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From looking at the FOV tables on this page, I can see that FOV is greatly reduced as sensor size decreases. It seems that 1/2.3 sensors would produced fields-of-view that are tiny, even at the shortest focal lengths.

And yet cameras with 1/2.3 sensors seem to be able to produce photographs at wide angle setting that have resonably wide FOV.

But is it impossible for a small sensor camera to achieve the same FOV as a full-frame camera? What about an APS-C sensor? Can APS-C sensors achieve the same FOV as a full-frame sensor using the shortest focal length lens available (and excluding fisheye lenses)?

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It seems that 1/2.3 sensors would produced fields-of-view that are tiny, even at the shortest focal lengths.

You're not completely wrong — you're just underestimating the limit.

The relationship between sensor size, focal length, and field of view is simple — see What is "angle of view" in photography? for details, or simply What is crop factor and how does it relate to focal length?.

A 1/2.3" sensor has a crop factor of around 5.6. That is, to have the same focal length as a full-frame camera, the lens needs to be 5.6× shorter. That means to give the field of view equivalent to a 24mm wide-angle lens on a full frame camera, you need a lens of about 4.3mm. The widest-angle consumer camera with a 1/2.3" sensor I'm aware of is the Canon SX60, with a superzoom lens which goes from 3.8 to 247mm. At the short/wide end, that's equivalent to the field of view of a 21mm lens on full-frame.

So, as you note:

And yet cameras with 1/2.3 sensors seem to be able to produce photographs at wide angle setting that have resonably wide FOV.

Yes, "reasonably wide" is possible. But wider than that seems to be an engineering challenge not commonly met.

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The angle of view realized by the camera is a product of sensor size and lens focal length.

The venerable 35mm full frame film camera sports an image that measures 24mm height by 36mm length. The corner-to-corner (diagonal) measure of this rectangle is 43mm.

The 1/2.3 digital chip produces an image 4.55mm height by 6.17mm length. The diagonal measure of this rectangle is 7.67mm.

The angle of view most often published is actually the diagonal angle of view. This value is the widest. Mount a 7.67mm on your 1/2.3 camera or mount a 43mm on a full frame and the angle of view for both will be 53°. Yes, both yield the same. (TV sets are sold by their diagonal measure because this value is the longest). Perhaps the diagonal measure is the least value and the horizontal is the most valuable but bigger is better in the world of advertising.

Also, mounting a lens with a focal length about equal to the diagonal measure yields an angle of view of 45°. Such a lash-up is considered the “normal” lens for that format. Mount a shorter lens then “normal” and you are in wide-angle territory (technically about 70% of “normal”. Mount a longer then “normal” lens, you are in telephoto territory (about 2X of “normal” or longer).

Bottom line: Angle of view and wide-angle – normal – telephoto is dependent o the combination of sensor size and lens focal length.

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Field of view is primarily determined by the actual lens(es), although the sensor does matter, as well. The (effective) focal length is the primary indicator. Given a specific lens, though, a smaller sensor does yield a smaller FOV, but an actual 35mm lens on a full frame sensor and a 35mm-equivalent lens on a cell phone sensor should have approximately the same FOV, aside from the rounding-off that all lens/sensor/camera/phone manufacturers do.

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