied by the fxA if the relative dimensions of the FF rectangle, 3x2 or width x height, are maintained, as they are with APS-C. Therefore, the efA of an APS-C sensor is (2^2/3^2)864= (4/9)864= 384mm^2.
In the case of the Leica Q that has a 28mm fixed lens and a FF 24MP sensor, cropping to a 35mm angle of view can be analyzed to find the MP equivalent (MPeq) to area of the cropped portion of the sensor. 28cf=35, cf= 35/28= 5/4= 1.25; And, dr= 4/5= 0.8; then efA= MPeq= (dr^2)(fxA)= 4^2/5^2(fxA)= 0.8^2(fxA)= 16/25(fxA)= 0.64(fxA)= 0.64(24MP)= 15.36MP. Simplified, cropping any FF sensor with a fixed lens of Xmm to a known focal length or equivalent field of view, the efA= (1st focal length/2nd focal length)^2(fxA); in the case of the Leica Q, efA= (28/X)^2(24MP), where X is the known new focal length (or equivalent field of view). For X= 50mm, then efA=(28/50)^2(24MP)= 7.526MP
A word or two on crop factor:
Over the years the camera has shrunk. As film evolved its resolving power improved and smaller film formats became commonplace. In the early 1900’s a still camera using 35mm motion picture film was introduced. The 35mm film format (image size) measures 24mm height by 36mm length. Millions of 35mm cameras were sold. Because of the popularity of the 35mm, it evolved as the gold standard as to camera and lens design. This is because each format size has a “normal” when it comes to lens focal length.
Now the camera lens projects an image of the outside world on the surface of film or digital sensor. The size of the image of objects projected and the angle of view seen is a function of the focal length of the lens being used. Focal length is a measurement taken from about the center of the lens to the image plane (surface of film or chip), when the camera is imaging a far distant subject (at infinity ∞ as far as the eye can see). This distance is by tradition measured in millimeters.
An experienced photographer, familiar with his/her camera has a feel for how big or small images of objects will be and how wide or narrow view will be. These understandings are intertwined with focal length of lens being used and the size of the format. Because the venerable 35mm film camera has been around from almost 100 years, the view this format delivers with any given focal length has become the gold standard. Stated differently, the crop factor is an aid to help make a comparison as to how a vista will image using the 35mm format as the benchmark.
Each film or digital sensor format size has one focal length that will be labeled as “normal”. We compute normal by obtaining the diagonal measure of the image size (format size) of film or chip. As an example, the 35mm frame measures 24mm by 36mm, and the diagonal of this rectangle is about 45mm. If we mount a lens about equal to this value, the view delivered is said to be “normal” (matches the human experience) as to perspective. The angle of view will be about 45⁰ with the camera in the horizontal (landscape) positon. The 45mm “normal” for the 35mm format is a peculiar value, so lens makers have chosen to round this value up to 50mm for their convenience. The difference is of little consequence.
If we mount a lens shorter than normal, the image of objects will be condensed and the angle of view expanded. As a general rule, the realm of wide-angle is 70% of “normal” or shorter. For the 35mm, this will be a focal length of 35mm or shorter. If we mount a lens 200% of “normal” or longer, we enter the realm of telephoto.
The meat of this discussion is: time marches on and modern digital image sensors are shrinking. Popular today is the compact digital -- also called APS-C format. This was a failed film format of the 1980’s. It measures about 16mm height by 24mm length. The diagonal measure is 30mm. Thus a lens of 30mm focal length is “normal” as it delivers an angle of view of 45⁰ when the camera held horizontal. Wide-angle is 20mm or shorter and telephoto is 60mm or longer.
Now to compute the crop factor, we divide the two diagonal measures; thus 45 ÷ 30 = 1.5. This is the crop factor. What do we do with a crop factor? Say your buddy has a full frame camera (a nickname for the 35mm format as is Fx). You have a compact digital -- nicknamed Dx format. Your buddy mounts a 100mm lens. Your camera is 1/1.5 = 0.66 or 66% of the size of the Fx. You mount a 100 ÷ 1.5 = 66mm, and your camera delivers the same view. In other words your smaller camera requires lenses that are 66% of the lenses required by the venerable 35mm full frame.
I wish crop factor would go away. They are useless to most except gray haired 35mm knowledgeable photographers. Tomorrow’s cameras will get smaller and smaller as imaging chip technology evolves. The smaller the imaging chip, the larger the crop factor value. In future the crop factor will disappear.
The 1/2.3 image sensor is a super small imaging chip. A crop factor of 6.22 is 1/6.22 = 0.1608 x100 = 16% of the size of a full frame. If a 100mm is mounted on a full frame, then a 100 ÷ 6.22 = 16mm. In other words, a 16mm focal length on this minute format delivers the same angle of view as a 100mm mounted on a full frame.
