Field of view (FoV), often expressed as an angle, is the extent of the observable world that is seen at a given distance from the observer. Technically, when stated as an angle we are describing the Angle of View (AoV) or AFoV, which may be confusing. To separate them, we define the FoV as a horizontal distance covered by the width of what can be seen by the sensor/person. The AoV is the angle that the sensor sees out from the center. One can derive the FoV from the AFoV by using a little bit of trigonometry and understanding the relationship between focal length, magnification, and aperature.
The trigonometry used to describe FoV is broken down into right triangles in the following diagram with angle of view degrees, Distance from the observer, and width (FoV) where is the small opposite side of the right triangle.
One can calculate the FoV of any optical system using the AfoV angle and Distance to target.
To solve for angle we split it down the middle and work with angle . By definition, tangent of an angle is equivalent to Opposite over Adjacent sides and we begin with:
Multiplying both sides by gives us
Finally, we double both sides because we’re solving for = FoV
Now we are left with a general formula for calculating field of view at Distance D with Angle A. With two of the three values we can always find the third. Using this formula we’ve created a small table of Fields of View for reference. It’s a linear relationship independent of units, we used yards for our example but it can be feet, meters, miles, or any other unit of distance.
|AfoV||FoV @ 50 yards||FoV @ 100 yards||FoV @ 150 yards|
|3 degrees||2.62 yards||5.24 yards||7.86 yards|
|5 degrees||4.37 yards||8.74 yards||13.10 yards|
|12 degrees||10.51 yards||21.02 yards||31.53 yards|
|40 degrees||36.40 yards||72.80 yards||109.2 yards|
Not all optical systems use circular lenses, they can be rectangular or even oddly not-uniform at all. Note the square sensor below. The calculations for these systems get much more complicated quickly.
For sport optics, field of view depends on the ratios of the magnification and the focal length of the objective/eye piece. This is why we can change the FoV and thus magnification of optical systems by moving the lenses in particular ratios from each other.
**Note that the diameter of the objective lens does not affect FoV.