# How can I calculate vertical field of view from horizontal field of view?

I have an Axis M1011 (specs available here), which I need to find the vertical field of view. The online specifications tell me the horizontal field of view is 47 degrees, and the image size is 640x480. Does this mean that the vertical field of view will be 32.25 degrees?

• Great question Joel! I will have to say that I'm completely lost, and you might also get some good answers if you posted it on the math or physics SE. Mar 21 '12 at 17:49

The formula for this calculation is:

a = 2 arctan(d/2f)

Where 'a' is the angle, 'd' is the size of the sensor in the direction of interest, and 'f' is the focal length. The real angle of view of this combination is actually much better than what you get but, because it's video (which is technically off topic here, but the basic question isn't), you're lopping off some info.

The camera's sensor is 6.35mm (1/4 inch according to doc), so if it holds to standard 3:2, then the vertical is about 4.23mm. So...

If I do a little bit of math, it would appear that the sensor is about 167 pixels per mm, so the vertical is using about 2.87mm of the sensor. That leads to the formula returning: 36.13 degrees or thereabouts, call it probably 36. How did I arrive at the 167? Well, the formula to yield an angle of 47 degrees would imply the horizontal is using about 3.83mm of the sensor and if there are 640 pixels, the math works out to about 167 pixels per mm.

Mind you, without real specs on the sensor, this is all approximation. However, the formula I gave you is correct. :)

• For whatever reason, the "inch" in sensor formats isn't really an inch — it's roughly 16mm. A 1/4"-format sensor is 4mm diagonal, and is probably 3:4 (matching 640x480), so 3.2mm by 2.4mm. Mar 21 '12 at 2:43
• @mattdm - That 3.2mm on the horizontal wouldn't yield a 47 degree angle of view with a 4.4mm focal length though, that math doesn't change. Mar 21 '12 at 2:48
• That does imply, presuming the camera makes use of most of the real estate, that the numbers I ultimately gave are pretty close to the size you described... Geeze, you guys really need to get off this archaic measuring system, especially if you can't be consistent about it! ;) Mar 21 '12 at 2:54
• Yeah. Think maybe the specs are wrong and that's really the diagonal FoV? Mar 21 '12 at 2:55
• @mattdm - 4mm diagonal is about 49 degrees, so maybe, but that's a fairly large margin of error for a pretty precise formula. Either way, that's the formula for field of view, regardless of horizontal or vertical. :) Mar 21 '12 at 2:59

The formula John gives is correct, but as you can see from that answer and the comments, the precision and accuracy of the listed specs are in doubt.

That's not, actually, unreasonable of the camera manufacturer, given the state of the rest of the camera world. The sensor size is really a sensor class, and can't be relied on for calculating the actual size. In the comments to John's answer I guess that it's probably 4mm diagonal, but it could vary significantly from that and still be within the spec for a 1/4" sensor. Likewise, lens focal length is likely to rounded or approximate (although one could hope that with a number like 4.4mm, it's closer to 4.4 than it is to 4.0 or 5.0.). And we're unclear on whether 47° is really the horizontal field of view, but even if it is, expect it to be rounded at the very least.

Therefore, if this is important, you should actually measure the horizontal or vertical angle of view and work from there. (Some suggestions on how to do this are found at How to test actual focal length? — it basically comes down to the same formula.)

using trigonometric formulas you can do that:

where

``````a = 47
b is the value you are looking for
AB = 480/2
AC = 640/2
``````

I will not go into details of formulas and theorems (most af all because I do not remeber them...), but using this tool you can calculate the result easily.

The resulting angle is 36.2

The formula is:

FOVh = 2*atan((x/y)tan(FOVv/2))

(source)