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Focal length is the only factor that decides whether a lens is described as being normal or wide angle.

Below which focal length are lenses considered to be wide angle? Is this focal length the same for all types of camera (full frame, APS-C, etc.)?

Also, from the point of view of designing/building a lens, do normal lenses and wide angle lenses differ other than in focal length?

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Wide angle is in comparison to 'normal'. On a full frame camera normal is considered to be 50mm and wide angle is 35mm or less.

Using these numbers we can determine that the accepted ratio for wide angle is less than 0.7 times that of normal on a given format. On APS-C the normal focal length is 35mm, making wide angle start at 23mm.

Most SLR cameras have a mount to film plane distance of about 40mm. Any lens with a focal length less than this distance requires a reverse telephoto group to extend the distance between the lens and the focal plane. On a full frame sensor a normal lens can be much simpler than a wide angle lens since a normal lens (50mm) has a greater focal length than the mount to film plane distance (40mm). On APS-C the normal lens (35mm) has a distance less than the mount to film plane distance, so there is no fundamental differences between the construction of a normal and wide angle lens.

Source: Wide Angle Lens on Wikipedia

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Wikipedia gives a very specific — but unsourced! – definition: a lens is considered wide-angle when it covers the angle of view between 64° and 84°. I think they're back-calculating that from the field of view from 24mm to 35mm lenses on 35mm film / full-frame. (The lower bound because 24mm is traditionally the beginning of "ultra-wide".)

The equivalent focal lengths in different formats would be:

1":                   9mm – 13mm     
Four Thirds:         12mm – 18mm
APS-C (Canon):       15mm – 22mm
APS-C (Other):       16mm – 23mm
Full-Frame:          24mm – 35mm

And this seems like a reasonable-enough definition — but, Wikipedia notwithstanding, there's some wiggle room. The Focal Encyclopedia of Photography says:

Wide Angle Lens — A lens having significantly greater covering power than a normal lens [....] A wide angle lens may be defined as a lens having a focal length [providing an angle of view greater than] 53 degrees, the angle of coverage of a normal lens. A fisheye lens, although quite unique, is a wide angle lens. Common wide angle lenses might include a 20, 24, or 35mm lens.

This definition is based on the definition of "normal" as a focal length equal to the film or sensor diagonal — that's about 43mm on full-frame or 29mm on APS-C. By this definition, anything wider than that would be "wide angle".

I think it's important to recognize that this is a historically loose term — rather than saying (in full-frame terms), "Ultrawide is under 23.4999mmm, Wide is 23.5mm to 35.499, Normal is exactly 43.2666mm", it's best to say "Normal is around 40mm to 60mm; Wide is around 24mm to 35mm; Ultra-Wide is below 24mm, give or take". And, yeah, that leaves 36-39mm in an awkward, unclear area — sometimes life is like that!

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    +1. I like to think of ultra/wide/normal/tele/supertele as a spectrum or continuum, like colors in a crayon box. If wide is yellow, and normal is green, what do the yellow-green and green-yellow crayons map to? What about honeydew, where does that fit on the yellow/green spectrum? =) That's where the (highly scientific) "-ish" suffix comes into play: it's a wide-ish normal, or a long-ish wide. – scottbb Dec 30 '16 at 16:41
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The job of the lens is to project an image of the outside world onto the surface of film or digital image sensor. This projected image is brightest and sharpest in the center of the image. In other words, there is fall off so we look at the corners to see if the coverage will be adequate. Normally, the shorter the focal length, the closer the lens must be to film or sensor. Thus making a short lens with good coverage is challenging. For this reason, most cameras are principally fitted with a “normal” focal length lens. This will be a lens with a focal length about equal to the corner to corner measurement of the format frame (the diagonal measure).

For the full frame (Fx) the format measures 24mm height by 36mm length. The diagonal measure of the full frame is 43mm. Fitted with a 43mm focal length, the angle of view delivered is 31° height, 45° width, 53° diagonal. Because a 43mm is somewhat unusual, the industry settled on 50mm as being “normal”. When a 50mm is mounted, the angles of view for the full frame is 27° height, 40° length and 47° diagonal. Again the 50mm lash-up is considered “normal” for the Fx format.

As a rule of thumb, the realm of wide angle for any format is a lens about 70% of “normal = 35mm for the Fx. Such a lens will deliver 37° height, 54° length and 63° diagonal. A 28mm delivers 46° height, 66° length and 75° diagonal.

The Compact Digital or APS (Advanced Photo “System) format is 16mm height by 24mm length with a diagonal of 30mm. When fitted with a “normal” 30mm the angle of views are: 30° height, 44° length and 51° diagonal. Wide-angle 70% of “normal” = 20mm. Fitted with a 20mm, the angles of view are: 44° height, 61° length and 72° diagonal.

Again rule of thumb: “normal” = lens about equal to diagonal measure of the format. Wide-angle about 70% of “normal” or shorter. The realm of telephoto is 200% of “normal” and longer. Portrait is a moderate telephoto 200% thru 250% of “normal”.

The problem with wide-angle lenses design is accaptable coverage of the format. In other words, too much vignette will be unacceptable. To accomplish, the lens maker likely uses a retro focus design. This is a telephoto like lens mounted backwards. This design delivers the needed wide angle of view and because the distant lens to sensor is elongated the vignette is mitigated

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